Related papers: Domino tilings beyond 2D
We consider polygonal tilings of certain regions and use these to give intuitive definitions of tiling-based perimeter and area. We apply these definitions to rhombic tilings of Elnitsky polygons, computing sharp bounds and average values…
We show that the following problem is undecidable: given two polygonal prototiles, determine whether the plane can be tiled with rotated and translated copies of them. This improves a result of Demaine and Langerman [SoCG 2025], who showed…
Some relations between cohomological dimensions and depths of linked ideals are investigated and discussed by various examples.
We provide a proof of the Alpern multi-tower theorem for Z^d actions. We reformulate the theorem as a problem of measurably tiling orbits of a Z^d action by a collection of rectangles whose corresponding sides have no non-trivial common…
We study the problem of perfect tiling in the plane and exploring the possibility of tiling a rectangle using integral distinct squares. Assume a set of distinguishable squares (or equivalently a set of distinct natural numbers) is given,…
A polyomino is a polygonal region with axis parallel edges and corners of integral coordinates, which may have holes. In this paper, we consider planar tiling and packing problems with polyomino pieces and a polyomino container $P$. We give…
Helfgott and Gessel gave the number of domino tilings of an Aztec Rectangle with defects of size one on the boundary of one side. In this paper we extend this to the case of domino tilings of an Aztec Rectangle with defects on all boundary…
In dimensions greater than or equal to 3, the local structure of a singular holomorphic foliation conceals a globally defined foliation on the projective space of dimension one less. In this paper, we will discuss how the global dynamics of…
Since it first emerged in Wijsman's seminal work [29], the Wijsman topology has been intensively studied in the past 50 years. In particular, topological properties of Wijsman hyperspaces, relationships between the Wijsman topology and…
Nandakumar asked whether there is a tiling of the plane by pairwise non-congruent triangles of equal area and equal perimeter. Here a weaker result is obtained: there is a tiling of the plane by pairwise non-congruent triangles of equal…
Holographic multiple-input multiple-output (HMIMO) technology, which uses spatially continuous surfaces for signal transmission and reception, is envisioned to be a promising solution for improving the data rate and coverage of wireless…
The concept of star-duality is described for self-similar cut-and-project tilings in arbitrary dimensions. This generalises Thurston's concept of a Galois-dual tiling. The dual tilings of the Penrose tilings as well as the Ammann-Beenker…
In this paper, we continue the study of domino-tilings of Aztec diamonds. In particular, we look at certain ways of placing ``barriers'' in the Aztec diamond, with the constraint that no domino may cross a barrier. Remarkably, the number of…
Recently, the first author together with Jens Marklof studied generalizations of the classical three distance theorem to higher dimensional toral rotations, giving upper bounds in all dimensions for the corresponding numbers of distances…
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…
From descent theory to higher geometry, the idea of gluing has been embedded in many elegant and powerful techniques, proving instrumental for the solution of many problems. In this paper, we introduce a framework that allows to link…
Polygon spaces have been studied extensively, and yet missing from the literature is a simple property that every polygon has: dimension. This is distinct (possibly) from the dimension of the ambient space in which the polygon lives. A…
A generic massive Thirring Model in three space-time dimensions exhibits a correspondence with a topologically massive bosonized gauge action associated to a self-duality constraint, and we write down a general expression for this…
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…
In this survey, I suggest to approach the problem of functorial properties of quantum cohomology by drawing lessons from several versions of Mirror duality involving deformation spaces.