Related papers: Tilings of Rectangular Regions by Rectangular Tile…
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 edge-to-edge tilings of the sphere by congruent quadrilaterals of Type $a^2bc$ are classified as $3$ classes: a sequence of two-parameter families of $2$-layer earth map tilings with $2n$ $(n\ge3)$ tiles, a one-parameter family of…
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…
We introduce and study the number of tilings of unit height rectangles with irrational tiles. We prove that the class of sequences of these numbers coincides with the class of diagonals of N-rational generating functions and a class of…
An \emph{auspicious tatami mat arrangement} is a tiling of a rectilinear region with two types of tiles, $1 \times 2$ tiles (dimers) and $1 \times 1$ tiles (monomers). The tiles must cover the region and satisfy the constraint that no four…
A simple, yet unifying method is provided for the construction of tilings by tiles obtained from the attractor of an iterated function system (IFS). Many examples appearing in the literature in ad hoc ways, as well as new examples, can be…
This article shines new light on the classical problem of tiling rectangles with squares efficiently with a novel method. With a twist on the traditional approach of resistor networks, we provide new and improved results on the matter using…
In areas as diverse as contemporary art, play structures, climbing equipment, and modular construction toys, we see the presence of building block-like polyhedral complexes, which are generalizations of the pieces in the game Tetris. We…
Rhombus Penrose tilings are tilings of the plane by two decorated rhombi such that the decoration match at the junction between two tiles (like in a jigsaw puzzle). In dynamical terms, they form a tiling space of finite type. If we remove…
We determine all non-edge-to-edge tilings of the sphere by regular spherical polygons of three or more sides.
This article examines the tilings of a strip with equilateral triangles. The number of ways in which the lattices can be covered with a combination of tiles of the two types of triangles is related to Pell's numbers. Additionally, the…
In this paper we consider faultfree tromino tilings of rectangles and characterize rectangles that admit such tilings. We introduce the notion of {\it crossing numbers} for tilings and derive bounds on the crossing numbers of faultfree…
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…
In this paper a closed form expression for the number of tilings of an $n\times n$ square border with $1\times 1$ and $2\times1$ cuisenaire rods is proved using a transition matrix approach. This problem is then generalised to $m\times n$…
In this paper, we study the structure of the set of tilings produced by any given tile-set. For better understanding this structure, we address the set of finite patterns that each tiling contains. This set of patterns can be analyzed in…
We study tilings of rectangular boards using unit squares together with a single type of big tile shaped as a Ferrers diagram. We derive generating functions for these tilings, prove real-rootedness and interlacing properties of associated…
We consider tilings of quadriculated regions by dominoes and of triangulated regions by lozenges. We present an overview of results concerning tileability, enumeration and the structure of the space of tilings.
We introduce an elementary transformation called flips on tilings by squares and triangles and conjecture that it connects any two tilings of the same region of the Euclidean plane.
Tilings are around us everywhere, and our curiosity draws us to study their properties. A tiling is a way of arranging pieces on a board, such that there is no space left uncovered, nor any space covered by more than one tile. In…
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…