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**EINSTEIN** o@ca.on.york_county.toronto.globe_and_mail 2003-04-12 published

'He kept a little flame of geometry alive'

Superstar University of Toronto mathematician considered himself
an artist, but his seminal work inevitably found practical applications

By Siobhan
ROBERTS
Saturday,
April 12, 2003 - Page F11

Widely considered the greatest classical geometer of his time
and the man who saved his discipline from near extinction, Harold
Scott MacDonald
COXETER, who died on March 31 at 96, said of
himself, with characteristic modesty, "I am like any other artist.
It just so happens that what fills my mind is shapes and numbers."

Prof. COXETER's work focused on hyperdimensional shapes, specifically
the symmetry of regular figures and polytopes. Polytopes are
geometric shapes of any number of dimensions that cannot be constructed
in the real world and can be visualized only when the eye of
the beholder possesses the necessary insight; they are most often
described mathematically and sometimes can be represented with
hypnotically intricate fine-line drawings.

"I like things that can be seen," Prof.
COXETER once remarked.
"You have to imagine a different world where these queer things
have some kind of shape."

Known as Donald (shortened from MacDonald,) Prof.
COXETER had
such a passion for his work and unrivalled elegance in constructing
and writing proofs that he motivated countless mathematicians
to pick up the antiquated discipline of geometry long after it
had been deemed passé.

John Horton
CONWAY, the Von Neumann professor of mathematics
at Princeton University, never studied under Prof.
COXETER, but
he considers himself an honorary student because of the
COXETERian
nature of his work.

"With math, what you're doing is trying to prove something and
that can get very complicated and ugly.
COXETER always manages
to do it clearly and concisely," Prof.
CONWAY said. "He kept
a little flame of geometry alive by doing such beautiful works
himself.

"I'm reminded of a quotation from Walter Pater's book The Renaissance.
He was describing art and poetry, but he talks of a small, gem-like
flame: 'To burn always with this hard, gem-like flame, to maintain
this ecstasy, is success in life.' "

Prof. COXETER's oeuvre included more than 250 papers and 12 books.
His Introduction to Geometry, published in 1961, is now considered
a classic -- it is still in print and this year is back on the
curriculum at McGill University. His Regular Polytopes is considered
by some as the modern-day addendum to Euclid's Elements. In 1957,
he published Generators and Relations for Discrete Groups, written
jointly with his PhD student and lifelong friend Willy
MOSER.
It is currently in its seventh edition.

Prof. COXETER's self-image as an artist was validated by his
Friendship with and influence on Dutch artist M. C.
ESCHER, who,
when working on his Circle Limit 3 drawings, used to say, "I'm
Coxetering today."

They met at the International Mathematical Congress in Amsterdam
in 1954 and then corresponded about their mutual interest in
repeating patterns and representations of infinity. In a letter
to his son, Mr.
ESCHER noted that a diagram sent to him by Prof.
COXETER that inspired his Circle Limit 3 prints "gave me quite
a shock."

He added that "
COXETER's hocus-pocus text is no use to me at
all.... I understand nothing, absolutely nothing of it."

While Mr. ESCHER claimed total ignorance of math, Prof.
COXETER
wrote numerous papers on the Dutchman's "intuitive geometry."

Though Prof.
COXETER did geometry for its own sake, his work
inevitably found practical application. Buckminster
FULLER encountered
his work in the construction of his geodesic domes. He later
dedicated a book to Prof.
COXETER: "By virtue of his extraordinary
life's work in mathematics, Prof.
COXETER is the geometer of
our bestirring twentieth century. [He is] the spontaneously acclaimed
terrestrial curator of the historical inventory of the science
of pattern analysis."

Prof. COXETER's work with icosohedral symmetries served as a
template of sorts in the Nobel Prize-winning discovery of the
Carbon 60 molecule. It has also proved relevant to other specialized
areas of science such as telecommunications, data mining, topology
and quasi-crystals.

In 1968, Prof.
COXETER added to his list of converts an anonymous
society of French mathematicians, the Bourbakis, who actively
and internationally sought to eradicate classical geometry from
the curriculum of math education.

"Death to Triangles, Down with Euclid!" was the Bourbaki war
cry. Prof.
COXETER's rebuttal: "Everyone is entitled to their
opinion. But the Bourbakis were sadly mistaken."

One member of the society, Pierre
CARTIER, met Prof.
COXETER
in Montreal and became enamoured of his work. Soon, he had persuaded
his fellow Bourbakis to include Prof.
COXETER's approach in their
annual publication. "An entire volume of Bourbaki was thoroughly
inspired by the work of
COXETER," said Prof.
CARTIER, a professor
at Denis Diderot University in Paris.

In the 1968 volume, Prof.
COXETER's name was writ large into
the lexicon of mathematics with the inauguration of the terms
"COXETER number," "
COXETER group" and
"COXETER graph."

These concepts describe symmetrical properties of shapes in multiple
dimensions and helped to bridge the old-fashioned classical geometry
with the more au courant and applied algebraic side of the discipline.
These concepts continue to pervade geometrical discourse, several
decades after being discovered by Prof.
COXETER.

Prof. COXETER became a serious mathematician at the relatively
late age of 14, though family folklore has it that, as a toddler,
he liked to stare at the columns of numbers in the financial
pages of his father's newspaper.

He was born into a Quaker family in Kensington, just west of
London, on February 9, 1907. His mother, Lucy
GEE, was a landscape
artist and portrait painter, and his father, Harold, was a manufacturer
of surgical instruments, though his great love was sculpting.

They had originally named their son MacDonald Scott
COXETER,
but a godparent suggested that the boy's father's name should
be added at the front. Another relative then pointed out that
H.M.S. COXETER made him sound like a ship of the royal fleet
so the names were switched around.

