Related papers: Octagonal tilings with three prototiles
While it is a classical result dating back to Dehn (1903) that squares composing a perfect rectangle must have rational side lengths, the arithmetic complexity of these tilings, specifically the growth of the denominators of these rational…
The alternating (zigzag) numbers $A_n$, counting the ascending alternating permutations of $\left\{1,\cdots,n\right\}$ and defined by the exponential generating function $\tan x+\sec x$, admit several classical combinatorial and analytic…
Convex hexagons that can tile the plane have been classified into three types. For the generic cases (not necessarily convex) of the three types and two other special cases, we classify tilings of the plane under the assumption that all…
Given a graph $G$ and collection of subgraphs $T$ (called tiles), we consider covering $G$ with copies of tiles in $T$ so that each vertex $v\in G$ is covered with a predetermined multiplicity. The multinomial tiling model is a natural…
Given a collection of N rectangles such that the side ratio of each one is a quadratic irrationality, we find all rectangles which can be tiled by rectangles similar to one of the given ones. It means that each possible shape can be used…
A linear combination $aT_r(m)+bT_s(n)$ of an \mbox{$r$-gonal} number $T_r(m)$ and an $s$-gonal number $T_s(n)$ with mutually coprime positive integer coefficients $a$ and $b$ produces infinitely many primes as $m$ and~$n$ varies over the…
We consider two families of planar self-similar tilings of different nature: the tilings consisting of translated copies of the fractal sets defined by an iterated function system, and the tilings obtained as a geometrical realization of a…
Periodic tilings play a role in the decorative arts, in construction and in crystal structures. Combinatorial tiling theory allows the systematic generation, visualization and exploration of such tilings of the plane, sphere and hyperbolic…
We give a formula for the number of lozenge tilings of a hexagon on the triangular lattice with unit triangles removed from arbitrary positions along two non-adjacent, non-opposite sides. Our formula implies that for certain families of…
Some combinatorial properties of fixed boundary rhombus random tilings with octagonal symmetry are studied. A geometrical analysis of their configuration space is given as well as a description in terms of discrete dynamical systems, thus…
In this paper, we consider representations of integers as sums of at most four distinct $m$-gonal numbers (allowing a fixed number of repeats of each polygonal number occurring in the sum). We show that the number of such representations…
We briefly review the standard methods used to construct quasiperiodic tilings, such as the projection, the inflation, and the grid method. A number of sample Mathematica programs, implementing the different approaches for one- and…
For every positive integer $n$, we introduce a set $\mathcal{T}_n$ made of $(n+3)^2$ Wang tiles (unit squares with labeled edges). We represent a tiling by translates of these tiles as a configuration $\mathbb{Z}^2\to\mathcal{T}_n$. A…
It is shown that if n<7, then each tiling of R^n by translates of the unit cube [0,1)^n contains a column; that is, a family of the form {[0,1)^n+(s+ke_i): k \in Z}, where s \in R^n, e_i is an element of the standard basis of R^n and Z is…
The {\alpha}-Kakutani substitution rule splits the unit interval into two subintervals of lengths alpha and 1 - {\alpha}, for a fixed {\alpha} in (0,1). A simple inflation-substitution procedure produces tilings of the real line and their…
We give a simple proof of T. Stehling's result, that in any normal tiling of the plane with convex polygons with number of sides not less than six, all tiles except the finite number are hexagons.
A set of tiles for covering a surface is composed of two types of tiles. The base shape of each one of them is a diamond or rhombus, both with angles 60 and 120 degrees. They are distinguished by labeling one as an acute diamond with a base…
Given a tiling of a 2D grid with several types of tiles, we can count for every row and column how many tiles of each type it intersects. These numbers are called the_projections_. We are interested in the problem of reconstructing a tiling…
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…
In this work, we consider a class of substitutions on infinite alphabets and show that they exhibit a growth behaviour which is impossible for substitutions on finite alphabets. While for both settings the leading term of the tile counting…