Professor H. S. M. Coxeter

Geometer who pursued mathematics as art

Harold Scott Macdonald Coxeter, mathematician: born London 9 February 1907;
Fellow, Trinity College, Cambridge 1931-36, Honorary Fellow 2001; Assistant Professor of Mathematics, University of Toronto 1936-43, Associate Professor 1943-48, Professor 1948-80 (Emeritus); Editor, Canadian Journal of Mathematics 1948-57; FRS 1950; CC 1997; married 1936 Rien Brouwer (died 1999; one son, one daughter); died Toronto, Ontario 31 March 2003

H. S. M. Coxeter, Professor of Mathematics at the University of Toronto for more than 30 years, was an extraordinary mathematician. While he was one of the greatest and most successful members of the mathematical community, his work was inspired by the world around him, and he contributed to it. He pursued mathematics as a form of art.

It is not surprising that Coxeter was a great friend and admirer of the Dutch graphic artist M.C. Escher. In 1997 he published a paper, "The Trigonometry of Hyperbolic Tessellations", in which he proved that, despite knowing no mathematics, Escher had achieved mathematical perfection in his 1958 woodcut Circle Limit III. "Escher did it by instinct," Coxeter explained. "I did it by trigonometry."

Coxeter investigated symmetries, in nature and in the arts. He studied the mathematics of frieze patterns and crystals. When he was still in high school, he became fascinated with four-dimensional spaces and their symmetries. Later, his interest extended to even higher dimensions.

His research into symmetries led him to structures that are now called Coxeter groups. His results on these structures have turned out to be fundamental in many subsequent investigations in several areas of mathematics.

Coxeter was a passionate geometer, following the great tradition that is so intimately connected with the name of Euclid and that has inspired mathematicians for over 2,000 years. Even at times when the fashion in mathematics seemed to move in another direction, Coxeter was pursuing aesthetically appealing geometric problems. In the introduction to Geometry Revisited (1967), he and his co-author, S.L. Greitzer, write:

Let us revisit Euclid. Let us discover for ourselves a few of the newer results. Perhaps we may be able to recapture some of the wonder and awe that our first contact with geometry aroused.

One of Coxeter's books, Regular Complex Polytopes (1974), contains advanced mathematical theories of geometric objects, yet it does not fit into any bookcase. It displays many beautiful and sometimes intricate figures. It is meant to be a coffee-table book. Coxeter writes in the preface:

I have made an attempt to construct it like a Bruckner symphony, with crescendos and climaxes, little foretastes of pleasure to come, and abundant
cross-references.
An accomplished musician, Coxeter liked to point out relations between music and mathematics.

Harold Scott Macdonald ("Donald") Coxeter was born in 1907 in London, and educated at King Alfred School in north London, St George's School, Harpenden, and Trinity College, Cambridge. He got his PhD in 1931 at Cambridge, and was a research fellow at Trinity from 1931 to 1936, during which he spent two years, as a Rockefeller Foundation Fellow and then a J.E. Procter Fellow at Princeton University. In 1936 he joined the Faculty of the Department of Mathematics at Toronto University as an Assistant Professor. He stayed there all his life, serving as Professor of Mathematics
from 1948 to 1980.

Coxeter was welcomed as a visiting professor at universities in Britain, the Netherlands, Italy, Australia and the United States. He was an inspiring and popular lecturer at conferences and universities all over the globe. In July 2002, he gave an invited address at the conference in honour of Janos Bolyai on Hyperbolic Geometry in Budapest. He had a number of honorary doctorates from universities in Canada and in Germany.

He was a Fellow of the Royal Society (1950) and a Fellow of the Royal Society of Canada. In 1995, he received the CRM/Fields Institute Prize, and in 1997 the Distinguished Service Award of the Canadian Mathematical Society and the Sylvester Medal of the Royal Society of London. In the same year, Coxeter was named a Companion of the Order of Canada.

Donald Coxeter contributed greatly to mathematical knowledge, with over 200 scientific publications to his credit, including a number of well-received books, several of which were translated into many different languages. In 1970, he wrote an article on "Solids, geometric" for the Encyclopaedia Britannica.

He was also a vegetarian, he was active in saving the environment, and he promoted peace.

Erich Ellers

15 April 2003 Independent