Cyclic Cellular Automaton - Two or More Dimensions

Two or More Dimensions

In two dimensions, with no threshold and the von Neumann neighborhood or Moore neighborhood, this cellular automaton generates three general types of patterns sequentially, from random initial conditions on sufficiently large grids, regardless of n. At first, the field is purely random. As cells consume their neighbors and get within range to be consumed by higher-ranking cells, the automaton goes to the consuming phase, where there are blocks of color advancing against remaining blocks of randomness. Important in further development are objects called demons, which are cycles of adjacent cells containing one cell of each state, in the cyclic order; these cycles continuously rotate and generate waves that spread out in a spiral pattern centered at the cells of the demon. The third stage, the demon stage, is dominated by these cycles. Almost surely, every cell of the automaton eventually enters a repeating cycle of states, where the period of the repetition is either n or (for automata with n odd and the von Neumann neighborhood) n + 1. The same eventually-period behavior occurs also in higher dimensions. Small structures can also be constructed with any even period between n and 3n/2. Merging these structures, configurations can be built with a global super-polynomial period.

For larger neighborhoods, similar spiraling behavior occurs for low thresholds, but for sufficiently high thresholds the automaton stabilizes in the block of color stage without forming spirals. At intermediate values of the threshold, a complex mix of color blocks and partial spirals, called turbulence, can form. For appropriate choices of the number of states and the size of the neighborhood, the spiral patterns formed by this automaton can be made to resemble those of the Belousov-Zhabotinsky reaction in chemistry, although other cellular automata more accurately model the excitable medium that leads to this reaction.