Self-Organizing Cellular Automata:
the experimental mathematics of complex spatial systems

David Griffeath
Mathematics Dept., University of Wisconsin

A cellular automaton (CA) is a discrete dynamics, generated by repeated application of a parallel update rule, that evolves as a sequence of lattice configurations, or a digital movie if you will. This paradigm, which goes back to Ulam and von Neumann in the late 1940s, and was popularized by Conway's Game of Life in the 70s and Wolfram's experimental work in the 80s, is currently used to model the phenomenology of linear and nonlinear systems in many areas of applied science. The advent of interactive computer visualization enables a systematic investigation of CA rules in a way that was unthinkable without such technology. The method of analysis, which combines empirical observation (thoughtful movie watching) with traditional deductive reasoning, provides a stimulating case study of contemporary directions in mathematics.

My goal is to illustrate these themes in a series of three computer demonstrations. The titles of my presentations, the first of which is intended for a general audience, are as follows:

I. Cellular Automata as Toy Universes

II. The Mathematics of Excitable Cellular Automata

III. The Mathematics of Threshold Growth

For the past two years I have been working, with Bob Fisch, on the development of effective, interactive simulation tools for the study of CA rules. We are designing two platforms in parallel:

These platforms, especially the latter, will be the focus of in my lectures. Anyone interested in downloading WinCA, or previewing some of the computer graphics and research topics that I will discuss, should check out my home page on the World Wide Web, the Primordial Soup Kitchen:

That link contains an annotated archive of my colorful images, references to several of my articles in this area, and links to other sites on the Net that deal with cellular automata, interacting particles, and other complex spatially-distributed systems.