King of Infinite Space

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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29 Responses to King of Infinite Space

  1. Thomas Love says:

    It is hard to get away from the topic of string theory, it pops up in the most unexpected places, like the comic page of the Times. But then, perhaps the comic page is where string theory belongs.

    Thanks for the review, I’ll have to look up the book.

  2. Xerxes says:

    Your comment about spherical tetrahedra reminded me of a project I worked on where I needed to find the volumes of convex 3d polyhedra and 4d polytopes. I know how to work out simplex volumes and arbitrary convex 2d polygons, but I searched the literature for a long time on the higher-dimensional problem and never came up with anything. Eventually I resigned myself to estimating them using Monte Carlo techniques. Maybe somebody knows a better solution?

  3. D R Lunsford says:

    I remember reading “Regular Polytopes” as a kid and discovering empirically a fact about the Platonic solids. I was eating a lot of Dannon Yogurt at the time – the container was environment-friendly wax paper which however was rather weak. To strengthen the top, a circular cardboard disk was inserted. I pried out a bunch of these identical disks and used them to make the five solids by inscribing regular polygons in them etc. In the end, each face of each solid thus could be inscribed in the same circle. When I set them on my desk, I noticed that they paired up in altitudes, the cube and octahedron having the same altitude, likewise the icosahedron and the dodecahedron, while the tetrahedron was paired with itself, being self-dual! I went on to prove this little factoid by brute force. I often wondered if a simple proof could be had. I always thought Coxeter would have been delighted by that.


  4. Ben Recht says:


    Surprisingly, Monte Carlo techniques are your best bet for high dimensions. Computing the volume of polytopes is #P-Hard.

    On the other hand, there are Monte-Carlo techniques that yield approximations to the volume in time polynomial in the dimension. See

    Simonovits, M. “How to compute the volume in high dimension?” Math. Program., Ser. B 97: 337–374 (2003)

    for a survey and lots of references.


  5. D. Eppstein says:

    One point I found odd in the book’s discussion of the Coxeter-visual-geometry vs Bourbaki-formalist-algebra dichotomy was that it seemed to be placing Hilbert on the formalist side. The author of Geometry and the Imagination, an enemy of visual thinking in geometry?

  6. D R Lunsford says:

    Eppstein – yes, absolutely. Courant in fact to some degree rebelled against his teacher Hilbert. Hilbert once said “physics is too hard for physicists” – this was the beginning of a bad trend in math/physics, that Courant lamented in later life. Hilbert believed in axiomatics – Courant, like Weyl, had a more metaphysical approach and did not like drawing fine lines between the subjects.


  7. Yes, I agree that Bourbaki and Coxeter ended up having a lot in common. The treatment of Coxeter groups by Bourbaki (N. Bourbaki, Groupes et Algebras de Lie, Chap. 4, 5, et 6. Hermann, Paris, 1968.) is perhaps the best written chapter in his treatise (it even contains an illustration!). It was written by Pierre Cartier; a very revealing account of his work on the text is given in his interview:

    M. Senechal, The continuing silence of Bourbaki—An interview with Pierre Cartier, June 18, 1997. The Mathematical Intelligencer 1 (1998) 22–28.

  8. Another Weyl-Coxeter connection is furnished by the quincunx lattice. See “For Sir Thomas Browne” and “Geometry’s Tombstones.”

  9. Steve says:

    What’s E(10)? I thought there was only E(6), E(7) and E(8).

  10. Tony Smith says:

    A web page at says in part:

    “… Once Bourbaki had finally finished its first six books, the obvious question was “what next?”. … Pierre Cartier was working with Bourbaki … Its second series … consisted of two very successful books :
    Book VII Commutative algebra
    Book VIII Lie Groups
    … Bourbaki was now becoming involved in a battle with its publishing company over royalties and translation rights. The matter was settled in 1980 after a “long and unpleasant” legal process, where, as one Bourbaki member put it “both parties lost and the lawyer got rich” …
    In 1983 Bourbaki published its last volume : IX Spectral Theory. …”.

