View source for Harold Scott MacDonald Coxeter

{{Short description|Canadian geometer (1907–2003)}}
{{Use dmy dates|date=April 2022}}
{{redirect|Coxeter|an English literary antiquary|Thomas Coxeter}}
{{Infobox scientist
|name              = Harold Scott MacDonald Coxeter
|honorific_suffix  = {{postnom|CC|FRS|FRSC|size=100%}}
|image             = H.S.M. Coxeter.jpeg
|image_size        = 240px
|caption           =
|birth_date        = {{birth date|1907|02|09|df=y}}
|birth_place       = [[London]], England
|death_date        = {{death date and age|2003|03|31|1907|02|09|df=y}}
|death_place       = [[Toronto, Ontario]], Canada
|citizenship       =
|nationality       =
|ethnicity         =
|field             = [[Geometry]]
|work_institutions = [[University of Toronto]]
|alma_mater        = [[University of Cambridge]] (B.A., 1929; Ph.D., 1931)
|doctoral_advisor  = [[H. F. Baker]]<ref>{{MathGenealogy|id=12555}}</ref>
|doctoral_students = {{plainlist|1=
*[[W. G. Brown]]
*[[Norman Johnson (mathematician)|Norman Johnson]]
}}
|known_for         = [[Coxeter element]]<br>[[Coxeter functor]]<br>[[Coxeter graph]]<br>[[Coxeter group]]<br>[[Coxeter matroid]]<br>[[Coxeter notation]]<br>[[Coxeter's loxodromic sequence of tangent circles]]<br>[[Coxeter–Dynkin diagram]]<br>[[Coxeter–Todd lattice]]<br>[[Boerdijk–Coxeter helix]]<br>[[Goldberg–Coxeter construction]]<br>[[Todd–Coxeter algorithm]]*<br>[[Tutte–Coxeter graph]]<br>[[LCF notation]]<br>[[Regular skew apeirohedron|Regular skew apeirohedra]]
|author_abbrev_bot =
|author_abbrev_zoo =
|influences        =
|influenced        =
|prizes            = {{plainlist|class=nowrap|
* [[Smith's Prize]] (1931)
* [[Henry Marshall Tory Medal]] (1949)
* [[Fellow of the Royal Society|FPS]] (1950)
* [[Jeffery–Williams Prize]] (1973)
* [[CRM-Fields-PIMS prize]] (1995)
* [[Sylvester Medal]] (1997)
}}
|religion          =
|footnotes         =
|signature         =
|spouse            = Hendrina, died in 1999
|children          = a daughter, Susan Thomas, and a son, Edgar
}}

'''Harold Scott MacDonald''' "'''Donald'''" '''Coxeter''' {{Post-nominals|country=CAN|CC|FRS|FRSC|sep=}} (9 February 1907 &ndash; 31 March 2003)<ref name="Memoir">{{cite journal |last1=Roberts |first1=Siobhan |last2=Ivić Weiss |first2=Asia |author1-link=Siobhan Roberts |editor1-last=Longair |editor1-first=Malcolm |editor1-link=Malcolm Longair |title=Harold Scott MacDonald Coxeter. 9 February 1907 — 31 March 2003: Elected FRS 1950 |journal=Biographical Memoirs of Fellows of the Royal Society |date=2006 |volume=52 |pages=45–66 |doi=10.1098/rsbm.2006.0004 |issn=1748-8494 |doi-access=free}}</ref> was a British-Canadian [[geometer]] and mathematician. He is regarded as one of the greatest geometers of the 20th century.<ref>{{cite web |title=Geometry Revisited |url=https://www.maa.org/press/books/geometry-revisited |website=[[Mathematical Association of America]] |access-date=25 December 2018 |archive-date=25 December 2018 |archive-url=https://web.archive.org/web/20181225225114/https://www.maa.org/press/books/geometry-revisited |url-status=dead }}</ref>

Coxeter was born in England and educated at the [[University of Cambridge]], with student visits to [[Princeton University]]. He worked for 60 years at the [[University of Toronto]] in Canada, from 1936 until his retirement in 1996, becoming a full professor there in 1948. His many honours included membership in the [[Royal Society of Canada]], the [[Royal Society]], and the [[Order of Canada]].

