Related papers: Forcing nonperiodic tilings with one tile using a …
Kwasniewski's cobweb posets uniquely represented by directed acyclic graphs are such a generalization of the Fibonacci tree that allows joint combinatorial interpretation for all of them under admissibility condition. This interpretation…
Is there a fixed dimension $n$ such that translational tiling of $\mathbb{Z}^n$ with a monotile is undecidable? Several recent results support a positive answer to this question. Greenfeld and Tao disprove the periodic tiling conjecture by…
Motivated by the problem of domino tilings of the Aztec diamond, a weighted particle system is defined on $N$ lines, with line $j$ containing $j$ particles. The particles are restricted to lattice points from 0 to $N$, and particles on…
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…
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 provide a definitive classification of all finite sets of regular polygons that admit a tiling of the hyperbolic plane, thereby establishing the decidability of the Domino Problem for this class of prototiles. We show that admissibility…
This paper introduces two tiles whose tilings form a one-parameter family of tilings which can all be seen as digitization of two-dimensional planes in the four-dimensional Euclidean space. This family contains the Ammann-Beenker tilings as…
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…
We give a necessary and sufficient condition on a $d$-dimensional affine subspace of $\mathbb{R}^n$ to be characterized by a finite set of patterns which are forbidden to appear in its digitization. This can also be stated in terms of local…
Motivated by the enumeration of a class of plane partitions studied by Proctor and by considerations about symmetry classes of plane partitions, we consider the problem of enumerating lozenge tilings of a hexagon with ``maximal staircases''…
We study decision problems on geometric tilings. First, we study a variant of the Domino problem where square tiles are replaced by geometric tiles of arbitrary shape. We show that this variant is undecidable regardless of the shapes,…
An h-tiling on a finite simplicial complex is a partition of its geometric realization by maximal simplices deprived of several codimension one faces together with possibly their remaining face of highest codimension. In this last case, the…
A finite collection $P$ of finite sets tiles the integers iff the integers can be expressed as a disjoint union of translates of members of $P$. We associate with such a tiling a doubly infinite sequence with entries from $P$. The set of…
The Euclidean Steiner problem is the problem of finding a set $St$, with the shortest length, such that $St \cup A$ is connected, where $A$ is a given set in a Euclidean space. The solutions $St$ to the Steiner problem will be called…
The vertex corona of a vertex of some tiling is the vertex together with the adjacent tiles. A tiling where all vertex coronae are congruent is called monocoronal. We provide a classification of monocoronal tilings in the Euclidean plane…
Enumeration of tilings is the mathematical study concerning the total number of coverings of regions by similar pieces without gaps or overlaps. Enumeration of tilings has become a vibrant subfield of combinatorics with connections and…
We consider the problem of embedding the Steiner points of a Steiner tree with given topology into the rectilinear plane. Thereby, the length of the path between a distinguished terminal and each other terminal must not exceed given length…
Tilings of a surface of negative Euler characteristic by n-gons with n\ge 7 is a finite problem. We develop the algorithm for finding all the tilings for fixed number of tiles and present the calculation for tilings of surfaces of small…
To understand an aperiodic tiling (or a quasicrystal modeled on an aperiodic tiling), we construct a space of similar tilings, on which the group of translations acts naturally. This space is then an (abstract) dynamical system. Dynamical…
Crystallographic tilings of the Euclidean space $\mathbb{E}^n$ are defined as simple tilings whose group of isometric automorphisms is crystallographic. To classify crystallographic tilings by their automorphism groups it is necessary to…