    I am sad that Bourbaki did not continue. As Alexadre Borovik said, Book VIII Lie Groups contained some of the best material by Bourbaki, and it would have been nice to have seen further development of such material described by Bourbaki.
    Maybe such further description might have shown clearly how, as Peter said, “… Bourbaki and Coxeter …[had]… a lot in common …”.

    It is interesting to me that the law of “intellectual property” had a role in the end of Bourbaki.
    It reminds me of a legal conference (in the 90s) in which an audience member said to a panel something like “the internet is a medium for free exchange of ideas”,
    whereupon a panel member (corporate lawyer) said “Corporate interests are taking over the net and anarchists like you will be put down”.

    Tony Smith

  11. Aaron Bergman says:

    What’s E(10)? I thought there was only E(6), E(7) and E(8).

    Add two dots to your Dynkin diagram and hope for the best. You get an infinite dimensional Kac-Moody algebra which seems to be related to the compactification of M-theory down to one dimension on a T^10. This looks like an introduction.

  12. Thomas Larsson says:

    E(10) is not just infinite-dimensional, but very infinite-dimensional (of exponential growth). This makes it quite untractable mathematically, unlike the affine Kac-Moody algebras which only are of polynomial growth. To quote Victor Kac:

    “It is a well kept secret that the theory of Kac-Moody algebras has been a disaster.”

  13. Curious minds want to know says:
  14. Steve Myers says:

    A note on volumes in many dimensions: Hamming has an amusing proof (in “Numerical Analysis for Scientists & Engineers”) that as dimensions increase, volume goes to the surface.

  15. h says:


    You can subdivide any convex polytope to simplicies. There are trivial and not-so-trivial algorithms to do that, which are easy to find in the literature. 3 or 4 dimensions is low enough, so this should be fast, and also gives exact results (for rational polytopes).

  16. About the new maths and triangles : I’ve also been exposed to both, since i was about 10 at the end of the new math era, and had a teacher in high school that was in love with triangles (I suspect he had an affair with the Euler line). I did not find this boring at all, and both were very useful in learning rigour and the joy of discovery. As a child I remember a day when our teacher showed us how to count in the octal and binary systems. It was really exciting. In high school there was nothing more thrilling to me than working on a geometric problem for the all week end. Later, the French education changed and they got rid of both the new maths and almost all of elementary geometry. I had to teach in a high school for four years : now everything is more practical and “useful”, less abstract… and so damn boring !

  17. D R Lunsford says:

    Curious, no this one.


  18. Andy says:

    (OT: There’s an Op-Ed on ST by Brian Greene in today’s NYT.)

  19. Tony Smith says:

    Peter says “… Siobhan Roberts … quotes Amanda Peet as proposing that string theory become “a faith-based initiative” …”.

    However, on her web page cited by D R Lunsford, Amanda Peet herself says:
    “… I study … the fundamental dynamics of quantum gravity … using string theory …
    Thus far, significant progress has been made … [i]n the search for string theoretic mechanisms for resolution of spacetime singularities.  …
    I will also be very interested to see how new ideas about the Landscape may inform the old question of vacuum selection in string theory …”.

    Peet’s web page says that it was “Last updated: 2006/09/22”, so it seems to me that it is probably a current statement of her views, and not an out-dated web page containing stuff about which she has changed her mind.

    Since I think that Peet herself is more likely to give an accurate description of her own opinions than a third party, it seems to me likely that
    Roberts has substantially distorted Peet’s view of string theory,
    that makes me very skeptical about the accuracy of other statements made by Roberts.

    Tony Smith

    PS – Maybe my opinions above are affected by my opinion of some journalists (the Amazon page says “Siobhan Roberts is a journalist”), which is exemplified by an interview of Joe Namath shortly after he became a New York Jet. A reporter asked him if, at the University of Alabama, he had majored in basket-weaving. Joe replied:
    “No, basket-weaving was too hard, so I majored in journalism.”

  20. Peter Woit says:


    The quote from Amanda Peet was clearly a joke by her.