He was an author of 12 books, including ''[[The Fifty-Nine Icosahedra]]'' (1938) and ''[[Regular Polytopes (book)|Regular Polytopes]]'' (1947). Many concepts in geometry and [[group theory]] are named after him, including the [[Coxeter graph]], [[Coxeter group]]s, [[Coxeter's loxodromic sequence of tangent circles]], [[Coxeter–Dynkin diagram]]s, and the [[Todd–Coxeter algorithm]].

== Biography ==
Coxeter was born in [[Kensington]], England, to Harold Samuel Coxeter and Lucy ({{nee|Gee}}). His father had taken over the family business of Coxeter & Son, manufacturers of surgical instruments and compressed gases (including a mechanism for anaesthetising surgical patients with [[nitrous oxide]]), but was able to retire early and focus on sculpting and baritone singing; Lucy Coxeter was a portrait and landscape painter who had attended the [[Royal Academy of Arts]]. A maternal cousin was the architect Sir [[Giles Gilbert Scott]].<ref name="ODNB">{{Cite ODNB|url=https://www.oxforddnb.com/view/10.1093/ref:odnb/9780198614128.001.0001/odnb-9780198614128-e-89876|doi = 10.1093/ref:odnb/89876|title = Oxford Dictionary of National Biography|year = 2004}}</ref><ref name="Memoir"/>

In his youth, Coxeter composed music and was an accomplished pianist at the age of 10.<ref name=Roberts>[[Siobhan Roberts|Roberts, Siobhan]], ''King of Infinite Space: Donald Coxeter, The Man Who Saved Geometry'', Walker & Company, 2006, {{ISBN|0-8027-1499-4}}</ref> He felt that [[mathematics and music]] were intimately related, outlining his ideas in a 1962 article on "Music and Mathematics" in the ''Canadian Music Journal''.<ref name=Roberts/>

He was educated at [[King Alfred School, London]], and [[St George's School, Harpenden]], where his best friend was John Flinders Petrie, later a mathematician for whom [[Petrie polygon]]s were named. He was accepted at [[King's College, Cambridge]], in 1925, but decided to spend a year studying in hopes of gaining admittance to [[Trinity College, Cambridge|Trinity College]], where the standard of mathematics was higher.<ref name="Memoir" /> Coxeter won an entrance scholarship and went to Trinity in 1926 to read mathematics. There he earned his BA (as [[Senior Wrangler]]) in 1928, and his doctorate in 1931.<ref name=Roberts/><ref name=MacTutor>{{MacTutor Biography|id=Coxeter}}</ref> In 1932 he went to [[Princeton University]] for a year as a [[Rockefeller Fellow]], where he worked with [[Hermann Weyl]], [[Oswald Veblen]], and [[Solomon Lefschetz]].<ref name=MacTutor/> Returning to Trinity for a year, he attended [[Ludwig Wittgenstein]]'s seminars on the [[philosophy of mathematics]].<ref name=Roberts/> Wittgenstein selected Coxeter and others to take notes of his lectures, the collection of which later became ''[[Blue and Brown Books|The Blue Book]]''.<ref>{{cite book |last1=Monkl |first1=Ray |title=Ludwig Wittgenstein: The Duty of Genius |date=2012 |publisher=Random House |page=336}}</ref> In 1934 he spent a further year at Princeton as a Procter Fellow.<ref name=MacTutor/>

