Fractal Structure of a Class of Two-Dimensional Two-State Cellular Automata

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

“…First, we have that S(6) = S(2 2 ) + 4 × 3 4 S(2), because we notice from Figure 6(a) that S(6) is divided by a square S(2 2 ) and four copies of 3 4 S(2). Next, S(22) in Figure 6(b) is represented by a square S(2 4 ) and four copies of 3 4 S(6), and we have that S(54) = S(2 5 ) + 4 × 3 4 S(22) by a similar argument. Using the same procedure, it is easy to see that S(n + 2 m+i ) = S(2 m+i ) + 3S(n) for any natural number i.…”

“…We have investigated the fractal structures generated by symmetrical two-dimensional cellular automata. In our previous studies [3,8], we had determined numerically that the spatio-temporal pattern has a fractal-like structure, and the proof of this result was presented in [4]. In this paper, we first provided an overview of the existence of a limit set for a cellular automaton and described the fractal dimension (box-counting dimension) of the boundary of the spatial pattern.…”

“…It was found that a family of unstable invariant sets exists for which the dynamics are conjugate to the one-dimensional elementary cellular automaton Rule 150 [3]. A further study demonstrated that the automaton admits a limit set starting from a certain initial configuration, and showed the box-counting dimension of its boundary [4]. This demonstrated the relation between a two-dimensional cellular automaton and a singular function.…”

“…Section 2 describes the prerequisites concerning cellular automata and singular functions. In Section 3, we provide an overview of previously derived results [4], and present a more precise calculation. In Section 4, numerical results are presented.…”

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

“…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.…”