ESCHER firstname.lastname@example.org_county.toronto.globe_and_mail 2003-04-04 published
Mathemetician touted as geometry genius
Friday, April 4, 2003 - Page R13
Toronto -- The Canadian mathemetician who was considered the greatest classical geometer of his generation has died. Professor H. S. M. COXETER was 96.
Prof. COXETER, who went by the Christian name Donald (a shortened version of Macdonald, one of his middle names), dominated the math department at the University of Toronto for 60 years and was legendary in the field of hyperdimensional geometries. In particular, his work on icosahedral symmetries laid the groundwork for a 1996 Nobel Prize won by two Texan scientists who discovered the Carbon 60 molecule.
British-born, Prof. COXETER came to Canada in 1936 to perform work that influenced such luminaries as the Dutch artist M.C. ESCHER and Buckminster FULLER, who described him as "the geometer of our bestirring 20th century, the spontaneously acclaimed terrestrial curator of the historical inventory of the science of pattern analysis."
Prof. COXETER, who at one time headed the Canadian Mathematical Society and in 1997 was appointed a Companion to the Order of Canada, died on Monday.
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ESCHER email@example.com_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 passť.
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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