## Donald Coxeter

### Taking geometry into the realms of art, footballing, kaleidoscopes and relativistic quantum field theory

©Donald Coxeter was one of the leading mathematicians, and perhaps the greatest geometer, of the last century. British by birth and training, he spent his working life in Canada, as a professor at the University of Toronto from 1936 until his retirement. His tremendous output, extending over 70 years, included 12 books -- at least four of them classics -- and some two hundred papers.
At two and a half millennia old, geometry has claims to be the oldest and noblest of the branches of mathematics. It was, with arithmetic and algebra, part of the core curriculum of school mathematics and especially of university mathematics until the middle of the last century, when it began to lose ground. Coxeter's best-known book, *Introduction to Geometry* (1961), was a deliberate and partly successful attempt to halt this erosion.

The Ancient Greeks were aware of the five "Platonic solids", solid figures whose faces were identical regular polygons. Three have (equilateral) triangular faces -- the tetrahedron (three faces meeting at each vertex), the octahedron (with four) and the icosahedron (with five). In addition, there are the cube (three squares meeting at each vertex), and dodecahedron (three regular pentagons at each vertex). Coxeter's *Regular Polytopes* (1948) gives a systematic account of these, and their relatives. One of these, involving both hexagons and pentagons, is now well-known for its use on soccer balls, and it is an illustration of the power and universal reach of mathematics that this figure has had a profound impact in two quite different fields.

The first is chemistry, where it led Sir Harry Kroto and collaborators to the discovery of carbon 60. The second is architecture, where it inspired Buckminster Fuller to create his famous geodesic dome: the carbon 60 molecule is accordingly called Buckminsterfullerene or the Buckyball.

The ordinary geometry of the world, studied by the Ancient Greeks, is called Euclidean geometry. But in the 19th century mathematics received a profound and liberating shock when it was discovered that other geometries exist, in which, for instance, parallel lines meet, and the angles of a triangle do not add up to two right angles. Coxeter's *Non-Euclidean Geometry* (1942) was a classic treatment of this field. *The Real Projective Plane* (1949) was an equally important treatment of the mathematics of perspective (and now of computer graphics and virtual reality).

When Coxeter spoke at the International Congress of Mathematicians in 1954, he attended the special exhibition there of the graphic work of the Dutch artist M. C. Escher (1989-1972). The two men became friends, and there was a remarkable cross-fertilisation. Coxeter's mathematical knowledge of non-Euclidean geometry inspired Escher's series of prints Circle Limit I-IV. These are based on the standard model of non-Euclidean geometry, represented in the interior of a circle where as one approaches the circumference one "goes off to infinity". This represents one of the three basic types of geometry (negative curvature). The other two (ordinary plane geometry, with zero curvature) and spherical geometry (with positive curvature) are also represented in Escher's work, the first in his honeycomb patterns and the second in his woodcuts on spheres.

Escher wrote to Coxeter in 1958 thanking him for his booklet *A Symposium on Symmetry* and adding "the text of your article on *Crystal Symmetry and its Generalisations* is much too learned for a simple, self-made pattern man like me." Coxeter's comment on this collaboration was: "Escher did it by instinct, I did it by trigonometry."

Much of Coxeter's time was devoted to group theory, or ways of measuring symmetry. This concerns the geometry of, for instance, kaleidoscopes and reflections in different planes, now known as Coxeter groups. His book *Generators and Relations for Discrete Groups* (with W. O. J. Moser, 1957) contains an introduction to his extensive work on geometrical figures.

Another strand of his thinking became influential in theoretical physics, where his ideas played a role in such areas as relativistic quantum field theory, the marriage of quantum theory with Einstein's special theory of relativity. Coxeter numbers, Coxeter diagrams and the like play their part in the physics of elementary particles and their classification.

Although he never used computers in his own mathematics, similar ideas are crucial to the information age, specifically, in the theory of error-correcting codes, and Coxeter's work abounds in such marvellous illustrations of the power of mathematics in areas apparently far removed from it.

Harold Scott MacDonald Coxeter was born in Kensington in 1907. His father was in manufacturing, his mother was an artist. He was a mathematical prodigy, and also highly musical, becoming a fine pianist and composing an opera at the age of 12. His achievements at St George's School, Harpenden, led his father to consult Bertrand Russell about his son's future. The recommendation that he should leave school and use a private maths tutor led to a scholarship to Trinity College, Cambridge.

There he attended lectures by Ludwig Wittgenstein, took a first, and in 1931 obtained a doctorate under H. F. Baker, who was then the leading figure in geometry in Britain. He became a research fellow in Cambridge, and spent two years at Princeton as a visiting fellow.

The year 1936 was a crucial one in Coxeter's life. He accepted the offer of an assistant professorship at the University of Toronto, where he was to stay for the rest of his life. In the same year he married Hendrina Brouwer.

Coxeter was widely honoured. He was elected a Fellow of the Royal Society of Canada in 1948 and a Fellow of the Royal Society in 1950. He was president of the Canadian Mathematical Society (1962-63), vice-president of the American Mathematical Society (1968), president of the International Congress of Mathematicians in Vancouver (1974), a foreign member of the American Academy of Arts and Sciences, and a Companion of the Order of Canada (1997).

Coxeter, whose hobby was magic, attributed his longevity to vegetarianism and his love of mathematics. His happy family life and the esteem of students and colleagues worldwide doubtless contributed too.

His wife died in 1999. He is survived by his son and daughter.

[Donald Coxeter, mathematician, was born on February 9, 1907.He died on March 31, 2003, aged 96.]

© The Times, 2003