Pre-Publicación 2025-20
Thierry Coulbois, Anahi Gajardo, Pierre Guillon, Victor H. Lutfalla:
Aperiodic monotiles: from geometry to groups
Abstract:
In 2023, two striking, nearly simultaneous, mathematical discoveries have excited their respective communities, one by Greenfeld and Tao, the other (the Hat tile) by Smith, Myers, Kaplan and Goodman-Strauss, which can both be summed up as the following: there exists a single tile that tiles, but not periodically (sometimes dubbed the einstein problem). The two settings and the tools are quite different (as emphasized by their almost disjoint bibliographies): one in Euclidean geometry, the other in group theory. Both are highly nontrivial: in the first case, one allows complex shapes; in the second one, also the space to tile may be complex. We propose here a framework that embeds both of these problems. From any tile system in this general framework, with some natural additional conditions, we exhibit a construction to simulate it by a group-theoretical tiling. We illustrate this by transforming the Hat tile into a new aperiodic group monotile, and we describe the symmetries of both the geometrical Hat tilings and the group tilings we obtain.
