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 . This example illustrates nicely some themes in probability and group dynamics, and lends itself easily to visualization.

## Bill Thurston 1946-2012

August 22, 2012 in Commentary, Uncategorized | Tags: Bill Thurston, obituary | by Danny Calegari | 11 comments

This morning I heard the awful news that Bill Thurston died last night. Many of us knew that Bill was very ill, but we all hoped (or imagined?) that he would still be with us for a while yet, and the suddenness of this is very harsh. As Sarah Koch put it in an email to me, “Although this was not unexpected, it is still shocking.” On the other hand, I am glad to hear that he was surrounded by family, and died peacefully.

I counted Bill as my friend, as well as my mentor, and I have many vivid and happy memories of time I spent with him. I hope that writing down a few of these reminiscences will be cathartic for me, and for others who are coping with this loss.

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