A Local Characterization of Combinatorial Multihedrality in Tilings

Tight fans and their canonical scalings

A new class of full fans in an euclidean space - tight fans - is introduced. Such fans are defined using a property of local symmetry in a face of a tiling. Tight fans are related to the theory of parallelotopes in an euclidean space. A theorem is proved that a fan of cells meeting in a given face of a tiling by parallelotopes is a tight fan. A new proof was given for a theorem by Delone on 5 combinatorial types of parallelotopes meeting in a common face of codimension 3. Canonical scalings of an euclidean space tiling are special functions defined on hyperfaces of the tiling. Existance of suc…

具有无限局部复杂性与 n 重旋转对称性的镶嵌，n=13,17,21

Tilings with Infinite Local Complexity and n-Fold Rotational Symmetry, n=13,17,21

本文通过构造非 Pisot 替换规则并利用 Danzer 算法，证明了存在具有无限局部复杂性的镶嵌，且其镶嵌空间中包含 n=13、17、21 重全局旋转对称性的镶嵌。

A metric characterisation of repulsive tilings

A tiling of $\mathbb{R}^d$ is repulsive if no $r$-patch can repeat arbitrarily close to itself, relative to $r$. This is a characteristic property of aperiodic order, for a non repulsive tiling has arbitrarily large local periodic patterns. We consider an aperiodic, repetitive tiling $T$ of $\mathbb{R}^d$, with finite local complexity. From a spectral triple built on the discrete hull $Ξ$ of $T$, and its Connes distance, we derive two metrics $d_{sup}$ and $d_{inf}$ on $Ξ$. We show that $T$ is repulsive if and only if $d_{sup}$ and $d_{inf}$ are Lipschitz equivalent. This generalises previous …

平面八边形铺砌的弱局部规则

Weak local rules for planar octagonal tilings

本文刻画了允许弱局部规则的平面八边形铺砌，并证明它们都基于二次无理数，从而证实了 Thang Le 的猜想。

Metrics on tiling spaces, local isomorphism and an application of Brown's Lemma

We give an application of a topological dynamics version of multidimensional Brown's lemma to tiling theory: given a tiling of an Euclidean space and a finite geometric pattern of points $F$, one can find a patch such that, for each scale factor $λ$, there is a vector $\vec{t}_λ$ so that copies of this patch appear in the tilling "nearly" centered on $λF+\vec{t}_λ$ once we allow "bounded perturbations" in the structure of the homothetic copies of $F$. Furthermore, we introduce a new unifying setting for the study of tiling spaces which allows rather general group "actions" on patches and we di…

An Approximation Theorem for Maps Between Tiling Spaces

We show that every continuous map from one translationally finite tiling space to another can be approximated by a local map. If two local maps are homotopic, then the homotopy can be chosen so that every interpolating map is also local.

Generalized $β$-expansions, substitution tilings, and local finiteness

For a fairly general class of two-dimensional tiling substitutions, we prove that if the length expansion $β$ is a Pisot number, then the tilings defined by the substitution must be locally finite. We also give a simple example of a two-dimensional substitution on rectangular tiles, with a non-Pisot length expansion $β$, such that no tiling admitted by the substitution is locally finite. The proofs of both results are effectively one-dimensional and involve the idea of a certain type of generalized $β$-transformation.

A Game of Life on Penrose tilings

We define rules for cellular automata played on quasiperiodic tilings of the plane arising from the multigrid method in such a way that these cellular automata are isomorphic to Conway's Game of Life. Although these tilings are nonperiodic, determining the next state of each tile is a local computation, requiring only knowledge of the local structure of the tiling and the states of finitely many nearby tiles. As an example, we show a version of a "glider" moving through a region of a Penrose tiling. This constitutes a potential theoretical framework for a method of executing computations in no…

A General Framework for tilings, Delone sets, functions and measures, and their interrelation

We define a general framework that includes objects such as tilings, Delone sets, functions and measures. We define local derivability and mutual local derivability (MLD) between any two of these objects in order to describe their interrelation. This is a generalization of the local derivability and MLD (or S-MLD) for tilings and Delone sets which are used in the literature, under a mild assumption. We show that several canonical maps in aperiodic order send an object P to one that is MLD with P. Moreover we show that, for an object P and a class S of objects, a mild condition on them assures …