Related papers: Tiling with Small Tiles
Conway and Lagarias showed that certain roughly triangular regions in the hexagonal grid cannot be tiled by shapes Thurston later dubbed tribones. Here we study a two-parameter family of roughly hexagonal regions in the hexagonal grid and…
The study of knot mosaics is based upon representing knot diagrams using a set of tiles on a square grid. This branch of knot theory has many unanswered questions, especially regarding the efficiency with which we draw knots as mosaics.…
We consider three-dimensional domino tilings of cylinders $\mathcal{R}_N = \mathcal{D} \times [0,N]$ where $\mathcal{D} \subset \mathbb{R}^2$ is a fixed quadriculated disk and $N \in \mathbb{N}$. A domino is a $2 \times 1 \times 1$ brick. A…
Translational tiling problems are among the most fundamental and representative undecidable problems in all fields of mathematics. Greenfeld and Tao obtained two remarkable results on the undecidability of translational tiling in recent…
The Wang tiling is a classical problem in combinatorics. A major theoretical question is to find a (small) set of tiles which tiles the plane only aperiodically. In this case, resulting tilings are rather restrictive. On the other hand,…
If n is a multiple of 4, then a square of side n can be tiled with T-tetrominos, using a well-known construction. If n is even but not a multiple of four, then there exists an equally well-known construction for tiling a square of side n…
In this paper, we study tilings of $\mathbb Z$, that is, coverings of $\mathbb Z$ by disjoint sets (tiles). Let $T=\{d_1,\ldots, d_s\}$ be a given multiset of distances. Is it always possible to tile $\mathbb Z$ by tiles, for which the…
It has been common knowledge since 1950 that seven colours can be assigned to tiles of an infinite honeycomb with cells of unit diameter such that no two tiles of the same colour are closer than $d(7)=\frac{\sqrt{7}}{2}$ apart. Various…
Two new series of substitution tilings are introduced in which the tiles appear in infinitely many orientations. It is shown that several properties of the well-known pinwheel tiling do also hold for these new examples, and, in fact, for…
We give a computer-based proof of the following fact: If a square is divided into seven or nine convex polygons, congruent among themselves, then the tiles are rectangles.
As a continuation to our previous work [9, 10], we consider the domino tiling problem with impurities. (1) if we have more than two impurities on the boundary, we can compute the number of corresponding perfect matchings by using the…
We study tilings of polygons $R$ with arbitrary convex polygonal tiles. Such tilings come in continuous families obtained by moving tile edges parallel to themselves (keeping edge directions fixed). We study how the tile shapes and areas…
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…
We consider the tiling generating functions of semi-hexagons and quartered hexagons with dents on their sides. In general, there are no simple product formulas for these generating functions. However, we show that the modification in the…
There exist tilings of the plane with pairwise noncongruent triangles of equal area and bounded perimeter. Analogously, there exist tilings with triangles of equal perimeter, the areas of which are bounded from below by a positive constant.…
A finite set of integers $A$ tiles the integers by translations if $\mathbb{Z}$ can be covered by pairwise disjoint translated copies of $A$. Restricting attention to one tiling period, we have $A\oplus B=\mathbb{Z}_M$ for some…
Tile-based self-assembly systems are capable of universal computation and algorithmically-directed growth. Systems capable of such behavior typically make use of "glue cooperation" in which the glues on at least $2$ sides of a tile must…
We obtain tilings with a singular point by applying conformal maps on regular tilings of the Euclidean plane, and determine its symmetries. The resulting tilings are then symmetrically colored by applying the same conformal maps on…
The well-known problem stated by A. Meir and L. Moser consists in tiling the unit square with rectangles (details), whose side lengths equal $1/n\times 1/(n+1)$, where indices~$n$ range from 1 to infinity. Recently, Terence Tao has proved…
Classical results on aperiodic tilings are rather complicated and not widely understood. Below, an alternative approach is discussed in hope to provide additional intuition not apparent in classical works.