Harold Scott MacDonald Coxeter

Coxeter was born in Kensington, England, to Harold Samuel Coxeter and Lucy (née 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.[4][2]

In his youth, Coxeter composed music and was an accomplished pianist at the age of 10.[5] 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.[5]

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.[2] 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.[5][6] 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.[6] Returning to Trinity for a year, he attended Ludwig Wittgenstein's seminars on the philosophy of mathematics.[5] In 1934 he spent a further year at Princeton as a Procter Fellow.[6]

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,[7] 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.[6] Coxeter, M. S. Longuet-Higgins and J. C. P. Miller were the first to publish the full list of uniform polyhedra (1954).[8]

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

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, peach schnapps, and soy milk.[4]

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.[6] In 1990, he became a Foreign Member of the American Academy of Arts and Sciences[9] and in 1997 was made a Companion of the Order of Canada.[10]

In 1973 he received the Jeffery–Williams Prize.[6]

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.[11] A second such volume, The Coxeter Legacy, was published in 2006 based on a Toronto Coxeter symposium held in 2004.[12]

• 1940: "Regular and Semi-Regular Polytopes I", Mathematische Zeitschrift 46: 380–407, MR 2,10 doi:10.1007/BF01181449
• 1942: Non-Euclidean Geometry (1st edition),[13] (2nd ed, 1947), (3rd ed, 1957), (4th ed, 1961), (5th ed, 1965), University of Toronto Press (6th ed, 1998), MAA.
• 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
• 1949: The Real Projective Plane[14]
• 1957: (with W. O. J. Moser) Generators and Relations for Discrete Groups[15] 1980: Second edition, Springer-Verlag ISBN 0-387-09212-9
• 1961: Introduction to Geometry[16][17]
• 1963: Regular Polytopes (2nd edition), Macmillan Company
• 1967: (with S. L. Greitzer) Geometry Revisited
• 1970: 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
• 1981: (with R. Frucht and D. L. Powers), Zero-Symmetric Graphs, Academic Press.
• 1985: "Regular and Semi-Regular Polytopes II", Mathematische Zeitschrift 188: 559–591
• 1987 Projective Geometry (1987) ISBN 978-0-387-40623-7
• 1988: "Regular and Semi-Regular Polytopes III", Mathematische Zeitschrift 200: 3–45
• 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 99-35678, ISBN 0-486-40919-8

• Coxeter–James Prize
• Spiral similarity

• Davis, Chandler; Ellers, Erich W, eds. (2006). The Coxeter Legacy: Reflections and Projections. Providence, R.I.: American Mathematical Society. ISBN 978-0-8218-3722-1. OCLC 62282754.
• Roberts, Siobhan (2006). King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry. New York: Walker & Company. ISBN 978-0-8027-1499-2. OCLC 71436884.

Archives at
Location University of Toronto Archives & Records Management Services  Identifiers 1183  Source Harold Scott MacDonald Coxeter fonds
How to use archival material

• Harold Scott MacDonald Coxeter archival papers held at the University of Toronto Archives and Records Management Services
• Harold Scott MacDonald Coxeter at the Mathematics Genealogy Project
• H. S. M. Coxeter (1907–2003), Erich W. Ellers, Branko Grünbaum, Peter McMullen, Asia Ivic Weiss Notices of the AMS: Volume 50, Number 10.
• www.donaldcoxeter.com www.math.yorku.ca/dcoxeter webpages dedicated to him (in development)
• Jaron's World: Shapes in Other Dimensions, Discover mag., Apr 2007
• The Mathematics in the Art of M.C. Escher video of a lecture by H.S.M. Coxeter, April 28, 2000.