A self-ruling monotile for aperiodic tiling

TL;DR

This paper introduces HexSeed, a single, self-ruling monotile made of 16 hexagons that, through a substitution rule, guarantees aperiodic tiling of the plane without additional motifs.

Contribution

It presents a novel self-ruling monotile that enforces aperiodicity solely through its shape and substitution rule, advancing the solution to the ein Stein problem.

Findings

HexSeed guarantees aperiodic tiling by design.

The monotile is composed of 16 identical hexagons with binary markings.

A proof of aperiodicity is provided using motifs.

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.