When Prof.
COXETER was 12, he 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.

But more than anything he fancied himself a composer, writing
several piano concertos, a string quartet and a fugue. His mother
took her son and his musical compositions to Gustav
HOLST.
His
advice: "Educate him first."

He was then sent to boarding school, where he met John Flinders
PETRIE, son of Egyptologist Sir Flinders
PETRIE.
The two were
passing time at the infirmary contemplating why there were only
five Platonic solids -- the cube, tetrahedron, octahedron, dodecahedron
and icosahedron. They then began visualizing what these shapes
might look like in the fourth dimension. At the age of 15, Prof.
COXETER won a school prize for an English essay on how to project
these geometric shapes into higher dimensions -- he called it
"Dimensional Analogy."

Prof. COXETER's father took his son along with his essay to meet
friend and fellow pacifist Bertrand
RUSSELL.
Mr.
RUSSELL recommended
Prof. COXETER to mathematician E.H.
NEVILLE, a scout, of sorts,
for mathematics prodigies. He was impressed by Prof.
COXETER's
work but appalled by some inexcusable gaps in his mathematical
knowledge. Prof.
NEVILLE arranged for private tutelage in pursuit
of a scholarship at Cambridge. During this period, Prof.
COXETER
was forbidden from thinking in the fourth dimension, except on
Sundays.

He entered Trinity College, Cambridge, in 1926 and was among
five students handpicked by Ludwig
WITTGENSTEIN for his philosophy
of mathematics class. During his first year at Cambridge, at
the age of 19, he discovered a new regular polyhedron that had
six hexagonal faces at each vertex.

After graduating with first-class honours in 1929, he received
his doctorate under H. F.
BAKER in 1931, winning the coveted
Smith's Prize for his thesis.

Prof. COXETER did fellowship stints back and forth between Princeton
and Cambridge for the next few years, focusing on the mathematics
of kaleidoscopes -- he had mirrors specially cut and hinged together
and carried them in velvet pouches sewn by his mother. By 1933,
he had enumerated the n-dimensional kaleidoscopes -- that is,
kaleidoscopes operating up to any number of dimensions.

The concepts that became known as
COXETER groups are the complex
algebraic equations he developed to express how many images may
be seen of any object in a kaleidoscope (he once used a paper
triangle with the word "nonsense" printed on it to track reflections).

In 1936, Prof.
COXETER was offered an assistant professorship
at the University of Toronto. He made the move shortly after
the sudden death of his father and following his marriage to
Rien BROUWER.
She was from the Netherlnds and he met her while
she was on holiday in London.

As a professor, Prof.
COXETER was known to flout set curriculum.
Ed BARBEAU, now a professor at the U of T, recalled that at the
start of his classes, Prof.
COXETER would spread out a manuscript
on the desks at the front of the room. During his lecture, he
would often pause for minutes at a time to make notes when a
student offered something that might be relevant to his work
in progress. When the work was later published, students were
pleasantly surprised to find that their suggestions had been
duly credited.

Prof. COXETER was also known to show up to class carrying a pineapple,
or a giant sunflower from his garden, demonstrating the existence
of geometric principles in nature. And he was notorious for leaping
over details, expecting students to fill in the rest.

The Canadian Broadcasting Corporation's resident intellectual, Lister
SINCLAIR, was one of
Prof. COXETER's earliest students. He once recounted that Prof.
COXETER would "write an expression on the board and you could
see it talking to him. It was like Michelangelo walking around
a block of marble and seeing what's in there."

Asia Ivic WEISS, a professor at York University, Prof.
COXETER's
last PhD student and the only woman so honoured, describes an
incident that perfectly exemplifies Prof.
COXETER's math myopia.
Going into labour with her first child, she called him to cancel
their weekly meeting. Prof.
COXETER, who never acknowledged her
pregnancy, said not to worry, he would send over a stack of research
to keep her busy when she got home from the hospital.

Despite several offers from other universities, Prof.
COXETER
stayed at University of Toronto throughout his career.

Like his father, he was a pacifist. In 1997, he was among those
who marched a petition to the university president's office to
protest against an honorary degree being conferred on George
BUSH Sr. Prof.
COXETER recalled with disdain Robert
PRITCHARD's
telling him, "Donald, I have more important things to worry about."

After his official retirement in 1977, Prof.
COXETER continued
as a professor emeritus, making weekly visits to his office.
These subsided only in the past several months. On the weekend
before his death, he finished revisions on his final paper, which
he had delivered the previous summer in Budapest.

In his last five years, he survived a heart attack, a broken
hip (he sprung himself from the hospital early to drive to a
geometry conference in Wisconsin) and, most recently, prostate
cancer.

Considering his 96 years of vegetarianism and a strict exercise
regime, he felt betrayed by his body. "I feel like the man of
Thermopylae who doesn't do anything properly," he commented
recently after an awkward evening out, quoting nonsense poet
Edward LEAR.

Prof. COXETER died in his home, with three long last breaths,
just before bed on the last day of March.

His brain is now undergoing study at McMaster University, along
with that of Albert
**EINSTEIN**.
Neuroscientist
Sandra
WITELSON
is tryng to determine whether his brain's extraordinary capacities
are associated with its structure.

Prof. COXETER met with her at the beginning of March and learned
that the atypical elements of Einstein's brain, compared with
an average brain, were symmetrical on both right and left sides.

Prof. WITELSON said she wondered whether there might be similar
findings with Prof.
COXETER's brain. "Isn't that nice," he said.
"I suppose that would indicate all my interest in symmetry was
well founded."

Prof. COXETER leaves his daughter Susan and son Edgar. His wife
died in 1999.

Siobhan ROBERTS is a Toronto writer whose biography of Donald
COXETER will be published by Penguin in 2005.

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