Jamie Walton - Hexagonal aperiodic tiles - Hatfest скачать в хорошем качестве

Jamie Walton - Hexagonal aperiodic tiles - Hatfest

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+ε+ε^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. A talk from Hatfest at the Mathematical Institute, University of Oxford. See https://sites.google.com/view/thegrimmnetw... for more information.