Related papers: A note on discrete lattice-periodic sets with an a…
An irregular vertex in a tiling by polygons is a vertex of one tile and belongs to the interior of an edge of another tile. In this paper we show that for any integer $k\geq 3$, there exists a normal tiling of the Euclidean plane by convex…
We study tilings of the plane composed of two repeating tiles of different assigned areas relative to an arbitrary periodic lattice. We classify isoperimetric configurations (i.e., configurations with minimal length of the interfaces) both…
Tilings of the plane resemble the simplicial and other complexes from algebraic topology, but have not been studied from this perspective. We construct finite categories corresponding to polygons with labeled directed edges, and introduce…
We study tilings of the plane that combine strong properties of different nature: combinatorial and algorithmic. We prove existence of a tile set that accepts only quasiperiodic and non-recursive tilings. Our construction is based on the…
A group-theoretical approach to the construction of quasiperiodic tilings of a Euclidean plane, possessing five-fold symmetry, is applied. Of the infinitely many of variants of quasiperiodic partitions of the plane, possessing the dihedral…
We present an alternative geometric representation for the eleven Archimedean lattices, in which each site and bond is uniquely labeled by an ordered pair of integers and characterized via a modular function. This structured labeling…
\noindent The algebraic characterization of classes of locally isomorphic aperiodic tilings, being examples of quantum spaces, is conducted for a certain type of tilings in a manner proposed by A. Connes. These $2$-dimensional tilings are…
Which polygons admit two (or more) distinct lattice tilings of the plane? We call such polygons double tiles. It is well-known that a lattice tiling is always combinatorially isomorphic either to a grid of squares or to a grid of regular…
In the present paper, as we did previously in [7], we investigate the relations between the geometric properties of tilings and the algebraic properties of associated relational structures. Our study is motivated by the existence of…
A tiling (edge-to-edge) of the plane is a family of tiles that cover the plane without gaps or overlaps. Vertex figure of a vertex in a tiling to be the union of all edges incident to that vertex. A tiling is $k$-vertex-homogeneous if any…
The periodic tiling conjecture asserts that any finite subset of a lattice $\mathbb{Z^d}$ which tiles that lattice by translations, in fact tiles periodically. We announce here a disproof of this conjecture for sufficiently large $d$, which…
The Ammann A2 tiling is a simple aperiodically ordered tiling of the plane. We consider the graph derived from this tiling, by treating each corner of each tile as a vertex and each side of each tile as an edge. We present a closed-form…
Tiling models are classical statistical models in which different geometric shapes, the tiles, are packed together such that they cover space completely. In this paper we discuss a class of two-dimensional tiling models in which the tiles…
It is well-known that plane partitions, lozenge tilings of a hexagon, perfect matchings on a honeycomb graph, and families of non-intersecting lattice paths in a hexagon are all in bijection. In this work we consider regions that are more…
An aperiodic tile set was first constructed by R.Berger while proving the undecidability of the domino problem. It turned out that aperiodic tile sets appear in many topics ranging from logic (the Entscheidungsproblem) to physics…
An aperiodic tile set was first constructed by R. Berger while proving the undecidability of the domino problem. It turned out that aperiodic tile sets appear in many topics ranging from logic (the Entscheidungsproblem) to physics…
Aperiodic tiling --- a form of complex global geometric structure arising through locally checkable, constant-time matching rules --- has long been closely tied to a wide range of physical, information-theoretic, and foundational…
We study periodic tessellations of the Euclidean space with unequal cells arising from the minimization of perimeter functionals. Existence results and qualitative properties of minimizers are discussed for different classes of problems,…
We introduce and study the notion of orthosymmetric spaces over an Archimedean vector lattice as a generalization of finite-dimentional Euclidean inner spaces. A special attention has been paid to linear operators on these spaces.
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…