The Game of Life Rules on Penrose Tilings: Still Life and Oscillators

John Horton Conway’s Game of Life [1, 4] is a simple two-dimensional, two state cellular automaton (CA), remarkable for its complex behaviour [1, 13]. That behaviour is known to be very sensitive to a change in the CA rules. Here we continue our investigations [7, 10, 11] into its sensitivity to changes in the lattice, by the use of an aperiodic Penrose tiling lattice [5, 12]. Section 18.1 describes Penrose tilings; Sect. 18.2 generalises the concepts of neighbourhood and outer totalistic CA rules (which include the Game of Life) to aperiodic lattices, and introduces a naming convention for Penrose Life oscillators. Section 18.3 presents various Penrose lattice still life configurations; Sects. 18.4– 18.7 present various oscillators with periods from 2 to 15. Section 18.8 presents an algorithm to detect oscillators, and a means to classify them based on their underlying neighbourhood graph.