# Harold Scott MacDonald Coxeter

Harold Scott MacDonald "Donald" Coxeter CC FRS FRSC (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century.
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 (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.
In his youth, Coxeter composed music and was an accomplished pianist at the age of 10. 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.
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. 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. 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. Returning to Trinity for a year, he attended Ludwig Wittgenstein's seminars on the philosophy of mathematics. In 1934 he spent a further year at Princeton as a Procter Fellow.
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, 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. Coxeter, M. S. Longuet-Higgins and J. C. P. Miller were the first to publish the full list of uniform polyhedra (1954).
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.

## 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. In 1990, he became a Foreign Member of the American Academy of Arts and Sciences and in 1997 was made a Companion of the Order of Canada.
In 1973 he received the Jeffery–Williams Prize.
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. A second such volume, The Coxeter Legacy, was published in 2006 based on a Toronto Coxeter symposium held in 2004.

## Works

* 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), (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.
* 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
* 1957: (with W. O. J. Moser) Generators and Relations for Discrete Groups 1980: Second edition, Springer-Verlag ISBN 0-387-09212-9
* 1961: Introduction to Geometry, (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: 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.
* 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
* 2011: The Fifty-Nine Icosahedra, Tarquin Group, ISBN 978-1-907550-08-9