Penrose’s Ants

Quasicrystals, cellular automata, old-hat … A fresh fashionable composition – is reversible cellular automata on the quasicrystals (Penrose tilings). Here advantage of the second-order CA is clear – an alternative with Margolus block CA hardly could be developed for such an irregular lattice. The idea to consider two kinds of neighboring cells also works here. Each rhombus always has four closest cells, yet terms like “up”, “down”, “left” and “right” may be not quite justified due to different orientations of the rhombuses. Number of “far” neighboring cells adjacent only in corners may vary from three to seven (or even to ten for generalized Penrose tilings).

The simplest idea is to define rules by describing of two numbers: live cells in closest and far neighboring positions, but for two closest live cells two different cases also may be considered: if the cells are opposite or not.

Let us consider two-states irreversible CA with the rule: current state of a cell is not taken into account and next state of the cell is alive if two conditions are satisfied: (1) number of live cells in four closest positions is either one or two, but these two live cells should be opposite, (2) all far neighboring cells are empty. The reversible second-order CA with four states derived from that is an analogue of RCA on square lattice discussed earlier. It was used for preparation of all the animated pictures.