In 1936 Coxeter moved to the University of Toronto. In 1938 he and [[P. Du Val]], H. T. Flather, and [[John Flinders Petrie]] published ''[[The Fifty-Nine Icosahedra]]'' with [[University of Toronto Press]]. In 1940 Coxeter edited the eleventh edition of ''Mathematical Recreations and Essays'',<ref>{{cite journal|author=Frame, J. S.|title=Review: ''Mathematical Recreations and Essays'', 11th edition, by W. W. Rouse Ball; revised by H. S. M. Coxeter|journal=Bull. Amer. Math. Soc.|year=1940|volume=45|issue=3|pages=211–213|url=https://www.ams.org/journals/bull/1940-46-03/S0002-9904-1940-07170-8/S0002-9904-1940-07170-8.pdf|doi=10.1090/S0002-9904-1940-07170-8}}</ref> originally published by [[W. W. Rouse Ball]] in 1892. He was elevated to [[professor]] in 1948. He was elected a Fellow of the [[Royal Society of Canada]] in 1948 and a [[Fellow of the Royal Society]] in 1950. He met [[M. C. Escher]] in 1954 and the two became lifelong friends; his work on geometric figures helped inspire some of Escher's works, particularly the [[Circle Limit III|''Circle Limit'']] series based on [[hyperbolic geometry|hyperbolic]] [[tessellation]]s. He also inspired some of the innovations of [[Buckminster Fuller]].<ref name=MacTutor/> Coxeter, [[M. S. Longuet-Higgins]] and [[J. C. P. Miller]] were the first to publish the full [[list of uniform polyhedra]] (1954).<ref>Harold Coxeter, [[Michael S. Longuet-Higgins]] and [[J. C. P. Miller]]. "Uniform Polyhedra", [[Philosophical Transactions of the Royal Society A]] 246: 401–50 {{doi|10.1098/rsta.1954.0003}}</ref>

He worked for 60 years at the [[University of Toronto]] and published twelve books.

==Personal life==

Coxeter was a [[Vegetarianism|vegetarian]]. He attributed his longevity to his vegetarian diet, daily exercise such as fifty press-ups and standing on his head for fifteen minutes each morning, and consuming a nightly cocktail made from [[Kahlúa]] (a coffee liqueur), [[peach schnapps]], and [[soy milk]].<ref name="ODNB"/>

==Awards==
Since 1978, the [[Canadian Mathematical Society]] have awarded the [[Coxeter–James Prize]] in his honor.

He was made a [[Fellow of the Royal Society]] in 1950 and in 1997 he was awarded their [[Sylvester Medal]].<ref name=MacTutor/> In 1990, he became a Foreign Member of the [[American Academy of Arts and Sciences]]<ref>{{cite web|url=https://www.amacad.org/person/harold-scott-macdonald-coxeter|title=Harold Scott MacDonald Coxeter|work=Member directory|date=9 February 2023 |publisher=American Academy of Arts and Sciences|access-date=2024-06-07}}</ref> and in 1997 was made a [[Order of Canada#Grades|Companion]] of the [[Order of Canada]].<ref>{{Canadian honour|Type=orc|ID=3762|accessdate=26 May 2010}}</ref>

In 1973 he received the [[Jeffery–Williams Prize]].<ref name=MacTutor/>

A [[festschrift]] in his honour, ''The Geometric Vein'', was published in 1982. It contained 41 essays on geometry, based on a symposium for Coxeter held at Toronto in 1979.<ref>{{cite journal|last=Edge|first=W. L.|date=June 1983|doi=10.1017/s0013091500017016|issue=2|journal=Proceedings of the Edinburgh Mathematical Society|pages=284–285|title=Review of ''The Geometric Vein''|volume=26|doi-access=free}}</ref>  A second such volume, ''The Coxeter Legacy'', was published in 2006 based on a Toronto Coxeter symposium held in 2004.<ref>{{Cite book|title=The Coxeter Legacy|year=2006 |publisher=American Mathematical Society|place=Providence, Rhode Island|editor1-last=Davis|editor1-first=Chandler|editor2-last=Ellers|editor2-first=Erich|isbn=978-0821837221}}</ref>

