A self-ruling monotile for aperiodic tiling

Abstract: Can the entire plane be paved with a single tile that forces aperiodicity? This is known as the ein Stein problem (in German, ein Stein means one tile). This paper presents a monotile that delivers aperiodic tiling by design. It is based on the monotile developed by Taylor and Socolar (whose aperiodicity is forced by means of a non-connected tile that is mainly hexagonal) and motif-based hexagonal tilings that followed this major discovery. Here instead, a single substitution rule makes its shape, and when applying it, forces the tiling to be aperiodic. The proposed monotile, called HexSeed, is self-ruling. It consists of 16 identical hexagons, called subtiles, all with edgy borders representing the same binary marking. No motif is needed on the subtiles to make it work. Additional motifs can be added to the monotile to provide some insights. The proof of aperiodicity is presented with the use of such motifs.