An interview with Siobhan Roberts
Einstein wrote that "pure mathematics is, in its way, the poetry of logical ideas." If that's so, then Donald Coxeter (1907–2003) was one of the foremost such poets of the 20th century. Born in London, the mathematician studied under Ludwig Wittgenstein at Cambridge and did postgraduate studies at Princeton, but he established his name during a remarkable 60-year tenure at the University of Toronto, during which he revitalized the fallow field of classical geometry, driven by an almost obsessive love of its beauty.
Coxeter was 94 when journalist Siobhan Roberts profiled him in an article for Toronto Life, following him to the last geometry conference he would attend, in Budapest in the summer of 2002. The article, "Figure Head," won a National Magazine Award, and Roberts has expanded it into a book-length biography, King of Infinite Space (Walker & Co.). She's also developing a documentary film on Coxeter. American Scientist Online managing editor Greg Ross interviewed Roberts by e-mail in November 2006.
How was geometry regarded when Coxeter came to the field?
When I was first getting to know Coxeter and followed him to geometry conferences in Banff and Budapest, his colleagues and admirers kept telling me, "This is the man who saved geometry from extinction." So it's well known among the cognoscenti that classical geometry was considered old-fashioned and in decline, if not nearly dead, by the time Coxeter embarked on his career in the 1930s—it had been relegated to a recreation, a trivial tinkering with toys and something fun to do on a Sunday afternoon, but certainly not worth staking a career on. The fashionable trends in mathematics tended toward the abstract and algebraic—prickly, undulating equations but no shapes and diagrams. The epitome of this trend was the Bourbakis, a secret society of crème-de-la-crème French mathematicians founded in the 1930s. When Bourbaki was at its peak, circa 1960, one of the founding members, Jean Dieudonné, the Bourbaki scribe, famously declared, "Death to triangles!"—that sums it up rather graphically, so to speak.
Did Coxeter see it differently? What attracted him?
He did truly did "see" it differently—he was attracted by the very visual and hands-on quality of geometry. He loved polytopes, a class of shapes that reside in any number of dimensions. Polytopes include two-dimensional polygons, like the triangle, and three-dimensional polyhedra, like the five Platonic solids (the cube and the tetrahedron are perhaps the most common, but there's also the octahedron, the icosahedron and the dodecahedron). Coxeter was particularly fond of trying to envision higher-dimensional polytopes, and the fourth dimension, he said, was his favorite. Usually the fourth dimension is thought of as time, but dimensions can be thought of simply as extra coordinates that quantify existence—say, the daily temperature, or President Bush's poll numbers. Geometers, naturally, prefer to think of the fourth dimension in terms of space. Coxeter was so preoccupied with the fourth dimension when he was a teenager that he was dismally behind on some of the mathematical basics. To compensate, when he was cramming for the entrance exams to Cambridge, his tutor forbade him from thinking in four dimensions except on Sundays.
Polytopes became an enduring passion, and he spent much of his career investigating and classifying the symmetries of these shapes. Today his work on symmetries is applied in pretty much every realm of science—for example, his "Coxeter groups" are being used by string theorists in conjunction with Einstein's gravity equations in the search for supersymmetry. Of course, these applications were inadvertent as far as Coxeter was concerned. He was a huge fan of the rational nonsense that runs through Alice in Wonderland, and when I asked him to explain why, he said, "It's like reading about a part of mathematics that you know is beautiful but that you don't quite understand. Like string theory. That's as much a mystery to me as it is to anyone who can't make head nor tail of the 11th or 16th dimension."
So he was interested more in pure mathematics than in applications?
Coxeter was a pure mathematician, yes. But he didn't observe the rigid divide that sometimes separates the pure and applied sciences. His curiosity was piqued by applications in nature. He was particularly fond of a phenomenon called "phyllotaxis," which describes the "leaf arrangement" of certain plants whose leaves or buds or petals grow in patterns governed by the golden ratio (1:1.618)—pineapples and sunflowers are two examples, and Coxeter was known to grow his own sunflowers and bring them to class with the whorl of their seeds highlighted with dots of red nail polish. He was also interested in the spacetime continuum and the shape of viruses (the common cold, which I'm fighting at the moment, is the shape of an icosahedron—a spheroid structure comprised of 20 triangles). It is interesting to note, though, that Coxeter was never interested in applications pertaining to computers; computers were the bridge too far for him. There is a funny story about him collaborating with a communications man, Gord Lang, providing some sphere-packing research for Lang's work to develop a modem, but when Coxeter was presented with the application he was a tad horrified. He always thought computers would lure students away from pure mathematics.
Ironically, as it turns out, the computer has been a big force behind geometry's renaissance. The antigeometry trend of the early to mid-20th century was largely due to the fact that geometry, with its shapes and diagrams, was visual, and the visual sense, being subjective, was fallible and allowed the geometer to be led astray and to succumb to erroneous visual reasoning. Modern mathematicians sought a more rigorous and rational methodology, and algebra, with equations either balancing or not, provided that certainty. When geometry merged with the computer, however, the uncertainty of the visual component was eliminated—because the computer has all the algebraic axioms programmed in and won't allow the user to take an errant path or make any of those visual mistakes. So computers provide the geometer with better tools, with a souped-up geometry kit.
