{{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 }} |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 apeirohedra |author_abbrev_bot = |author_abbrev_zoo = |influences = |influenced = |prizes = {{plainlist|class=nowrap| * Smith's Prize (1931) * Henry Marshall Tory Medal (1949) * FRS (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''' ({{IPAc-en|ˈ|k|ɒ|k|s|i|t|ər}}) {{Post-nominals|country=CAN|CC|FRS|FRSC|sep=}} (9 February 1907 – 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'' (1947). Many concepts in geometry and group theory are named after him, including the Coxeter graph, Coxeter groups, Coxeter's loxodromic sequence of tangent circles, Coxeter–Dynkin diagrams, 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>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 polygons 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, 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 ''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'' series based on hyperbolic tessellations. 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 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 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), 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'', (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 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 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 Faculty of Arts and Science 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 Category:British fellows of the Royal Society Category:Mathematicians of the University of Toronto