Related papers: The groups $G_{n}^{k}$ and $2n$-gon tilings
The study of tilings is a major problem in many mathematical instances, which is studied in two main different approaches: when considering the existence (or obstructions to the existence) of a tiling with a given tile and the other…
We compute the number of rhombus tilings of a hexagon with sides $N,M,N,N,M,N$, which contain a fixed rhombus on the symmetry axis. A special case solves a problem posed by Jim Propp.
Given a finite collection of two-dimensional tile types, the field of study concerned with covering the plane with tiles of these types exclusively has a long history, having enjoyed great prominence in the last six to seven decades. Much…
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…
Three types of geometric structure---grid triangulations, rectangular subdivisions, and orthogonal polyhedra---can each be described combinatorially by a regular labeling: an assignment of colors and orientations to the edges of an…
A tiling of a topological disc by topological discs is called monohedral if all tiles are congruent. Maltby (J. Combin. Theory Ser. A 66: 40-52, 1994) characterized the monohedral tilings of a square by three topological discs. Kurusa,…
In this paper, we give some sufficient conditions for a $n$-dimensional rectangle to be tiled with a set of bricks. These conditions are obtained by using the so-called Frobenius number.
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…
A tiling is a decomposition of a polygon into finitely many non-overlapping triangles. We prove that if a regular n-gon, $n \geq 5$, $n \neq 28$, can be tiled with similar right triangles, then one of the angles of these triangles is in…
A study of 7-fold tilings that use a set of three proto-rhombs in a substitution scheme to tile a large area. A set is discovered that is thought to be the most minimal or smallest one. The scheme uses 11 proto-rhombs to tile the next…
Two-dimensional random tilings of rhombi can be seen as projections of two-dimensional membranes embedded in hypercubic lattices of higher dimensional spaces. Here, we consider tilings projected from a $D$-dimensional space. We study the…
We determine all non-edge-to-edge tilings of the sphere by regular spherical polygons of three or more sides.
We apply the theory of Groebner bases to the study of signed, symmetric polyomino tilings of planar domains. Complementing the results of Conway and Lagarias we show that the triangular regions T_N=T_{3k-1} and T_N=T_{3k} in a hexagonal…
We propose the first algebraic determinantal formula to enumerate tilings of a centro-symmetric octagon of any size by rhombi. This result uses the Gessel-Viennot technique and generalizes to any octagon a formula given by Elnitsky in a…
We present a geometrical description of new canonical $d$-dimensional codimension one quasiperiodic tilings based on generalized Fibonacci sequences. These tilings are made up of rhombi in 2d and rhombohedra in 3d as the usual Penrose and…
Square-triangle-rhombus ($\mathcal{STR}$) tilings are encountered in various self-organized multi-component systems. They exhibit a rich structural diversity, encompassing both periodic tilings and long-range ordered quasicrystals,…
The local structure of a tiling is described in terms of a multiplicative structure on its pattern classes. The groupoid associated to the tiling is derived from this structure and its integer group of coinvariants is defined. This group…
We give an explicit and effective construction for rhombus cut-and-project tilings with global n-fold rotational symmetry for any n. This construction is based on the dualization of regular n-fold multigrids. The main point is to prove the…
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…
We develop a recursive formula for counting the number of rectangulations of a square, i.e the number of combinatorially distinct tilings of a square by rectangles. Our formula specializes to give a formula counting generic rectangulations,…