Medley of Spirals from Cyclic cellular automata

Medley of Spirals from Cyclic Cellular Automata Auxiliary Materials

A cyclic cellular automaton (CCA) is defined as an automaton where each cell takes one of N states 0, 1,..., N-1 and a cell in state i changes to state i+1 mod N at the next time step if it has a neighbor that is in state i+1 mod N, otherwise it remains in state i at the next time step. Classically CCA are applied on the 2-dimensional integer lattice with von Neuman neighborhoods (nearest 4 neighbors). However, this rule can be applied to any configuration of cells and any definition of neighborhood in any dimension. In fact, it can be applied to any graph.

Here we explore CCA using various sub-neighborhoods of a 3 by 3 neighborhood around points labeled as below.

0 1 2
3 4 5
6 7 8

We also explore the automata on arrangements of cells that are crystals and quasi-crystals created by canonical projection. In each case, there is an expected long term periodicity and the movies below show a frame per period plus 1.

If the movies do not display in your browser, try saving the target file and running them locally.

Also available is a paper [preprint].