==Works==

===Books===
* 1942: [https://books.google.com/books/about/Non_Euclidean_Geometry.html?id=usKZpDAH0WUC ''Non-Euclidean Geometry''] (1st edition),<ref>{{cite journal|author=Blumenthal, L. M.|author-link=Leonard Blumenthal|title=Review: ''Non-euclidean geometry'' by H. S. M. Coxeter|journal=Bull. Amer. Math. Soc.|year=1943|volume=49|issue=9|pages=679–680|url=https://www.ams.org/journals/bull/1943-49-09/S0002-9904-1943-07977-3/S0002-9904-1943-07977-3.pdf|doi=10.1090/s0002-9904-1943-07977-3|doi-access=free}}</ref> (2nd ed, 1947), (3rd ed, 1957), (4th ed, 1961), (5th ed, 1965), [[University of Toronto Press]] (6th ed, 1998), [[Mathematical Association of America|MAA]], {{isbn|978-0-88385-522-5}}.
* 1949: ''The Real Projective Plane''<ref>{{cite journal|author=DuVal, Patrick|author-link=Patrick du Val|title=Review: ''The real projective plane'' by H. S. M. Coxeter|journal=Bull. Amer. Math. Soc.|year=1950|volume=56|issue=4|pages=376–378|url=https://www.ams.org/journals/bull/1950-56-04/S0002-9904-1950-09414-2/S0002-9904-1950-09414-2.pdf|doi=10.1090/s0002-9904-1950-09414-2|doi-access=free}}</ref>
* 1961: ''Introduction to Geometry'',<ref>{{cite journal|author=Freudenthal, H.|author-link=Hans Freudenthal|title=Review: ''Introduction to geometry'' by H. S. M. Coxeter|journal=Bull. Amer. Math. Soc.|year=1962|volume=68|issue=2|pages=55–59|url=https://www.ams.org/journals/bull/1962-68-02/S0002-9904-1962-10714-9/S0002-9904-1962-10714-9.pdf|doi=10.1090/s0002-9904-1962-10714-9|doi-access=free}}</ref><ref>{{cite journal|author= Levi, H.|author-link=Howard Levi|title=Review: ''Introduction to Geometry'' by H. S. M. Coxeter|journal=The Journal of Philosophy|year=1963|volume=60|issue=1|pages=19–21|doi=10.2307/2023059|jstor=2023059}}</ref> (2nd paperback edition 1989, {{isbn|978-0-471-50458-0}}.)
* 1963: ''Regular Polytopes'' (2nd edition), Macmillan Company
* 1967: (with [[S. L. Greitzer]]) ''Geometry Revisited''
* 1970: [https://books.google.com/books/about/Twisted_Honeycombs.html?id=qwXrfwL9ByQC ''Twisted honeycombs''] (American Mathematical Society, 1970, Regional conference series in mathematics Number 4, {{isbn|0-8218-1653-5}})
* 1973: ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition), Dover edition, {{isbn|0-486-61480-8}}
* 1974: ''Projective Geometry'' (2nd edition)
* 1974: ''Regular Complex Polytopes'', [[Cambridge University Press]], {{isbn|978-0-521-20125-4}}.
* 1981: (with R. Frucht and D. L. Powers), ''Zero-Symmetric Graphs'', [[Academic Press]], {{isbn|978-0-12-194580-0}}.
* 1987 ''[https://www.springer.com/us/book/9780387406237 Projective Geometry]'' (1987) {{isbn|978-0-387-40623-7}}
* 1995: F. Arthur Sherk, Peter McMullen, Anthony C. Thompson and Asia Ivić Weiss, editors: ''Kaleidoscopes — Selected Writings of H. S. M. Coxeter''. [[John Wiley and Sons]], {{isbn|0-471-01003-0}}.
* 1999: ''The Beauty of Geometry: Twelve Essays'', Dover Publications, {{LCCN|99035678}}, {{isbn|0-486-40919-8}}
* 2011:  The Fifty-Nine Icosahedra, Tarquin Group, {{isbn|978-1-907550-08-9}}
===Selected Papers===
* 1940: [https://doi.org/10.1007%2FBF01181449 "Regular and Semi-Regular Polytopes I"], ''[[Mathematische Zeitschrift]]'' 46: 380–407, MR 2,10 {{doi|10.1007/BF01181449}}
* 1954: (with [[Michael S. Longuet-Higgins]] and [[J. C. P. Miller]]) "Uniform Polyhedra", ''[[Philosophical Transactions of the Royal Society A]]'' 246: 401–50 {{doi|10.1098/rsta.1954.0003}}
* 1957: (with W. O. J. Moser) ''Generators and Relations for Discrete Groups''<ref>{{cite journal|author=Hall Jr., Marshall|author-link=Marshall Hall (mathematician)|title=Review: ''Generators and relations for discrete groups'' by H. S. M. Coxeter and W. O. J. Moser|journal=[[Bulletin of the American Mathematical Society]]|year=1958|volume=64, Part 1|issue=3|pages=106–108|url=https://www.ams.org/journals/bull/1958-64-03/S0002-9904-1958-10178-0/S0002-9904-1958-10178-0.pdf|doi=10.1090/S0002-9904-1958-10178-0|doi-access=free}}</ref> 1980: Second edition, Springer-Verlag {{isbn|0-387-09212-9}}
* {{cite journal | last=Coxeter | first=H. |author-mask=1| title=Frieze patterns | journal=[[Acta Arithmetica]] | volume=18 | date=1971 | issn=0065-1036 | doi=10.4064/aa-18-1-297-310 | doi-access=free | pages=297–310 | url=https://www.impan.pl/shop/publication/transaction/download/product/97697?download.pdf | access-date=2025-05-24}}
* 1985: [https://doi.org/10.1007%2FBF01161657 "Regular and Semi-Regular Polytopes II"], ''Mathematische Zeitschrift'' 188: 559–591
* 1988: [https://doi.org/10.1007%2FBF01161745 "Regular and Semi-Regular Polytopes III"], ''Mathematische Zeitschrift'' 200: 3–45

