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I was saddened to hear the news that Geoff Mess recently passed away, just a few days short of his 54th birthday. I first met Geoff as a beginning graduate student at Berkeley, in 1995; in fact, I believe he gave the first topology seminar I ever attended at Berkeley, on closed 3-manifolds which non-trivially cover themselves (the punchline is that there aren’t very many of them, and they could be classified without assuming the geometrization theorem, which was just a conjecture at the time). Geoff was very fast, whip-smart, with a daunting command of theory; and the impression he made on me in that seminar is still fresh in my mind. The next time I saw him might have been May 2004, at the N+1st Southern California Topology Conference, where Michael Handel was giving a talk on distortion elements in groups of diffeomorphisms of surfaces, and Geoff (who was in the audience), explained in an instant how to exhibit certain translations on a (flat) torus as exponentially distorted elements. Geoff was not well even at that stage — he had many physical problems, with his joints and his teeth; and some mental problems too. But he was perfectly pleasant and friendly, and happy to talk math with anyone. I saw him again a couple of years later when I gave a colloquium at UCLA, and his physical condition was a bit worse. But again, mentally he was razor-sharp, answering in an instant a question about (punctured) surface subgroups of free groups that I had been puzzling about for some time (and which became an ingredient in a paper I later wrote with Alden Walker).

Geoff Mess in 1996 at Kevin Scannell’s graduation (photo courtesy of Kevin Scannell)

Geoff published very few papers — maybe only one or two after finishing his PhD thesis; but one of his best and most important results is a key step in the proof of the Seifert Fibered Theorem in 3-manifold topology. Mess’s paper on this result was written but never published; it’s hard to get hold of the preprint, and harder still to digest it once you’ve got hold of it. So I thought it would be worthwhile to explain the statement of the Theorem, the state of knowledge at the time Mess wrote his paper, some of the details of Mess’s argument, and some subsequent developments (another account of the history of the Seifert Fibered Theorem by Jean-Philippe Préaux is available here).

My eldest daughter Lisa recently brought home a note from her school from her computer class teacher. Apparently, the 5th grade kids have been learning to program in Logo, in the MicroWorlds programming environment. I have very pleasant memories of learning to program in Logo back when I was in middle school. If you’re not familiar with Logo, it’s a simple variant of Lisp designed by Seymour Papert, whereby the programmer directs a turtle cursor to move about the screen, moving forward some distance, turning left or right, etc. The turtle can also be directed to raise or lower a pen, and one can draw very pretty pictures in Logo as the track of the turtle’s motion.

Let’s restrict our turtle’s movements to alternating between taking a step of a fixed size S, and turning either left or right through some fixed angle A. Then a (compiled) “program” is just a finite string in the two letter alphabet L and R, indicating the direction of turning at each step. A “random turtle” is one for which the choice of L or R at each step is made randomly, say with equal probability, and choices made independently at each step. The motion of a Euclidean random turtle on a small scale is determined by its turning angle A, but on a large scale “looks like” Brownian motion. Here are two examples of Euclidean random turtles for A=45 degrees and A=60 degrees respectively.

The purpose of this blog post is to describe the behavior of a random turtle in the hyperbolic plane, and the appearance of an interesting phase transition at $\sin(A/2) = \tanh^{-1}(S)$. This example illustrates nicely some themes in probability and group dynamics, and lends itself easily to visualization.