# Donald Coxeter

### Guardian obituary

Energetic scholar drawing young people into geometric circles

To generations of schoolchildren, myself included, the name of HSM Coxeter stood for the wonder of geometry. The spectacular images -- polyhedra radiating out in star shapes, and rotating rings of tetrahedra -- to be found in his revised 11th edition of WW Rouse Ball's marvellous Mathematical Recreations And Essays (1939; 13th edition, 1987) pointed to a mind of outstanding imagination and analytical power.

Donald Coxeter, as he was known, has died at the age of 96. Based for most of his life in Toronto, he was the greatest of the remarkable geometers inspired by HF Baker at Cambridge University before the second world war; others included P Du Val, W L Edge and W V D Hodge.

Born into a Quaker family in Kensington, west London, Coxeter showed early brilliance in music and mathematics, and went to King Alfred's school, Hampstead. His interest in geometry blossomed while he was boarding at St George's school, Harpenden, and, at 14, was confined to the sick bay with flu, in the next bed to John Petrie, son of the Egyptologist Sir Flinders Petrie.

As Coxeter recalled to his half-sister Eve: "To begin with, we both felt too ill to think of anything but groaning, but I was not quite deranged. We had just been learning about the first polyhedron shapes, and I thought 'This is too elementary, theorems cannot stop here.' Round and round in my head, behind my eyes and on to my retina, the visual images of our geometric models spun wildly.

"Suddenly, I sat up and shouted to John. 'If these models could go beyond the third dimension, which we can see in the shapes, what about a fourth dimension? It could be an inversion, with kinds of cave-like spaces inside each facet.' A day or two later, John was well enough to draw me a realistic model based on my ideas, but using tonal shading in pencil to make it look more three-dimensional.

"Suddenly, we could both see the extra dimension. I knew then that mathematics, geometry in particular, was to be my main nourishment."

On the advice of Bertrand Russell, Coxeter left school for private tuition, and in 1926 won a scholarship to Trinity College, Cambridge. After gaining his PhD in 1931, he was a research fellow of Trinity until 1936, during which time he spent two separate years at Princeton University (1932-33 and 1934-35).

In 1936, he was appointed an assistant professor at the University of Toronto, where he remained until retirement. He served as full professor from 1948 to 1980, and was creatively active to the end; he published 12 books and more than 200 articles, several in collaboration with others.

"Coxeter groups", generated by reflection patterns, as in kaleidoscopes, have played an important part in many areas of mathematics and physics. Coxeter's work on polyhedra and their symmetries -- such as the tesselation of a sphere by triangles -- helped the American engineer and architect Buck minster Fuller to develop geodesic domes in the late 1940s, and led Sir Harry Kroto and two scientists at Rice University, Houston, to the 1996 Nobel-prizewinning discovery of the carbon 60 molecule, known popularly as the buckyball.

Coxeter was appointed a fellow of the Royal Society of Canada in 1948, and of the Royal Society in 1950. In 1974, he chaired the international congress of mathematicians' meeting in Vancouver.

At an earlier ICM, at Amsterdam in 1954, he met the artist Maurits Escher, each having an enormous influence on the other, art and mathematics becoming one. Coxeter inspired the Dutchman's circle limit etchings, in which motifs become smaller towards a limiting circle; in 1996, Coxeter demonstrated that the third of the set had arrived at a trigonometrical truth by purely intuitive means.

From 1989, John Robinson developed Coxeter's conceptions in amazing abstract sculptures, such as Firmament, built from mutually tangential spheres -- the artist presented this piece to the mathematician for his 90th birthday in 1997, the year that Coxeter received the Royal Society's Sylvester medal, and was made a Companion of the Order of Canada.

After the death of his wife Hendrina (Rien) Brouwer in 1999 -- they had married in 1936 -- Coxeter undertook a gruelling regime, speaking at conferences all over the world.

For instance, in September 2000, he travelled from Sweden to Cambridge, to lecture on Five Spheres In Mutual Contact, the subject of the Robinson sculpture, at the Isaac Newton Institute. Trinity College then made him an honorary fellow, and, a few days later, he not only repeated his lecture at Liverpool University, but spent the following morning presenting more elementary geometry to schoolchildren. Though frail, he remained young at heart.

