Yet Another Aperiodic Tile Set

We present here an elementary construction of an aperiodic tile set. Although there already exist dozens of examples of aperiodic tile sets we believe this construction introduces an approach that is different enough to be interesting and that the whole construction and the proof of aperiodicity are hopefully simpler than most existing techniques. Aperiodic tile sets have been widely studied since their introduction in 1962 by Hao Wang [7]. It was initially conjectured by Wang that it was impossible to enforce the aperiodicity of a coloring of the discrete plane Z with a finite set of local constraints (there either was a valid periodic coloring or none at all). This would imply that it was decidable whether there existed a valid coloring of the plane for a given set of local rules. This last problem was introduced as the domino problem, and eventually proved undecidable by Robert Berger in 1964 [1, 2]. In doing so, Berger produced the first known aperiodic tile set: a set of local rules that admitted valid colorings of the plane, none of them periodic. Berger’s proof was later made significantly simpler by Raphael M. Robinson in 1971 [12] who created the set of Wang tiles now commonly known as the Robinson tile set. Since then many other aperiodic tile sets have been found, not only on the discrete plane [3, 4, 8, 9], but also on the continuous plane [5, 11]. In this article we will describe yet another construction of an aperiodic tile set. Although the resulting tile set will produce tilings quite similar to those of Robinson’s (an infinite hierarchical structure of embedded squares) the local constraints will be presented in a (hopefully) more natural way: we will start from simple geometrical figures and organize them step by step by adding new rules progressively.