Dances, Billiards, and Pretzels
When I came to the Heidelberg Laureate Forum, I expected a feast for my mind. I didn’t expect a feast for my eyes! Take a look at this incredible video, by Diana Davis, which was featured in today’s lecture by Fields medalist Curtis McMullen.
Davis, who is now at Brown University, submitted this video to a competition called Dance Your Ph.D., and won first prize in the Physics category. (Yes, mathematics is a subfield of physics for the purpose of this competition.) She has not only created an entertaining (and humorous!) video but also explained, with only a few words, all of the main elements of a theorem in her thesis. In fact, I would say that the theorem can be better understood through the dance than by normal mathematical language!
You be the judge. The problem is to predict what happens to a billiard ball on a table in the shape of a regular pentagon. In particular, you want to know how to shoot the ball in such a way that it returns to its original spot after a certain number of bounces. Davis’s theorem gives a complete description of the dynamics of such a shot.
She starts by reflecting the billiard table through a mirror placed on one side, creating a non-convex octagon. The advantage in this is that the billiard ball never bounces — it just goes in a straight line, passing right through the side where the mirror is.
But what happens when the ball gets to a non-mirrored side? Well, then it does the Pac-Man thing — it re-emerges through a parallel side. This is a total head-scratcher for non-mathematicians (or non-Pac-Man players), but it is beautifully illustrated in Davis’s video. Davis then considers which sides the ball passes through in order, and gets a sequence of colors or letters, A, B, C, D, E. Then the question becomes: Which sequences are possible? She shows a simple but cool reduction algorithm, which involves distorting the table with a shear transformation, cutting it apart, and gluing it together again to get a simpler sequence. In this manner, she shows that you can reduce any legal sequence of bounces to a repeating sequence with just two alternating colors or letters: A, B, A, B, …
Her video ends, “Q.E.D.,” which in this case I think means, “Quod Erat Dance-andum.” (That Which Was To Be Danced.)
McMullen’s talk explained in beautiful fashion how this billiards problem relates to a question in the theory of Riemann surfaces. In my last post, about Keenan Crane’s work, I mentioned that tori (or doughnuts) come in different conformal classes that are described by something called a complex structure on the torus. What I didn’t tell you is that you can do the same thing with toruses that have two or three holes. (The latter are not doughnuts but pretzels — a particularly apt topic because my hotel here in Germany serves pretzels for breakfast every morning!)
What are all possible pretzel shapes? Well, there are infinitely many. But many of them are conformally equivalent. In fact, Bernhard Riemann proved in the 19th century that the set of all conformal classes of pretzels is 6-dimensional. This means that it takes only 6 parameters to describe the shape of any pretzel, up to conformal equivalence. These parameters themselves form a complicated, 6-dimensional surface called a Teichmuller space, and mathematicians would like to understand its structure.
What McMullen showed* was that the Teichmuller space contains certain special families of pretzels that form a 1-dimensional subset of this 6-dimensional space. You can think of them as being like meridians on a globe. These special pretzels are obtained by gluing together the edges of a polygon, just as Davis’s dance is performed on an octagon whose edges are glued together in a particular way. In these special polygons, every billiard trajectory is either perfectly periodic (like the orbits that Davis studied) or perfectly chaotic. So Davis’s theorem is relevant to (although I won’t say it exactly solves) this 150-year-old problem of understanding the “space of all pretzels”!
Listening to McMullen’s lecture was like having the light dawn on me after 30 years. I first heard about Teichmuller spaces when I was in graduate school, and my brain shut down. They were too abstract for me to comprehend. If I had only known that you could explain them by following the path of a billiard ball on a regular polygon, how much easier life would have been! Unfortunately, the billiard ball connection wasn’t known then — it has only been appreciated since 1989 through the work of W.A. Veech. (See this Wikipedia entry for more.)
What a thrill it was, through the work of Veech and Davis and the lecture of McMullen, to feel as if I understood something about Teichmuller space for the first time!
* P.S. For those attendees who don’t remember McMullen talking about pretzels, that’s correct. He actually talked about the Teichmuller space of two-holed tori or “genus two Riemann surfaces.” However, I think it would have been more fun if he had talked about three-holed tori (pretzels) since we are in Germany, the birthplace of the pretzel!