Back when I was a graduate student trying to figure out how to define and calculate topological charge in lattice gauge theory, at one point I went over to the math department to ask some people I knew if they had any idea about how to calculate the volumes of spherical tetrahedra. I was taken to the math department lounge to consult with the master of 3 dimensions, Bill Thurston. Thurston explained to me that this could be done by breaking the tetrahedra into “double-rectangular tetrahedra”, whose volumes were then expressed in terms of the angles defining them using something called Schläfli functions, defined back in 1860. This experience helped cure me of my prejudice that modern mathematicians were probably ignorant of the older more concrete mathematics of the 19th century.
Thurston also pointed me to a more modern reference for this, a paper by H.S.M. Coxeter from 1935 entitled The Functions of Schläfli and Lobatschefsky. I ultimately found a much simpler way of computing topological charges, but I always wondered about this early 20th century mathematician, whose parents had given him a set of initials reminiscent of a British naval vessel. Later on in life, I learned a bit about some important algebraic constructions called Coxeter groups, and also heard that there was an active mathematician in Toronto named Donald Coxeter. I assumed that there were at least two and maybe three mathematicians named Coxeter out there, perhaps relatives.
It turns out that these are all the same Coxeter (the M. is for MacDonald), and there’s a very nice new biography of him that has recently appeared, writtten by Siobhan Roberts and entitled King of Infinite Space. Coxeter only died quite recently, in 2003 at the age of 96, and Roberts was able to get to know him while writing the book. It contains a wealth of information about pieces of mathematical history I was not aware of, often buried in the very extensive footnotes.
Coxeter’s main interest was in “classical” geometry, the geometry of figures in two and three dimensional space and he wrote a very popular and influential college-level textbook on the subject, Introduction to Geometry. Much of this subject can be thought of as group theory, thinking of these figures in terms of their discrete symmetry groups. This subject has always kind of left me cold, perhaps mainly because these groups play little role in the kind of physics I’ve been interested in, where what is important are continuous Lie groups, both finite and infinite-dimensional, not the kind of 0-dimensional discrete groups that Coxeter mostly investigated.
One theme of the book is to set Coxeter, as an exemplar of the intuitive, visual and geometric part of mathematics, up against Bourbaki, exemplifying the formal, abstract and algebraic. Bourbaki is blamed for the New Math, and I certainly remember being subjected by the French school system in the late sixties to an experimental math curriculum devoted to things like set theory and injective and surjective mappings. On the other hand, I also remember a couple years later in the U.S. having to sit through a year-long course devoted to extraordinarily boring facts about triangles, giving me a definite sympathy for the Bourbaki rallying cry of “A bas Euclide! Mort aux triangles!”. To this day, both of these seem to me like thoroughly worthless things to be teaching young students.
Actually Bourbaki and Coxeter ended up having a lot in common. They both pretty much ignored modern differential geometry, that part of mathematics that has turned out to be the fundamental underpinning of modern particle physics and general relativity. Coxeter’s most important work probably was the notion of a Coxeter group, which turns out to be a crucial algebraic construction, and ended up being a main topic in some of the later Bourbaki textbooks. A Coxeter group is a certain kind of group generated by reflections, and Weyl groups are important examples. Coxeter first defined and studied them back in the 1930s, part of which he spent in Princeton. Weyl was there at the same time giving lectures on Lie groups, and used Coxeter’s work in his analysis of root systems and Weyl groups.
Coxeter groups and associated Coxeter graphs pop up unexpectedly in all sorts of mathematical problems, and Roberts quotes many mathematicians (including Ravi Vakil, Michael Atiyah and Edward Witten) on the topic of their significance. There are quite a few places where one can learn more about this. These include various expository pieces by John Baez (see for example here, based to some extent on this), as well as a web-site set up by Bill Casselman. The AMS Notices had an interesting series of articles about Coxeter and his work, written shortly after his death. The proceedings of a recent conference at the Fields Institute in Toronto entitled The Coxeter Legacy – Reflections and Projections have recently been published. In a couple weeks there will be a special program in Princeton about Coxeter, aimed at the general public.
One reason I’d started reading the book about Coxeter was to get away from thinking about string theory, but this was definitely not a success, since the book contains a rather extensive discussion of string theory. Coxeter was aware of string theory, it seems it reminded him of Jabberwocky, and he’s quoted as follows:
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 else who can’t make head nor tails of the eleventh or sixteenth dimension.
Roberts quotes Witten (who she says is known as the “pope of strings”) about the possible relevance of Coxeter groups and E(10) to string theory. She describes string theory in somewhat skeptical terms:
But rumblings are that if a bigger breakthrough doesn’t occur soon, and in the form of streams of empirical evidence, string theory will at best be a branch of mathematics or philosophy, but not part of physics.
She quotes Amanda Peet as proposing that string theory become “a faith-based initiative”, and Susskind as “There’s nothing to do except hope the Bush administration will keep paying us.”
Update: Siobhan Roberts has set up a web-site for the book, and she tells me that she’ll soon be starting up a blog there.
Update: There’s a very good expository paper by Igor Dolgachev that discusses Coxeter groups, and generally the way reflection groups appear in algebraic geometry.
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