Donald Coxeter, mathematician; Telegraph obit
Hyfler/Rosner
Donald Coxeter
(Filed: 03/04/2003)
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 n-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."
Rich Clancey
+ Another great Telegraph obit. Even though I didn't understand a word of it.
Better known as H.S.M.Coxeter. Just yesterday I took his
"Regular Polytopes" off the shelf to decide if I should make
one more attempt to tackle it.
You could tell from his writing, even if you couldn't follow
the logic, that he proceded from an aesthetic concept of
geometry. I don't think they teach that as a methodology
any more...
--
rich clancey r...@world.std.com