Harold Scott MacDonald "Donald" Coxeter CC FRS FRSC (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. ^[2][3]

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Harold Scott MacDonald Coxeter CC FRS FRSC
Born	(1907-02-09)9 February 1907 London, England
Died	31 March 2003(2003-03-31) (aged 96) Toronto, Ontario, Canada
Alma mater	University of Cambridge (B.A., 1929; Ph.D., 1931)
Known for	Coxeter element Coxeter functor Coxeter graph Coxeter group Coxeter matroid Coxeter notation Coxeter's loxodromic sequence of tangent circles Coxeter–Dynkin diagram Coxeter–Todd lattice Boerdijk–Coxeter helix Goldberg–Coxeter construction Todd–Coxeter algorithm* Tutte–Coxeter graph LCF notation Regular skew apeirohedra
Spouse(s)	Hendrina, died in 1999
Children	a daughter, Susan Thomas, and a son, Edgar
Awards	Smith's Prize (1931) Henry Marshall Tory Medal (1949) FPS (1950) Jeffery–Williams Prize (1973) CRM-Fields-PIMS prize (1995) Sylvester Medal (1997)
Scientific career
Fields	Geometry
Institutions	University of Toronto
Doctoral advisor	H. F. Baker[1]
Doctoral students	W. G. Brown Norman Johnson

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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][3]

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.^[3] 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