Related papers: Tilings, tiling spaces and topology

Recently Taylor and Socolar introduced an aperiodic mono-tile. The associated tiling can be viewed as a substitution tiling. We use the substitution rule for this tiling and apply the algorithm of \cite{AL} to check overlap coincidence. It…

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…

We study the cohomology and hence $K$-theory of the aperiodic tilings formed by the so called 'cut and project' method, i.e., patterns in $d$ dimensional Euclidean space which arise as sections of higher dimensional, periodic structures.…

Moir\'e patterns of twisted and scaled bilayers have recently emerged as a fertile source of quasiperiodic order in two-dimensional materials. Inspired by these systems, we introduce the \emph{near-coincidence method} for generating…

The cohomology of a tiling or a point pattern has originally been defined via the construction of the hull or the groupoid associated with the tiling or the pattern. Here we present a construction which is more direct and therefore easier…

Traditionally a tiling is defined with a finite number of finite forbidden patterns. We can generalize this notion considering any set of patterns. Generalized tilings defined in this way can be studied with a dynamical point of view,…

A new method for constructing self-referential tilings of Euclidean space from a graph directed iterated function system, based on a combinatorial structure we call a pre-tree, is introduced. In the special case that we refer to as…

Model sets (or cut and project sets) provide a familiar and commonly used method of constructing and studying nonperiodic point sets. Here we extend this method to situations where the internal spaces are no longer Euclidean, but instead…

We briefly review the standard methods used to construct quasiperiodic tilings, such as the projection, the inflation, and the grid method. A number of sample Mathematica programs, implementing the different approaches for one- and…

Quasi-periodic structures of quasicrystals yield novel effects in diverse systems. However, there is little investigation on employing quasi-periodic structures in the morphology control. Here, we show the use of quasi-periodic surface…

The space of topological decompositions into triangulations of a surface has a natural graph structure where two triangulations share an edge if they are related by a so-called flip. This space is a sort of combinatorial Teichm\"uller space…

We obtain structural results on translational tilings of periodic functions in $\mathbb{Z}^d$ by finite tiles. In particular, we show that any level one tiling of a periodic set in $\mathbb{Z}^2$ must be weakly periodic (the disjoint union…

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 consider a certain tiling problem of a planar region in which there are no long horizontal or vertical strips consisting of copies of the same tile. Intuitively speaking, we would like to create a dappled pattern with two or more kinds…

In aperiodic order, non-periodic but "ordered" objects such as tilings, Delone sets, functions and measures are investigated. In this article we depict the common structure of these objects by using the general framework of abstract pattern…

We compute the Cech cohomology with integer coefficients of one-dimensional tiling spaces arising from not just one, but several different substitutions, all acting on the same set of tiles. These calculations involve the introduction of a…

Quasiperiodic patterns described by polyhedral "atomic surfaces" and admitting matching rules are considered. It is shown that the cohomology ring of the continuous hull of such patterns is isomorphic to that of the complement of a torus…

Identity-homotopic self-homeomorphisms of a space of non-periodic 1-dimensional tiling are generalizations of orientation-preserving self-homeomorphisms of circles. We define the analogue of rotation numbers for such maps. In constrast to…

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…

Aperiodic tilings are non-periodic tilings characterized by local constraints. They play a key role in the proof of the undecidability of the domino problem (1964) and naturally model quasicrystals (discovered in 1982). A central question…