Donald Coxeter

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Donald Coxeter, who died on Monday aged 96, made fundamental contributions in the study of multi-dimensional geometric shapes and was regarded as the greatest classical geometer of his generation.

Coxeter published in the geometrical field for 70 years, worked professionally at the University of Toronto for 60 years and wrote 12 books and more than 200 articles. He was best known for his work in hyperdimensional geometries and regular polytopes -- complicated geometric shapes of any number of dimensions that cannot be constructed in the real world but can be described mathematically and can sometimes be drawn.

In 1926, at the age of 19, he discovered a new regular polyhedron, having six hexagonal faces at each vertex. He went on to study the mathematics of kaleidoscopes and, by 1933, had enumerated the nn-dimensional kaleidoscopes (kaleidoscopes operating up to any number of dimensions). His complex algebraic equations expressing how many images of an object may be seen in a kaleidoscope are now known as Coxeter groups.

Coxeter's work on icosahedral symmetries played an important role in the discovery by scientists at Rice University, Texas, of the Carbon 60 molecule, for which they won the 1996 Nobel Prize in Chemistry. Carbon 60 is now being tested as a superconductor for use in everything from chemotherapy and telecommunications to Aids research.

He was guided by a profound and almost artistic appreciation of the beauty of symmetry and his work inspired many people outside the field of mathematics. Buckminster Fuller, the philosopher and architectural theorist, was inspired by Coxeter when he designed his famous geodesic dome. In a somewhat florid dedication to Coxeter in his book Synergetics, Buckminster Fuller described him as "the geometer of our bestirring twentieth century, the spontaneously acclaimed terrestrial curator of the historical inventory of the science of pattern analysis".

Coxeter also became a close friend of the Dutch graphic artist Maurits Escher, whom he first met in 1954 at an international mathematics conference in Amsterdam. Escher had been growing tired of repeating birds and fish on a flat plane. He was aware of Coxeter's work on the reflections of shapes in multi-dimensional space and wanted to know more.

Coxeter later sent Escher a copy of his paper Crystal Symmetry and Its Generalisations, which contained a series of complex geometric figures, including a pattern in which the motifs become ever smaller towards a limiting circle. Inspired by these designs, Escher went on to create a series of "Circle Limit" etchings, some of which he presented to Coxeter.

In 1996 Coxeter published a paper in which he proved that, despite knowing no mathematics, Escher had achieved mathematical perfection in his etching Circle Limit III. Coxeter showed that the arabesques of intersecting arcs that form the backbones of the fish in the design are based on an arcane formula involving the cosine of an angle and the hyperbolic sine of a logarithmic function; "Escher did it by instinct," Coxeter explained, "I did it by trigonometry."

Harold Scott MacDonald Coxeter, always known as Donald, was born into a Quaker family at Kensington, west London, on February 9 1907. His mother was a landscape artist and portrait painter, and his father a manufacturer of surgical instruments and anaesthetics. They had originally named their son MacDonald Scott Coxeter, but a godparent suggested that the boy's father's name, Harold, should be added at the front. Another relative pointed out that HMS Coxeter sounded too much like a battleship, so the names were switched around.

Donald was fascinated by the patterns of numbers from an early age. His mother noticed that, when he was two or three, he became entranced with the columns of numbers printed on the financial pages of the newspapers. This juvenile fascination was soon replaced by an interest in cones, triangles and symmetrical geometric objects of all sorts.

Yet it seemed, at first, that young Donald's talents lay elsewhere. He became an accomplished pianist and, as a child, composed piano pieces, a string quartet and, when he was 12, an opera. He also created his own language -- "Amellaibian" -- a cross between Latin and French, and filled a 126-page notebook with information on the imaginary world where it was spoken.

At St George's School at Harpenden, he harboured hopes of becoming a composer. But his appreciation of the beauties of symmetry turned him towards mathematics. Convalescing in the school sanatorium with the chicken pox, he found himself lying next to John Flinders Petrie, son of the Egyptologist Sir William Matthew Flinders Petrie.

