Fractal Patterns Created by Ulam´s Cellular Automaton

Ulam´s cellular automaton, a nonlinear two-dimensional cellular automaton, was introduced by Stanislaw Ulam for emulating crystalline growths. In this paper we give two numerical results in which the particular orbit of the automaton has some fractal structures. First result is that the boundaries of the spatio patterns are fractal curves as time approaches infinity. Second, we study the number of cells consisting the spatio patterns for each time step. We show that the dynamics of the number can be represented by Lebesgue´s singular function.