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

Abstract

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.

Singular function emerging from one-dimensional elementary cellular automaton Rule 150

“…x o for m = 0, 1, and 4, respectively. All of the numbers are the same, num 150 (2 15. Hence, we find that there exist fifteen S 4 s on the interval…”

“…There are several works discussed the relationship between the function and cellular automata. For the one-dimensional elementary cellular automaton Rule 90 the limit set is characterized by Salem's function [12], and for a two-dimensional automaton that is a mathematical model of a crystalline growth the limit set is also characterized by Salem's one (numerical result was given in [13], proofs were given in [14,15]).…”

Singular function emerging from one-dimensional elementary cellular automaton Rule 150

“…The second avenue for future work concerns the relationship between the other cellular automata and singular functions. We previously obtained the relationships among Rule 90, a two-dimensional automaton, and Salem's function [9,10,11,12]. In addition, the present study investigated the relationship between Rule 150 and the new singular function.…”

“…Recently, we studied the relationship between singular functions and self-similar patterns generated by elementary cellular automata. The limit set of Rule 90 is characterized by Salem's function [9], and for a two-dimensional automaton that is a mathematical model of crystalline growth, its limit set is also characterized by Salem's one (the numerical result is reported in [10], and proofs are reported in [11,12]). In the case of these previous works, the number of nonzero states in a spatial or spatio-temporal pattern of the cellular automaton is represented by functional equations that are equivalent to those of Salem's singular function.…”

Discontinuous Riemann integrable functions emerging from cellular automata

“…A singular function is a function that is monotonically increasing (or decreasing) and continuous everywhere, with a zero derivative almost everywhere; for example, Salem's singular function [1,2,3,4]. We studied the relationship between Salem's singular function and elementary cellular automata, Rule 90 and two two-dimensional elementary cellular automata [5,6,7,8,9], and that between another new singular function and Rule 150 [10]. This paper presents new Riemann integrable functions with countable discontinuous points from two-dimensional elementary cellular automata.…”