Cyclic cellular automaton

One-dimensional cyclic cellular automata can be interpreted as systems of interacting particles, while cyclic cellular automata in higher dimensions exhibit complex spiraling behavior.

The one-dimensional cyclic cellular automaton has been extensively studied by Robert Fisch, a student of Griffeath.

[1] Starting from a random configuration with n = 3 or n = 4, this type of rule can produce a pattern which, when presented as a time-space diagram, shows growing triangles of values competing for larger regions of the grid.

Additionally, a boundary between regions with values i and i + 2 (mod n) can be viewed as a third type of particle, that remains stationary.

However, for n ≥ 5, random initial configurations tend to stabilize quickly rather than forming any non-trivial long-range dynamics.

[3] 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.[4] At first, the field is purely random.

[5] 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.

[6] 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, or other systems of autowaves, although other cellular automata more accurately model the excitable medium that leads to this reaction.