Constructing New Aperiodic Self-simulating Tile Sets

Abstract

Wang tiles are unit size squares with colored edges. By using a fixed-point theorem a la Kleene for tilings we give novel proofs of classical results of tilings problems' undecidability by way of diagonalization on tilings (made possible by this theorem). Then, we present a general technique to construct aperiodic tile sets, i.e. , tile sets that generate only aperiodic tilings of the plane. Our last construction generalizes the notion of self-simulation and makes possible the construction of tile sets that self-simulate via self-similar tilings, showing how complex the self-simulation can be.