The two began chatting about H G Wells's Time Machine and about why there were only five Platonic solids, and passed the time contemplating the possibility of other dimensions. A few years later, Donald won a school prize for an essay on how to project geometric shapes into higher dimensions.
Impressed with his son's talents, Coxeter's father took him to meet the philosopher Bertrand Russell, who concluded he was brilliant and put him in contact with the mathematician E H Neville. Neville met the young prodigy, deemed his school inadequate, and suggested that he drop all subjects save mathematics and German (as the best mathematicians were German) and recommended him a private tutor in mathematics.

Coxeter won a scholarship to Trinity College, Cambridge, where he was one of only five students selected by Ludwig Wittgenstein to attend his philosophy of mathematics classes. After graduating with a First, he took a doctorate under H F Baker in 1931 then remained at Cambridge as a research fellow. During this period, he spent two years as a research visitor at Princeton University, as a Rockefeller Fellow in 1932-33 and Procter Fellow in 1934-35.

In 1936, Coxeter received an invitation from Sam Beatty at the University of Toronto offering him an assistant professorship there. His father, foreseeing the coming war, advised him to go. He remained in Toronto for the rest of his life.

In the Second World War, Coxeter was asked by the American government to work in Washington as a code-breaker. He accepted, but then backed out, partly because of his pacifist views and partly for aesthetic reasons: "The work didn't really appeal to me," he explained; "it was a different sort of mathematics."

Coxeter's best-known works include The Real Projective Plane (1955); Introduction to Geometry (1961); Regular Polytopes (1963); Non-Euclidian Geometry (1965); and Geometry Revisited (with S L Greitzer, 1967). He also published a famous work on group presentations, Generators and Relations for Discrete Groups (written jointly with W O J Moser, 1957).

A gaunt, bird-like, ascetic-looking man, Coxeter attributed his longevity to his vegetarianism, a daily exercise regime of 50 press-ups, a nightly cocktail of Kahlua, peach schnapps and soya milk, and an abiding fascination with his subject.

Despite, or perhaps because of, his appreciation of the aesthetics of mathematics, he never used a calculator or computer and wrote all his papers in pencil so that he could go back and correct them. He travelled to work by bus and could often be seen wandering around the university campus carrying a pineapple, which he used in his classes to illustrate natural symmetry.

His students adored him, though they were sometimes surprised by his other-worldliness. When a female student announced that she would not be attending one of their regular meetings because she was about to give birth, he gave her a complex 50-page draft of a paper for her to look through if she "had nothing else to do in the labour room".

Coxeter served as president of the Canadian Mathematical Society (1962-3); as vice president of the American Mathematical Society (1968); and as president of the International Congress of Mathematicians in Vancouver in 1974. He was elected a Fellow of the Royal Society of London in 1950 and a Fellow of the Royal Society of Canada in 1948; he was a foreign member of the American Academy of Arts and Sciences. He was appointed a Companion of the Order of Canada in 1997.
On his 90th birthday that year, Coxeter was presented with Firmament, a sculpture by the British sculptor John Robinson, illustrating a geometrical progression Coxeter had discovered, whereby spheres of certain diameters are mutually tangent.

Donald Coxeter married, in 1936, Hendrina Brouwer, who died in 1999; they had a son and a daughter, Susan, who looked after her father after her mother's death and accompanied him to mathematical conferences.

Last July, after Coxeter had given a talk at a conference in Budapest, she commented: "To think we've come all this way to talk about circles touching circles when there are so many more important things going on in the world. Dad would hate to be equated with Elvis Presley, but Elvis gave people some moments of joy, happiness, inspiration. And if that's what Dad's work does for these people, that's wonderful. Personally," she added, "I get more from Elvis Presley."

03/04/2003 © Telegraph Group Limited.