Related papers: Hexagonal parquet tilings: k-isohedral monotiles w…
This paper is intended to provide an introduction to the theory of substitution tilings. For our purposes, tiling substitution rules are divided into two broad classes: geometric and combinatorial. Geometric substitution tilings include…
Motivated by a question of R.\ Nandakumar, we show that the Euclidean plane can be dissected into mutually incongruent convex quadrangles of the same area and the same perimeter. As a byproduct we obtain vertex-to-vertex dissections of the…
A famous problem in discrete geometry is to find all monohedral plane tilers, which is still open to the best of our knowledge. This paper concerns with one of its variants that to determine all convex polyhedra whose every cross-section…
Does a given a set of polyominoes tile some rectangle? We show that this problem is undecidable. In a different direction, we also consider tiling a cofinite subset of the plane. The tileability is undecidable for many variants of this…
We consider a certain tiling problem of a planar region in which there are no long horizontal or vertical strips consisting of copies of the same tile. Intuitively speaking, we would like to create a dappled pattern with two or more kinds…
Icosahedral tilings, although non-periodic, are known to be characterized by their configurations of some finite size. This characterization has also been expressed in terms of a simple alternation condition. We provide an alternative proof…
We show it is possible to tile three-dimensional space using only tetrahedra with acute dihedral angles. We present several constructions to achieve this, including one in which all dihedral angles are less than $77.08^\circ$, and another…
We introduce a tiling problem between bounded open convex polyforms $\hat{P}\subset\mathbb{R}^2$ with directed and uniquely colored edges. If there exists a tiling of the polyform $\hat{P}_2$ by $\hat{P}_1$, we show that one can construct a…
We show that given any tiling of Euclidean space, any geometric patterns of points, we can find a patch of tiles (of arbitrarily large size) so that copies of this patch appear in the tiling nearly centered on a scaled and translated…
Decorating the Spectre tile with hexagons reveals triangular hexagonal clusters whose structure we study. In the process we reprove that the Spectre tilings exist and are uniquely hierarchical. The proof is not computer-assisted.
The present article studies combinatorial tilings of Euclidean or spherical spaces by polytopes, serving two main purposes: first, to survey some of the main developments in combinatorial space tiling; and second, to highlight some new and…
There is a sufficiently large $N\in h\mathbb{N}$ such that the following holds. If $G$ is a tripartite graph with $N$ vertices in each vertex class such that every vertex is adjacent to at least $2N/3+2h-1$ vertices in each of the other…
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…
This paper presents a tileset of 3 squares with local constraints on their borders and corners that enforce non-periodic tiling. We start with a description of the tileset and we demonstrate that it can tile the entire plane…
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…
We prove that for any point set P in the plane, a triangle T, and a positive integer k, there exists a coloring of P with k colors such that any homothetic copy of T containing at least ck^8 points of P, for some constant c, contains at…
The inclusion relation between simple objects in the plane may be used to define geometric set systems, or hypergraphs. Properties of various types of colorings of these hypergraphs have been the subject of recent investigations, with…
This article, written for undergraduate mathematics students, provides an accessible introduction to a few key problems in tiling theory: Heesch's problem, the isohedral number problem, and the existence of an aperiodic monotile. I…
We show how to determine if a given simple rectilinear polygon can be tiled with rectangles, each having an integer side.
In two series of papers we construct quasi regular polyhedra and their duals which are similar to the Catalan solids. The group elements as well as the vertices of the polyhedra are represented in terms of quaternions. In the present paper…