Post #2923561

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)