Complex Systems

Inherent Generation of Fractals by Cellular Automata Download PDF

Bruno Martin
Laboratoire d'Informatique, Signaux et Systèmes de Sophia Antipolis,
I3S-U.R.A. 1376-C.N.R.S., Université de Nice-Sophia Antipolis,
650, route des Colles, B.P. 145,
06903 Sophia Antipolis Cedex, France


In this paper we propose a method for generating fractal patterns using "classical'' cellular automata. Although the problem of fractal generation for linear cellular automata has been studied recently, this is not the case for classical cellular automata.

We first exhibit some basic techniques for the construction of the transition function, which draws a Cantor set, and show how this method can be generalized to cellular spaces of greater dimension. Then we give a method for embedding the configurations into a closed interval to obtain fractal patterns. We also define discrete dynamical systems for counting the minimum number of balls required to cover the fractal pattern.