    Maybe I’ll add a couple comments about Brian’s op-ed to the previous posting.

  21. LDM says:

    (OT: There’s an Op-Ed on ST by Brian Greene in today’s NYT.)

    The interesting thing about this letter, is, that in another 20+ years, Briane will be able resubmit it to the NYT almost verbatim.

  22. Tony,

    Yes indeed, Amanda Peet has a good sense of humour. She made the comment at the Strings ’05 conference in Toronto at a public session on “The Next Strings Revolution,” which was dubbed as an “airing of our dirty laundry” session— downloadable proceedings are available at the Fields Institute website,, though if memory serves I don’t believe Peet’s comment, which was made during the Q&A afterward, was included on that recording. Her comment, and the jocularity of the event in general, was nicely covered by Dennis Overbye in the NYT Science Times, “Lacking Hard Data, Theorists Turn to Democracy,” August 2, 2005 —

    As far being a journalist goes, accurate observation, documentation, and portrayal of events — as with science — is of utmost importance!


  23. DT says:

    How, in a group this nerdy (in a good way) could you mention Lobatschefsky in the post and have nobody comment on the eponymous Tom Lehrer song? Here’s a link for the lyrics:

    If you don’t know the song, you should. NB: same guy, different spelling.

  24. There is also a Lehrer song about the Bourbaki-inspired New Math!

  25. Tony Smith says:

    Siobhan and Peter,
    my apologies for being too dense to realize that Peet’s “faith-based initiative” remark was merely a joke intended to reflect her true views.
    In light of that, I will consider Siobhan to be one of the (all-too-few) good-guy journalists,
    not in the class of “some journalists” such as the Joe Namath interviewer (and all-too-many of the partisan talking-heads on TV “news”).

    As a concrete manifestation of my changed opinion, I have ordered a copy of “King of Infinite Space”, and Amazon says I should get it on Monday.

    A question:
    Could a student today get a PhD and a job (and grants) by following in Coxeter’s footsteps ?

    If so, would the student’s success be more likely:
    1 – by working in string theory physics ?
    2 – by working in a pure math department ?
    3 – by some other path (if so what)?

    If not,
    does that mean that it is now unlikely that we will see any more ideas that are as out-of-mainstream as Coxeter’s have sometimes been considered ?

    Tony Smith

  26. Tony Smith says:

    typo correction:
    In my previous post
    “… merely a joke intended to reflect her true views …”
    should have been
    “… merely a joke NOT intended to reflect her true views …”.

    My apologies again.

    Tony Smith

  27. Tony,

    That’s an interesting question. A number of people remarked when I was researching the book (and I may or may not have included it, can’t recall) that Coxeter’s students, all very talented mathematicians (Willy Moser, Asia Ivic Weiss, Barry Monson, Chris Fisher, to name a few), have not approached anything near Coxeter’s stature. And Coxeter’s position at the UofT was not filled with a “pure geometer” when he retired — and you’d think if anywhere, then there. Coxeter was one of a kind in many ways. But I’d be inclined to say 1) or 3), and I’m not sure what form the the latter category would take — perhaps a computer animation scientist at Pixar, or a mathematician in residence at Microsoft or At&T (and in that, their Coxeterian interests might take the shape of a nicely subsidized sideline for the most part, occasionally breaking through with an application). I talk about this in part two of the book, discussing where Coxeter’s style of classical geometry pops up in “modern” and applied contexts. Hope you enjoy it!


  28. Addendum:

    Re 2) can Coxeterian geometers finding positions in pure math depts: John Conway at Princeton is in many ways considered Coxeter’s spiritual successor as a geometer, in terms of the breadth and depth of his interests and work, and then there’s the aforementioned Bill Thurston at Cornell. Many “Coxeterian” geometers find positions in math depts, but the nature of Coxeter’s classical geometery, so to speak, has transcended its origins in terms of current research paths.


  29. yasir says:

    Its not the mutha-string – its the cosmic humor you seem to be striking upon repeatedly that is at the bottom of everything, and keeps everything going !




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