Related papers: Tiling groupoids and Bratteli diagrams

In this note we use techniques in the topology of 2-complexes to recast some tools that have arisen in the study of planar tiling questions. With spherical pictures we show that the tile counting group associated to a set $T$ of tiles and a…

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…

Regular ring lattices (RRLs) are defined as peculiar undirected circulant graphs constructed from a cycle graph, wherein each node is connected to pairs of neighbors that are spaced progressively in terms of vertex degree. This kind of…

We show that given any tiling of Euclidean space, any geometric patterns of points, we can find a patch of tiles (of arbitrarily large size) so that copies of this patch appear in the tiling nearly centered on a scaled and translated…

A vertex-transitive graph X is called local-to-global rigid if there exists R such that every other graph whose balls of radius R are isometric to the balls of radius R in X is covered by X. Let $d\geq 4$. We show that the 1-skeleton of an…

Let Lambda be a tiled R-order. We give a description of Aut_R(Lambda) as the semidirect product of Inn(Lambda) and a certain subgroup of Aut(Q(Lambda)), where Q(Lambda) is the link graph of Lambda. Additionally, we give criteria for…

The algebraic connectivity of a graph $G$ in a finite dimensional real normed linear space $X$ is a geometric counterpart to the Fiedler number of the graph and can be regarded as a measure of the rigidity of the graph in $X$. We analyse…

We introduce the notion of (abelian) similarity scheme, as a constructive model for topological self-similar fractals, in the same way in which the notion of iterated function system furnishes a constructive notion of self-similar fractals…

These notes derive aperiodic monotiles (arXiv:2303.10798) from a set of rhombuses with matching rules. This dual construction is used to simplify the proof of aperiodicity by considering the tiling as a colouring game on a Rhombille tiling.…

Let $k$ be a non-archimedean complete valued field and let X be a smooth Berkovich analytic $k$-curve. Let $F$ be a finite locally constant \'{e}tale sheaf on $k$ whose torsion is prime to the residue characteristic. We denote by $|X|$ the…

We extend rotation theory of circle maps to tiling spaces. Specifically, we consider a 1-dimensional tiling space $\Omega$ with finite local complexity and study self-maps $F$ that are homotopic to the identity and whose displacements are…

Let $\mathbb{X}$ be a weighted projective line of tubular type and $\operatorname{coh}\mathbb{X}$ the category of coherent sheaves on $\mathbb{X}$. The main purpose of this note is to show that the subgraph of the tilting graph consisting…

We prove a general width duality theorem for combinatorial structures with well-defined notions of cohesion and separation. These might be graphs and matroids, but can be much more general or quite different. The theorem asserts a duality…

Given a resolution of rational singularities $\pi\colon \tilde{X} \to X$ over a field of characteristic zero we use a Hodge-theoretic argument to prove that the image of the functor $\mathbf{R}\pi_*\colon \mathbf{D}(\tilde{X}) \to…

An interval translation map (ITM) is a map $T \colon I \to I$ defined as a piecewise translation on a finite partition of an interval $I$ into $r \ge 2$ subintervals. Unlike classical interval exchange transformations (IETs), the images of…

The Truchet tiles are a pair of square tiles decorated by arcs. When the tiles are pieced together to form a Truchet tiling, these arcs join to form a family of simple curves in the plane. We consider a family of probability measures on the…

Let $\mathcal{H}$ be a connected hereditary abelian category with tilting objects. It is proved that the cluster-tilting graph associated with $\mathcal{H}$ is always connected. As a consequence, we establish the connectedness of the…

Shift radix systems form a collection of dynamical systems depending on a parameter $\mathbf{r}$ which varies in the $d$-dimensional real vector space. They generalize well-known numeration systems such as beta-expansions, expansions with…

Extremal length is a conformal invariant that transfers naturally to the discrete setting, giving square tilings as a natural combinatorial analog of conformal mappings. Recent work by S. Hersonsky has explored generalizing these ideas to…

We demonstrate that the proper homotopy equivalence relation for locally finite graphs is Borel complete. Furthermore, among the infinite graphs, there is a comeager equivalence class. As corollaries, we obtain the analogous results for the…