Aperiodic monotiles: from geometry to groups

TL;DR

This work provides a unifying framework to compare geometric aperiodic tilings with group-tiling problems by discretizing poly-$\mathcal{K}$ tiles into group tiles. It proves a key equivalence: for a grid-possessing finite tile set $\mathcal{K}$, poly-$\mathcal{K}$ tilings of a space $\mathcal{W}$ correspond to finite tilings of $G\times\mathcal{K}$ with the same cotilers, preserving aperiodicity properties. The authors apply this to the Hat tiling, translating it into a group-theoretic monotile called the Cucaracha in the group $\Gamma$, a virtually $\mathbb Z^2$ Coxeter-type group, and analyze its symmetries and stabilizers. They also develop a general discrete-continuous toolkit (orbit map, tile decomposition) to transfer tiling problems across settings, and discuss crystallographic groups to connect with Archimedean tilings and Cayley graphs. Overall, the paper provides a concrete bridge between geometric and group tilings, enabling explicit constructions of aperiodic monotiles in small groups and offering new directions for undecidability and cross-domain results.

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.

Figures (16)

• Figure 1: A Hat tiling of $\mathbb R^2$ and a corresponding Cucaracha tiling on group $\Gamma$. The colors correspond to signature of the symmetry, and the darkness correspond to the parity of angle modulo $\pi/6$. The red tricross is a center of $3$-fold rotational symmetry.
• Figure 2: Here we consider a single L shaped tile in $\mathbb{R}^2$ with its whole group of isometries, we present 3 different cotilers, we denote by $t_{u}$ the translation by vector $u$, $R_2^u$, the rotation by $\pi$ at point $u$, and $\rho_h, \rho_v$ the horizontal and vertical reflections at the origin: (a) strongly periodic cotiler: $C=\{t_{(3a,2b)}\}_{(a,b)\in\mathbb{Z}^2}\cup\{R_2^{(3a,2b)}\circ t_{(3a,2b)}\}_{(a,b)\in\mathbb{Z}^2}$; (b) mildly but not strongly periodic cotiler: $\text{Stab}(C)=\{t_{(3a,0)}\}_{a\in\mathbb{Z}}$; (c) weakly but not mildly periodic cotiler: $\text{Stab}(C)=\{id,R_2^{(0,0)},\rho_h,\rho_v\}$.
• Figure 3: The Kite (left) and Kitegrid (right).
• Figure 4: The Kitegrid (left) and Semikitegrid (right) in light grey, with their dual graph. The dual graphs induce the Archimedean tilings of vertex configurations $3.4.6.4$ (left) and $4.6.12$ (right).
• Figure 5: The Hat tile as a polykite.

Theorems & Definitions (42)

• Remark 1
• Example 2
• Definition 3
• Remark 4
• Remark 5
• Remark 6
• Theorem 7: Main result; poly-$\mathcal{K}$ version
• Corollary 8
• proof : Proof of Theorem \ref{['t:maink']}
• Definition 9: Kite and Kitegrid