Photonic topological insulators of the Rudner type in terms of of tricolor cellular automata

A topological insulator is a material that simultaneously exhibits the properties of a conductor on the surface and an insulator in volume. Rudner's game is a simplified model of a topological insulator, experimentally implemented on two-dimensional photonic lattices and described as a cellular automaton similar to Conway's game of life. A three-color cellular automaton is a regular lattice of cells. Each cell takes one of three colors according to the specified cell recoloring rule. In this work, two colors of the cellular automaton correspond to the absence and presence of a photon in the resonator, and the third color corresponds to the boundary of the region. The Rudner game is generalized to the case of a three-dimensional array of optical cavities and the defining properties of a topological insulator are demonstrated. A class of cellular automata is introduced that preserve the boundary and the number of photons, as well as zero group velocity for photons far from the surface. Physical realizations are expected in photonics, electronics, mechanics, and acoustics.