Fractal Patterns Created by Ulam's Cellular Automaton

Relation between spatio-temporal patterns generated by two-dimensional cellular automata and a singular function

“…From previous observations [3,8], we notice that for the Sym-2dECA K n represents the structure of T 2n−1 x o . If n = 2 k for each natural number k, then the pattern of T 2n−1 x o is simply in a basic square.…”

“…Remark 3.1. In the previous papers [3,8], it was suggested that the number of "0" states in T 2n−1 x o was related to the singular function L 1/4 . Theorem 3.1 proves these results.…”

“…T 0 0 0 1 0 1 0 0 0 1 0 1 LCR 000 000 001 000 101 101 010 010 011 010 111 111 analysis method for Sym-2dECA (5) presented in [3,8]. We count the number of cells that form the pattern for each time step.…”

“…However, singular functions are continuous everywhere and differentiable almost everywhere [5][6][7]. The number of "0" states in the configuration of a two-dimensional nonlinear cellular automaton has been studied numerically, as shown in Figure 1 [3,8]. We can observe that the number of "0" states at each time step is described well by a singular function, which comprises a self-affine function on a unit interval.…”

“…It is also shown that some of the patterns are related to the singular functions. For six of the patterns, the number of "0" states at each time step is described by a singular function, which represents a generalization of a result for a particular cellular automaton [3,8]. In addition, for two of the patterns the cumulative sum of the number of "0" states is also described by this function.…”