Aperiodic monotiles: from geometry to groups

TL;DR

This paper introduces a unified framework connecting geometric and group-theoretic tilings, demonstrating how complex aperiodic tiles like the Hat tile can be represented within group theory, advancing understanding of non-periodic tilings.

Contribution

It develops a general framework that embeds geometric tilings into group-theoretic models, enabling the simulation of complex aperiodic tiles within algebraic structures.

Findings

Transformation of the Hat tile into a group monotile

Construction of a framework linking geometry and group theory

Analysis of symmetries in geometric and group tilings

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 our setting 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.