Coxeter became friends with the graphic artist M. C. Escher. How did this inform Escher's work?
The two struck up a correspondence, and a collaboration of sorts, after they crossed paths in 1954 at the International Congress of Mathematicians in Amsterdam. One of Coxeter's geometric figures—a tiling of the hyperbolic plane—directly inspired the artist's Circle Limit III prints, depicting a sphere full of multicolored fish that become infinitely smaller and smaller as they swim off the orb's horizon. When Escher worked on his Circle Limit series he would say, "I'm Coxetering today!" But Escher accomplished his mathematically inspired art sheerly by intuition, relying on Coxeter's geometric pictures, because Escher had no training in mathematics. Coxeter was always delighted by Escher's pieces, and wrote several papers about them and sent the artist letters full of mathematical analysis. Escher always maintained he understood not a word of Coxeter's explanations, dismissing it all as Coxeter's "hocus-pocus math."
What of his relationship with Buckminster Fuller? Fuller wasn't a trained mathematician either.
Coxeter had all the patience in the world for amateurs, but his regard for "Bucky," as he came to call him, ran hot and cold. Even before he met Fuller in 1968, Coxeter was entranced by Fuller's geodesic domes—he first saw one at Expo '67 in Montreal, where a geodesic dome was built as the American Pavilion, and subsequently when Coxeter's cottage burned down he contemplated a geodesic dome as a replacement, though this never came to pass, probably because it was too expensive for the penny-pincher in Coxeter.
But at times Coxeter found it difficult to stomach Fuller's tendency to take perhaps more credit than was his due—Fuller, for example, liked to rename things, and he gave the name vector equilibrium to a solid that had long been known as the cube octahedron, which miffed Coxeter. Bucky, perhaps to compensate for his lack of training, also had great bravado and gave talks with a lot of hype. As a result, Fuller became more famous for his polyhedra work than the original polyhedral adventurers. And in turn, the carbon 60 molecule, when it was discovered in 1985, was named the buckminsterfullerene, which also dismayed Coxeter—the structure of C_{60} was one of the Archimedean solids, the truncated icosahedron, so Coxeter felt it would have been better named after the great Archimedes.
Coxeter didn't have much patience for this aspect of Bucky's persona, but he admired his work, and over the years they enjoyed a steady correspondence and exchanged intellectual gifts of various kinds, sending ideas and papers and models back and forth. And Fuller dedicated his book Synergetics: Explorations in the Geometry of Thinking to Coxeter, praising him as "the geometer of our bestirring twentieth century ... and in thanks to all the geometers of all time whose importance to humanity he epitomizes." Coxeter was flattered with the dedication but didn't think so highly of the book.
It's commonly said that math is a young man's game, but Coxeter was still active in his nineties. What accounts for this, do you think?
His daily headstands, perhaps—Coxeter often joked that taking up his father's habit of a headstand every morning before breakfast was his intellectual elixir, and he kept doing them into his 90s. I once asked him why he kept at geometry into old age, and he said that he considered himself an artist, and like any other artist he was driven by an all-consuming obsession—it was just that his obsession was for patterns and shapes. Whatever the reason—"use it or lose it" must also have made a difference—Coxeter simply had a superior brain. It's being studied by the neuroscientist Sandra Witelson at McMaster University in Hamilton, Ontario, and she's also studying Einstein's brain. Her results on Coxeter's brain so far indicate that he, like Einstein, had an abnormally large parietal lobe—and, fittingly enough, this is the region of the brain responsible for visual thinking and spatial reasoning. I went to visit Witelson a couple of months ago, and I not only saw Coxeter's brain, which Witelson said was remarkably plump and not nearly as atrophied as you would expect for a man his age (he was 96 when he died, although he had the brain of a much younger man), but I also held Coxeter's brain, and that experience defies description. Writing his biography, I'd spent four years trying to get into the man's head, reading all his diaries and letters, and then there was the essence of his being, the organ that guided his thoughts, resting in my hands.
Did he achieve everything he set out to do?
He once remarked that he had more problems percolating through his mind than he would ever have time to address, so his curiosity as a geometer would never be satiated. He regretted in retrospect that he had spent so much time on his career and had not given enough attention to his family. Though at the same time, at the end of his life he was frustrated and saddened that his body was betraying his mind, and that he couldn't keep going with geometry—a couple days before he died he was pulling books on "convexity" off his bookshelf and starting another paper for yet another conference he had been invited to, in Budapest. But in terms of one tangible thing, he once told me he wished he had collaborated directly with his longtime friend Paul Erdös and achieved the coveted status of an Erdös number of 1 [signifying a direct collaboration with the prolific Hungarian mathematician]. Coxeter had to settled for an Erdös 2, via his collaboration with John Horton Conway at Princeton, which of course is nothing to sniff at. Conway is a huge fan of Coxeter's and his work, and in many ways is considered Coxeter's spiritual successor as a geometer.