Pieter MostertPeter 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'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's shell.
[Edit: For more background about the paper, see Sébastien'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.]
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