Post #1952679

2026-04-28 21:44 UTC

Peter Selinger has come up with a great way of generating hat tilings by overlaying a triangular grid on a periodic pattern, and placing a tile at each point that is not white, with the orientation and handedness of the tile determined by the colour of the point. A more thorough explanation is given in this preprint https://arxiv.org/pdf/2604.20964, where he and Sébastien Labbé show that this is a Markov partition. As mentioned in the paper, I came up with a similar construction a few years ago, but it required separate steps for tiles of a given orientation modulo 120. In this series of posts, I&#39;ll attempt to explain the connection between the two constructions, and demonstrate the analogous constructions for the hats-in-turtles and turtles-in-hats versions of the Spectre tiling. The aim is to give a sense of the main ideas, rather than a rigorous proof that this works. Before I get into the details, here is a Markov partition for turtle tilings, where control / anchor points are located on the underside of the turtle&#39;s shell. [Edit: For more background about the paper, see Sébastien&#39;s blog post: http://www.slabbe.org/blogue/2026/03/a-construction-of-the-hat-tilings-by-a-markov-partition/ This includes some printable files that can be used to construct patches of hat tilings in practice.] (1/n) #TilingTuesday #aperiodicTilings #aperiodocMonotile

Replies (1)

@pieter@mathstodon.xyz 2026-04-28 23:39

As Arnaud Cheritat explains here https://www.math.univ-toulouse.fr/~cheritat/2023-monotile/4D-lift/page-4.html, given a tiling in the Hat or Spectre* family, you can construct a MLD tiling of hexagons and rhombs (see also @mathBlock@mathstodon.xyz (https://mathblock8128.wordpress.com/2023/09/16/spar-lines-hexagons-and-tiling-a-finite-boundary/, Fig 22) and @jsmith@mathstodon.xyz (https://arxiv.org/pdf/2403.01911, Fig 3.1) who point out that there is a aperiodic set of prototiles consisting of a rhomb and two types of comets with directed, coloured edges). Given a base vertex, you can assign to each vertex in a comet-rhomb tiling a pair of complex numbers in \(\Lambda^2\), where \(\Lambda = \mathbb{Z}[\xi]\) (\(\xi = e^{2\pi i/6}\)) is the set of Eisenstein integers. By including edges, we obtain a directed graph with vertices in \(\Lambda^2\). This is the 'lift' of the tiling. A tiling using a particular tile set in the Hat or Spectre family is then a projection of this graph, where the hat tile corresponds to projecting onto \((1,-1)\) and the turtle with projecting onto \((1,\xi^2)\). [You could also do this using the vertices of the hat tile, but I find the comet-rhomb version simpler since there aren't as many points, and projections to tiles in the hat family are all of the form \((1, \sigma) \) for \(\sigma \in \mathbb{C}\) of modulus 1.] *By the Spectre family, I mean tilings using Tile(\(a,b\)) and Tile(\(b,a\)) (in the notation of https://arxiv.org/pdf/2303.10798) with appropriately marked edges (2/n)

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