He is survived by his daughter Susan, who accompanied him on his recent travels, and his son Edgar.

25 April 2003 © Guardian Newspapers Limited 2003

Donald Coxeter, as he was known, has died at the age of 96. Based for most of his life in Toronto, he was the greatest of the remarkable geometers inspired by HF Baker at Cambridge University before the second world war; others included P Du Val, W L Edge and W V D Hodge.

Born into a Quaker family in Kensington, west London, Coxeter showed early brilliance in music and mathematics, and went to King Alfred's school, Hampstead. His interest in geometry blossomed while he was boarding at St George's school, Harpenden, and, at 14, was confined to the sick bay with flu, in the next bed to John Petrie, son of the Egyptologist Sir Flinders Petrie.

As Coxeter recalled to his half-sister Eve: "To begin with, we both felt too ill to think of anything but groaning, but I was not quite deranged. We had just been learning about the first polyhedron shapes, and I thought 'This is too elementary, theorems cannot stop here.' Round and round in my head, behind my eyes and on to my retina, the visual images of our geometric models spun wildly.

"Suddenly, I sat up and shouted to John. 'If these models could go beyond the third dimension, which we can see in the shapes, what about a fourth dimension? It could be an inversion, with kinds of cave-like spaces inside each facet.' A day or two later, John was well enough to draw me a realistic model based on my ideas, but using tonal shading in pencil to make it look more three-dimensional.

"Suddenly, we could both see the extra dimension. I knew then that mathematics, geometry in particular, was to be my main nourishment."

On the advice of Bertrand Russell, Coxeter left school for private tuition, and in 1926 won a scholarship to Trinity College, Cambridge. After gaining his PhD in 1931, he was a research fellow of Trinity until 1936, during which time he spent two separate years at Princeton University (1932-33 and 1934-35).

In 1936, he was appointed an assistant professor at the University of Toronto, where he remained until retirement. He served as full professor from 1948 to 1980, and was creatively active to the end; he published 12 books and more than 200 articles, several in collaboration with others.

"Coxeter groups", generated by reflection patterns, as in kaleidoscopes, have played an important part in many areas of mathematics and physics. Coxeter's work on polyhedra and their symmetries -- such as the tesselation of a sphere by triangles -- helped the American engineer and architect Buck minster Fuller to develop geodesic domes in the late 1940s, and led Sir Harry Kroto and two scientists at Rice University, Houston, to the 1996 Nobel-prizewinning discovery of the carbon 60 molecule, known popularly as the buckyball.

Coxeter was appointed a fellow of the Royal Society of Canada in 1948, and of the Royal Society in 1950. In 1974, he chaired the international congress of mathematicians' meeting in Vancouver.

At an earlier ICM, at Amsterdam in 1954, he met the artist Maurits Escher, each having an enormous influence on the other, art and mathematics becoming one. Coxeter inspired the Dutchman's circle limit etchings, in which motifs become smaller towards a limiting circle; in 1996, Coxeter demonstrated that the third of the set had arrived at a trigonometrical truth by purely intuitive means.

From 1989, John Robinson developed Coxeter's conceptions in amazing abstract sculptures, such as Firmament, built from mutually tangential spheres -- the artist presented this piece to the mathematician for his 90th birthday in 1997, the year that Coxeter received the Royal Society's Sylvester medal, and was made a Companion of the Order of Canada.

After the death of his wife Hendrina (Rien) Brouwer in 1999 -- they had married in 1936 -- Coxeter undertook a gruelling regime, speaking at conferences all over the world.

For instance, in September 2000, he travelled from Sweden to Cambridge, to lecture on Five Spheres In Mutual Contact, the subject of the Robinson sculpture, at the Isaac Newton Institute. Trinity College then made him an honorary fellow, and, a few days later, he not only repeated his lecture at Liverpool University, but spent the following morning presenting more elementary geometry to schoolchildren. Though frail, he remained young at heart.

He is survived by his daughter Susan, who accompanied him on his recent travels, and his son Edgar.

**Ian Porteous***Harold Scott MacDonald 'Donald' Coxeter, geometrician, born*9*February*1907;*died*31 March 200325 April 2003 © Guardian Newspapers Limited 2003