# Harold Scott MacDonald Coxeter

### Quick Info

London, England

Toronto, Canada

**Donald Coxeter**'s work was mainly in geometry. In particular he made contributions of major importance in the theory of polytopes, non-euclidean geometry, group theory and combinatorics.

### Biography

**Donald Coxeter**was always known as Donald which came from his third name MacDonald. This needs a little explanation. He was first given the name MacDonald Scott Coxeter, but a godparent suggested that his father's name should be added, so Harold was added at the front. Another relative noted that H M S Coxeter made him sound like a ship. A permutation of the names resulted in Harold Scott MacDonald Coxeter.

Donald Coxeter's father was Harold Coxeter (born London about 1878) who was a gas manufacturer and his mother was Lucy Coxeter (born Lincoln about 1872) who was a painter.

Donald was educated at the University of Cambridge, receiving his B.A. in 1929. He continued to study for a doctorate at Cambridge under H F Baker, and this was awarded in 1931. He then became a Fellow continuing his researches at Cambridge. During this period he spent two years as a research visitor at Princeton University working under Veblen. He was Rockefeller Fellow during 1932-33 and Procter Fellow during 1934-35.

In 1936 Coxeter took up an appointment at the University of Toronto. He remained on the faculty at Toronto until his death. A celebration was held in the department in 1996 to celebrate his 60 years at the University of Toronto.

Coxeter's work was mainly in geometry. In particular he made contributions of major importance in the theory of polytopes, non-euclidean geometry, group theory and combinatorics.

Coxeter polytopes are the fundamental domains of discrete reflection groups, now called

*Coxeter groups*, and they give rise to tesselations. In 1934 Coxeter classified all spherical and euclidean Coxeter groups.

His work was motivated by the beauty of mathematics. Robert Moody, proposing Coxeter for an honorary degree from York University in Toronto, said:-

Modern science is often driven by fads and fashions, and mathematics is no exception. Coxeter's style, I would say, is singularly unfashionable. He is guided, I think, almost completely by a profound sense of what is beautiful.York was not the only university to honour Coxeter. He received nine honorary doctorates and was a Fellow of the Royal Society of London and a Fellow of the Royal Society of Canada.

Among his most famous geometry books are

*The real projective plane*(1955),

*Introduction to geometry*(1961),

*Regular polytopes*(1963),

*Non-euclidean geometry*(1965) and, written jointly with S L Greitzer,

*Geometry revisited*(1967). He also published a famous work on group presentations, which was written jointly with his first doctoral student W O J Moser,

*Generators and relations for discrete groups*.

His 12 books and 167 published articles cover more than mathematical research. Coxeter met Escher in 1954 and the two became lifelong friends. Another friend, R Buckminister Fuller, used Coxeter's ideas in his architecture. In 1938 Coxeter revised and updated Rouse Ball's

*Mathematical recreations and essays*, a book which Rouse Ball first published in 1892.

Coxeter had many artistic gifts, particularly in music. In fact before he became a mathematician he wanted to become a composer. However his interest in symmetry took him towards mathematics and into a career which he loved throughout has life. Coxeter wrote:-

I am extremely fortunate for being paid for what I would have done anyway.He attributed his long life to vegetarianism, a regular regime of exercise that saw him do 50 push-ups a day at the age of 89, and, perhaps most importantly, as he related himself

I am never bored.Allow me [EFR] a personal note:-

A colleague and I visited Donald in Toronto in the 1970s and I will always remember his office full of mathematical models. I remember the extreme kindness and wonderful hospitality of Donald and his Dutch wife Rien. When I said I had never had pumpkin pie before, Donald vanished into the kitchen and staggered back with a huge pumpkin which his frail figure hardly looked able to carry.In 1997 Coxeter was made a Companion of the Order of Canada. This is the highest of the three levels of honours that Canada makes.

He taught me how to write a mathematics paper. He was a craftsman at constructing a paper, counting the symbols to make sure that formulas did not break across a line.

When Donald visited me and a colleague in St Andrews we took him a walk down the pier at the harbour. He insisted, much to our trepidation, on climbing an insecure rusty ladder at the end of the pier. He was certainly not as frail as he looked!

### References (show)

- Siobhan Roberts,
*King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry*(New York, 2006) - D J Albers and G L Alexanderson (eds.), Mathematical People: Profiles and Interviews (Boston, 1985), 51-64.
- H S M Coxeter: published works,
*The geometric vein*(New York-Berlin, 1981), 5-13. - G F D Duff , H. S. M. Coxeter Celebrates 90th Birthday,
*Notices of the American Mathematical Society***44**(3) (1997), 340-341. http://www.ams.org/notices/199703/comm-coxeter.pdf - L Fejes Tóth, Some researches inspired by H S M Coxeter,
*The geometric vein*(New York-Berlin, 1981), 271-277. - I Hargittai, Lifelong symmetry: a conversation with H S M Coxeter,
*The Mathematical Intelligencer***18**(4) (1996), 35-41. - Harold Scott MacDonald Coxeter,
*Bull. London Math. Soc.***11**(1) (1979), 111-112. - C Musès, A celebration of higher-dimensional systems and a man who notably explored them,
*Kybernetes***25**(5) (1996), 48-52.

### Additional Resources (show)

Other pages about Donald Coxeter:

### Honours (show)

### Cross-references (show)

Written by J J O'Connor and E F Robertson

Last Update October 1997

Last Update October 1997