==See also==
* [[Coxeter–James Prize]]
* [[Spiral similarity]]
*{{intitle|Coxeter}}

==References==
{{Reflist}}

==Further reading==
*{{Cite book|editor1-last= Davis|editor1-first= Chandler|editor1-link= Chandler Davis|editor2-last= Ellers|editor2-first= Erich W|title= The Coxeter Legacy: Reflections and Projections|year= 2006|location= Providence, R.I.|publisher= American Mathematical Society|isbn= 978-0-8218-3722-1|oclc= 62282754}}
*{{Cite book|last= Roberts|first= Siobhan|author-link= Siobhan Roberts|title= King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry|year= 2006|location= New York|publisher= Walker & Company|isbn= 978-0-8027-1499-2|oclc= 71436884|url-access= registration|url= https://archive.org/details/kingofinfinitesp00robe}}

== External links ==
{{Archival records|title=Harold Scott MacDonald Coxeter fonds}} 

*[https://discoverarchives.library.utoronto.ca/index.php/harold-scott-macdonald-coxeter-fonds Harold Scott MacDonald Coxeter archival papers] held at the [https://utarms.library.utoronto.ca/ University of Toronto Archives and Records Management Services]
* {{MathGenealogy|id=12555}}
* [https://www.ams.org/notices/200310/fea-coxeter.pdf H. S. M. Coxeter (1907–2003), Erich W. Ellers, Branko Grünbaum, Peter McMullen, Asia Ivic Weiss] Notices of the AMS: Volume 50, Number 10.
* [http://www.math.yorku.ca/dcoxeter/  www.donaldcoxeter.com www.math.yorku.ca/dcoxeter] webpages dedicated to him (in development)
* [http://discovermagazine.com/2007/apr/jarons-world-shapes-in-other-dimensions Jaron's World: Shapes in Other Dimensions], [[Discover (magazine)|Discover mag.]], Apr 2007
* [http://media.pims.math.ca/realvideo-ram/education/2000/CtC/coxeter/coxeter.ram The Mathematics in the Art of M.C. Escher] video of a lecture by H.S.M. Coxeter, April 28, 2000.

{{Authority control}}

{{DEFAULTSORT:Coxeter, H.S.M.}}
[[Category:1907 births]]
[[Category:2003 deaths]]
[[Category:Companions of the Order of Canada]]
[[Category:Fellows of the Royal Society]]
[[Category:Fellows of the Royal Society of Canada]]
[[Category:British geometers]]
[[Category:Chirality]]
[[Category:Polytopes]]
[[Category:20th-century English mathematicians]]
[[Category:Recreational mathematicians]]
[[Category:Alumni of Trinity College, Cambridge]]
[[Category:Academic staff of the University of Toronto]]
[[Category:Canadian mathematicians]]
[[Category:20th-century Canadian mathematicians]]
[[Category:People from Harpenden]]
[[Category:Academics of the University of East Anglia]]
[[Category:Senior Wranglers]]
[[Category:Presidents of the Canadian Mathematical Society]]
[[Category:Canadian fellows of the Royal Society]]