Three-state von Neumann cellular automata and pattern generation

Abstract

In this study, we consider the theoretical structure and classification of two-dimensional (2D) three-state uniform cellular automata (CA) based on their visual behavior. The basic features of a CA comprise their discrete dynamic structure and the fact that they are modeled locally, but their behavior can be similar to a continuous system at large time and spatial scales. The geometrical structures of the patterns produced by iterating CA can be considered according to some basic properties. After applying the rules iteratively, it has been shown that CA are capable of producing complex behaviors. Some types of CA exhibit remarkably regular behavior with finite configurations. With simple initial configurations, the patterns generated may be self-replicating (SR), self-similar (SS), or mixed. In this study, we consider the theory of 2D uniform periodic boundary (PB), adiabatic boundary (AB), and reflexive boundary (RB) CA with a von Neumann neighborhood, and applications of image analysis for generating patterns. We investigate a 2D CA under these boundary restrictions for three-state cases, i.e., the ternary field Z3. We also study the applications of SR and SS patterns that correspond to the linear CA rules of 2D uniform CA with different boundary cases over Z3. The von Neumann neighborhood CA rule (i.e., Rule 2460) is classified into SR and SS types depending on the non-zero boundary values a,b,c,d of the neighboring cells that influence the cells under consideration. Based on the visual appearance of the patterns, we also show that Rule 2460 is sometimes sensitive, depending on the boundary conditions and chaotic behavior. Finally, we conclude by analyzing some results in detail for CA defined by Rules 2460NB, 2460PB, 2460AB, and 2460RB for non-symmetric figures.