Conway is here. Conway is here. Conway, Conway, Conway. John Horton "Life" Conway. John "Horned-Sphere Beard" Conway. John "I have my own knot polynomial" Conway. John "Penrose Tiles before breakfast, Sphere Packings before lunch, Nim in the afternoon, and Group Theory in the evening" Conway. John "Come down to Jack Murphy Stadium and watch the MMMMOONNNNNSSSTTEEEERRR SSIMMMPLLEE GROUPPPPPS collide and explode!" Horton Conway. Conway, Conway, Conway. Conway is here.
He was at breakfast. And so, amazingly enough, was I. I heard the British accent (Cambridge accent, actually, without the stuffiness) from across the room and there he was, looking like a middle-age surfer. Birkensock shoes, Bermuda shorts, (well, ok, an Escher) T-Shirt, Beard (not as long as I had expected), slight belly, straight hair combed back, and a look of intense yet peaceful radiance on his face. He was sitting at a table full of high school teachers, explaining the classification of surfaces. They were entranced.
I grabbed my daily dose of Coke and fruit loops and sat at an adjacent table. I ate slowly and strained to hear the conversation. Eventually they left in a Brazil-esque scene of Conway walking and talking quickly, and the teachers following in a pack.
The first lecture was by Jeff Weeks, who presented his "snap-pea" program, which I guess is every knot theorist's wet dream. It finds the symmetry groups, it finds the homology groups of the link compliment, it finds canonical cell decompositions via Mostow-Presad, it will untangle the proverbial Gordian knot without so much as cutting a strand. Heck, it even draws pretty pictures of three-space with embedded horiballs. Jeff is a top-knotch topologist who I first heard about two years ago from Colin Adams, who also does knot topology stuff. Colin told me about some physicists who were doing "extended sky surveys." Basically, they were looking up through their telescopes along a thin cone, and counting the number of galaxies they found as a function of their distance from us. These physicists were very surprised to find that this distribution was periodic -- the same hump kept on appearing over and over again in their graph. Now, this got the topologist mathematicians like Colin very excited because this sort of behavior would be exactly what you would expect if the universe was a non-Euclidean hyperbolic 3-manifold (a 3-manifold is just something which looks locally like everyday three dimensional space. Similarly, a 2-manifold is something which looks locally like a plane, and a 1-manifold is something which looks locally like a line. This has been your topology lesson for today.) which is just a fancy way of saying that the universe might be like one of those old asteroid games where the sides of the screen wrap around, so if your ship goes off the edge of the screen to the right, it comes back on the left. Except the universe would presumably be a lot bigger.
Anyway, so Colin told me that all of this got the topologists thinking and scratching their heads about what hyperbolic 3-manifold the universe might be. And for various obscure reasons which totally allude me, one of the best candidates was this 3-manifold which Jeff Weeks had discovered, called (get this) the Weeks manifold. Of course, if all this turned out to be true, Jeff would be considerably more famous than he is now, and would presumably have to start dressing better. Also, as I pointed out to Colin, it would also confirm the ancient wisdom that, after all, the universe really was made in a Weeks.
[Yes, I actually made you wade through two whole paragraphs of topology just to get to that (admittedly Week) pun. Just deal with it.]
So after Jeff, came Conway, whose lecture was entitled "Symmetries, Lattices, and Tilings I", but really should have been called "There's Only 17 Ways to Tile a Plane", as in:
It's all inside your head he said to me. If you use my notation, it's very plain to see. And I said I'd think about it, But could he please explain, About the 17 ways to tile a plane. There's only 17 ways to tile a plane. So use a square lattice, Alice. Or tile with hexagons, Won Fong. Using triangles will work, Kirk, Just listen to me. It can look like an Escher print, Clint. Or like your bathroom floor, Eleanor. But you can't use Penrose Tiles, Miles, If you want to do it periodically.etc. Anyway, Conway gave quite a show on how to express symmetries in the language of orbifolds (which are just manifolds divided or as mathematicians say "moded out" by the symmetry). He would go around the lecture hall pointing out symmetric things (the brick pattern on the wall, the carpet pattern, chairs, T-shirt designs) and having the audience tell him what the orbifold was. He then finished with an ultra-slick proof of the classification of orbifolds, which immediately implied the aforementioned fact of there only being seventeen periodic ways of tiling the plane.
The rest of the day, after such a dramatic morning, was a bit of a let down. Rephael Wenger gave a talk on a beautiful little construction of his on how to turn a combinatorial plane (a plane where you are only allowed to ask combinatorial questions like, "Is this point between these two lines?" rather than topological questions like "How close are these two points?") into a topological plane, thus answering a thirty year old open question. Godfired Toussaint talked about algorithms for computing triangulations of polygons, a rather simple problem which is remarkably easy to screw up. As he said, "In three dimensions, triangulations do not necessarily exist, even though there are published algorithms for computing them."
The undergraduates also gave a talk on their research today, even though they didn't have much of the way of results. I feel rather sorry for them, as they were given some really lossy problems to work on. For example, one of their problems is to come up with a better (less surface-area) way of tiling three dimensional space than Kelvin's alpha polyhedra. In the opinion of many mathematicians, this is probably impossible. Their other problem for the summer is to come up with upper bounds of knot energies; this is a better, though still very difficult problem, which they've made some progress on for two bridge knots.
[Note (added Jan. 13, 1993): On Dec. 13, 1993, I got mail from Frank Morgan reporting that Denis Weaire and Robert Phelan of Trinity College beat Kelvin's least-area partitioning of space into unit volumes (by a whopping 1%). The Weaire-Phelan structure uses a dodecahedron and a tetrakaidecahedron. So maybe the undergraduates had a chance after all...]
Played Ultimate again today after dinner, and I have to say that it was a lot more fun now that the temperature has dropped out of the 90's. I had so much fun, in fact, that I came in late to Jeff Erickson's 8:30 pm talk on "Lower Bounds in Computational Geometry." Jeff's a CS grad student at Berkeley, and when I emailed Yarvin [there used to be so many Curtis's hanging around Brown's CS department a while back that I got into the habit of calling them all by their last names] to ask if he knew this guy Jeff who did theoretical computational geometry, he responded, "Theoretical computational geometry makes me ill." I'm beginning to come to a similar feeling, even though they tell me it's a good source of grant money for up and coming geometers such as myself. Ah, the trade-offs we make.