Related papers: Self-dual tilings with respect to star-duality
We define bicolored tilings as a disk with a collection of smooth curves with a coloring map on the tiles that these curves delimit. Using two transformations, we define an equivalence on tilings. We then define the Scott map which creates…
We construct a unilateral lattice tiling of $\mathbb{R}^n$ into hypercubes of two differnet side lengths $p$ or $q$. This generalizes the Pythagorean tiling in $\mathbb{R}^2$. We also show that this tiling is unique up to symmetries, which…
We establish a topological duality for bounded lattices. The two main features of our duality are that it generalizes Stone duality for bounded distributive lattices, and that the morphisms on either side are not the standard ones. A…
We develop a systematic method for computing the angle combinations at all vertices in an edge-to-edge tiling of the sphere by pentagons with the same five angles. The method is a useful and necessary step in many tiling problems about…
In the perfect tiling problem, we aim to cover the vertices of a hypergraph~$G$ with pairwise vertex-disjoint copies of a hypergraph $F$. There are three essentially necessary conditions for such a perfect tiling, which correspond to…
Recent work in percolation has led to exact solutions for the site and bond critical thresholds of many new lattices. Here we show how these results can be extended to other classes of graphs, significantly increasing the number and variety…
We discuss problems of simultaneous tiling. This means that we have an object (set, function) which tiles space with two or more different sets of translations. The most famous problem of this type is the Steinhaus problem which asks for a…
The pinwheel tiling is the paradigm for a substitution tiling with circular symmetry, in the sense that the corresponding autocorrelation is circularly symmetric. As a consequence, its diffraction measure is also circularly symmetric, so…
This article builds on Thurston's height functions. His tiling algorithm is reinterpreted using lattice theory and then generalized in order to generate any tiling of a hole-free region. Combined with a natural encoding of tilings by words,…
We consider tilings of the plane with 12-fold symmetry obtained by the cut and projection method. We compute their cohomology groups using the techniques introduced by the second author, Hunton and Kellendonk. To do this we completely…
We derive the exact dual theory of a lattice Ginzburg-Landau theory with an additional topological Chern-Simons (CS) term. It is shown that in the zero-temperature limit, the statistical parameter $\theta=1/2\pi$ corresponds to a fixed…
Let X be a smooth elliptic fibration over a smooth base B. Under mild assumptions, we establish a Fourier-Mukai equivalence between the derived categories of two objects, each of which is an O^* gerbe over a genus one fibration which is a…
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…
In this survey, we explain a version of topological $L^2$-Serre duality for singular complex spaces with arbitrary singularities. This duality can be used to deduce various $L^2$-vanishing theorems for the $\overline\partial$-equation on…
The purpose of this paper is to give the flavor of the subject of self-similar tilings in a relatively elementary setting, and to provide a novel method for the construction of such polygonal tilings.
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…
The results involving rotationally symmetric tilings with multiple types of rhombuses, discovered by Penrose, Ammann, Beenker, or Socolar, are converted to tilings with multiple types of pentagons are presented. The pentagons can be convex…
Penrose tilings are the most famous aperiodic tilings, and they have been studied extensively. In particular, patterns composed with hexagons (H), boats (B) and stars (S) were soon exhibited and many physicists published on what they later…
Two-, three- and four-dimensional representations of Penrose tilings of the plane are described. The vertices that occur in these representations lie on lattices. Symmetries and methods of visualizing these representations are discussed.…
Let $B_R$ denote the closed Euclidean ball of radius $R$ in the plane. In this paper we prove that, if $V$ is the set of vertices of any unit length rhombic Penrose tiling then, for $R\ge 2$, \[\#(V\cap B_R)=\pi C_P R^2 + O(R^{2/3}(\log…