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

In this study, we examine the relation between the spatio-temporal patterns generated by two-dimensional symmetrical elementary cellular automata and a singular function. In a previous paper, we proved that a specific cellular automaton admits a "limit set" (a limit on the series of spatio-temporal patterns contracted with time), and we calculated the fractal dimension of the boundary of this limit set. In this paper, we provide an overview of the previous results and a more precise analysis. Numerical simulations demonstrate that the essential fractal-like patterns created by two-dimensional cellular automata are also related to a singular function.