We give a constructive method that can decrease the number of prototiles needed to tile a space. We achieve this by exchanging edge to edge matching rules for a small atlas of permitted patches. This method is illustrated with Wang tiles, and we apply our method to present via these rules a single prototile that can only tile $R^3$ aperiodically, and a pair of square tiles that can only tile $R^2$ aperiodically.
We find explicit optimal vertex, edge and face coulourings for the chair tiling, the Ammann--Beenker tiling, the rational pinwheel tiling and the pinwheel tiling.
It is proved that homeomorphic images of certain two-dimensional aperiodic tilings, such as Ammann-A2 tilings, are recognizable, in both mathematical and practical senses. One implication of the results is that it is possible to search for distorted aperiodic structures in nature, where they may be hiding in plain sight.
We introduce a new type of aperiodic hexagonal monotile; a prototile that admits infinitely many tilings of the plane, but any such tiling lacks any translational symmetry. Adding a copy of our monotile to a patch of tiles must satisfy two rules that apply only to adjacent tiles. The first is inspired by the Socolar--Taylor monotile, but can be realised by shape alone. The second is a local growth rule; a direct isometry of our monotile can be added to any patch of tiles provided that a tree on the monotile connects continuously with a tree on one of its neighbouring tiles. This condition forc…
We show that translational tiling problems in a quotient of $\mathbb{Z}^d$ can be effectively reduced or ``simulated'' by translational tiling problems in $\mathbb{Z}^d$. In particular, for any $d \in \mathbb{N}$, $k < d$ and $N_1,\ldots,N_k \in \mathbb{N}$ the existence of an aperiodic tile in $\mathbb{Z}^{d-k} \times (\mathbb{Z} / N_1\mathbb{Z} \times \ldots \times \mathbb{Z} / N_k \mathbb{Z})$ implies the existence of an aperiodic tile in $\mathbb{Z}^d$. Greenfeld and Tao have recently disproved the well-known periodic tiling conjecture in $\mathbb{Z}^d$ for sufficiently large $d \in \mathb…
We present a single, connected tile which can tile the plane but only non-periodically. The tile is hexagonal with edge markings, which impose simple rules as to how adjacent tiles are allowed to meet across edges. The first of these rules is a standard matching rule, that certain decorations match across edges. The second condition is a new type of matching rule, which allows tiles to meet only when certain decorations in a particular orientation are given the opposite charge. This forces the tiles to form a hierarchy of triangles, following a central idea of the Socolar--Taylor tilings. Howe…
Jeandel and Rao proved that 11 is the size of the smallest set of Wang tiles, i.e., unit squares with colored edges, that admit valid tilings (contiguous edges of adjacent tiles have the same color) of the plane, none of them being invariant under a nontrivial translation. We study herein the Wang shift $Ω_0$ made of all valid tilings using the set $\mathcal{T}_0$ of 11 aperiodic Wang tiles discovered by Jeandel and Rao. We show that there exists a minimal subshift $X_0$ of $Ω_0$ such that every tiling in $X_0$ can be decomposed uniquely into 19 distinct patches of sizes ranging from 45 to 112…
In this paper, we propose an Integer Linear Model whose solutions are the aperiodic rhythms tiling with a given rhythm A. We show how this model can be used to efficiently check the necessity of the Coven-Meyerowitz's $(T2)$ condition and also to define an iterative algorithm that finds all the possible tilings of the rhythm A. To conclude, we run several experiments to validate the time efficiency of this model.
This paper aims to better understand the link better understand the links between aperiodicity in subshifts and pattern complexity. Our main contribution deals with substitutive subshifts, an equivalent to substitutive tilings in the context of symbolic dynamics. For a class of substitutive subshifts, we prove a quadratic lower bound on their pattern complexity. Together with an already known upper bound, this shows that this class of substitutive subshifts has a pattern complexity in $Θ(n^2)$. We also prove that the recent bound of Kari and Moutot, showing that any aperiodic subshift has patt…
