Related papers: Colouring monohedral tilings: defects and grain bo…
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…
We determine all non-edge-to-edge tilings of the sphere by regular spherical polygons of three or more sides.
In this work, we give a method to obtain nontrivial transitive perfect colorings of the non-regular Archimedean tilings using the least possible number $n$ of colors. We also look for other non-equivalent transitive perfect $n$-colorings of…
We consider here square tilings of the plane. By extending the formalism introduced in [3] we build a correspondence between plane maps endowed with an harmonic vector and square tilings satisfying a condition of regularity. In the case of…
The investigation of colour symmetries for periodic and aperiodic systems consists of two steps. The first concerns the computation of the possible numbers of colours and is mainly combinatorial in nature. The second is algebraic and…
We introduce a square tiling/tetragonal strip representation to the P, D, and G triply periodic minimal surfaces. This approach is useful in identifying mixtures and grain boundaries of these surfaces that might be useful for material…
Tilings and tiling systems are an abstract concept that arise both as a computational model and as a dynamical system. In this paper, we characterize the sets of periods that a tiling system can produce. We prove that up to a slight…
In contrast to many known results concerning periodic tilings of the Euclidean plane with pentagons, here tilings with rotational symmetry are investigated. A certain class of convex pentagons is introduced. It can be shown that for any…
We prove that for each partition of the Lobachevsky plane into finitely many Borel pieces one of the cells of the partition contains an unbounded centrally symmetric subset.
We study the minimal complexity of tilings of a plane with a given tile set. We note that every tile set admits either no tiling or some tiling with O(n) Kolmogorov complexity of its n-by-n squares. We construct tile sets for which this…
An h-tiling on a finite simplicial complex is a partition of its geometric realization by maximal simplices deprived of several codimension one faces together with possibly their remaining face of highest codimension. In this last case, the…
In the perfect tiling problem, we aim to cover the vertices of a hypergraph~$G$ with pairwise vertex-disjoint copies of a hypergraph $F$. There are three essentially necessary conditions for such a perfect tiling, which correspond to…
We conclude an investigation of Abrishami, Esperet, Giocanti, Hamman, Knappe and M\"oller studying the existence of periodic colourings of locally finite graphs. A colouring of a graph $\Gamma$ is periodic if the resulting coloured graph…
For a fixed integer h>=1, let G be a tripartite graph with N vertices in each vertex class, N divisible by 6h, such that every vertex is adjacent to at least 2N/3+h-1 vertices in each of the other classes. We show that if N is sufficiently…
Is there a fixed dimension $n$ such that translational tiling of $\mathbb{Z}^n$ with a monotile is undecidable? Several recent results support a positive answer to this question. Greenfeld and Tao disprove the periodic tiling conjecture by…
Aperiodic tilings are non-periodic tilings characterized by local constraints. They play a key role in the proof of the undecidability of the domino problem (1964) and naturally model quasicrystals (discovered in 1982). A central question…
The first undecidability result on the tiling is the undecidability of translational tiling of the plane with Wang tiles, where there is an additional color matching requirement. Later, researchers obtained several undecidability results on…
The edge-to-edge tilings of the sphere by congruent polygons, where all edges are straight, have been completely classified. We classify the curvilinear version of the similar triangular tilings, where the edges may not be straight, and…
The Heesch problem 'grades' polygons that fail to tile the plane in terms of the number of layers (or corollas) of copies of it that can be formed around a central unit. We study the different topology of ' walls', which we define to be…
This is a chapter in an incoming book on aperiodic order. We review results about the topology, the dynamics, and the combinatorics of aperiodically ordered tilings obtained with the tools of noncommutative geometry.