Hexagonal Aperiodic Tilings

Abstract

Before the discovery of the hat monotile, several small aperiodic prototile sets were found which were based on the simple periodic tessellation of hexagons. These include Penrose’s $ 1+ \epsilon + \epsilon^2 $ tiles, the Taylor–Socolar tile and a tile based on ‘orientational matching rules’. These three fall short of solving the problem of finding a single tile that forces aperiodicity purely via tile geometry. In particular, the latter set is more faithfully considered (from the perspective of shifts of finite type) as a twin pair of tiles with charged edges, the two tiles being identical only up to a charge flip that 2-to-1 factors to aperiodic model sets. I will show the proof of aperiodicity of this tile set, which is notably very simple. The three sets above all factor to the aperiodic arrowed half-hex tilings, whilst the hat, although being related to hexagons, forces aperiodicity in a very different way. So, in discussing these other near misses to the strongest forms of the aperiodic monotile problem, I hope to frame what a remarkable discovery the hat was.