Wolfram Computation Meets Knowledge

Wolfram Summer School


EPaul-Jean Letourneau

Summer School

Class of 2005


Paul-Jean Letourneau grew up in Calgary and avidly pursued the arts almost exclusively. Around the age of 16, he underwent a phase transition and became interested in mathematics to learn more about things like fractal geometry. He devoted himself to learning the sciences, particularly physics, and in 1998 he enrolled in the honors physics program at the University of British Columbia. While there, he did a number of work-experience placements, including medical imaging at the Vancouver General Hospital, NMR at the University of Alberta, geophysics in Calgary, and biophysics at the National Institutes of Health in Bethesda, Maryland. He graduated with a B.Sc. in physics in December 2003. Paul-Jean is currently pursuing his master’s degree in physics, where he is elucidating the connections between fluctuations seen in simple programs and those present in physical systems.

Project: Manners of Fluctuation Present in Simple Programs

My plan is to discover and distinguish among the manners of fluctuation present in simple programs. I will do this by comparing the distributions produced by taking different measurements on simple programs, and looking for trends that correlate with their Wolfram classes, or other general features apparent from the detailed dynamics of the rules. Possible simple programs to study include cellular automata, Turing machines, and cyclic tag systems. Possible measurements on the output of the simple programs include lengths of runs of a single digit, the number of specific n-digit sequences produced, and correlations between sequences produced. Distributions to look for, which are familiar from statistical physics, include Maxwell-Boltzmann, Poisson, Gaussian, and power-law. Each arises due to differences in the character of the interactions, but they are often largely independent of the microscopic details of the constituents. I suspect this implies that there are classes of simple rules that give rise to the same types of distributions. Having found such classes, one can then analyze the detailed behaviors of the rules within them, and conjecture properties that physical systems giving rise to the distributions must have.

Favorite Four-Color, Nearest-Neighbor, Totalistic Rule

Rule chosen: 1572

I found it in a search for rules that produce “walls.” I searched the first 8000 or so rules more or less exhaustively to find it. A typical evolution of rule 1572 from random initial conditions is shown in the figure. What happens is that overall there is class 3 behavior in the digits 0 and 2 (white and red), with walls that form during the evolution. But regions of 1’s (yellow) “percolate” through the class 3 stuff, and in fact “corrode” the walls.