Pinwheel Tilings Research Articles

Multiscale substitution tilings

We introduce a new general framework for constructing tilings of Euclidean space, which we call multiscale substitution tilings. These tilings are generated by substitution schemes on a finite set of prototiles, in which multiple distinct scaling constants are allowed. This is in contrast to the standard case of the well-studied substitution tilings which includes examples such as the Penrose and the pinwheel tilings. Under an additional irrationality assumption on the scaling constants, our construction defines a new class of tilings and tiling spaces, which are intrinsically different from those that arise in the standard setup. We study various structural, geometric, statistical, and dynamical aspects of these new objects and establish a wide variety of properties. Among our main results are explicit density formulas and the unique ergodicity of the associated tiling dynamical systems.

Fusion: a General Framework for Hierarchical Tilings

One well studied way to construct quasicrystalline tilings is via in ate-and-subdivide (a.k.a. substitution) rules. These produce self-similar tilings the Penrose, octagonal, and pinwheel tilings are famous examples. We present a di erent model for generating hierarchical tilings we call fusion rules . In ate-and-subdivide rules are a special case of fusion rules, but general fusion rules are more exible and allow for defects, changes in geometry, and even constrained randomness. A condition that produces homogeneous structures and a method for computing frequency for fusion tiling spaces are discussed.

K-Theory of Crossed Products of Tiling C*-Algebras by Rotation Groups

Let $\Omega$ be a tiling space and let $G$ be the maximal group of rotations which fixes $\Omega$. Then the cohomology of $\Omega$ and $\Omega/G$ are both invariants which give useful geometric information about the tilings in $\Omega$. The noncommutative analog of the cohomology of $\Omega$ is the K-theory of a C*-algebra associated to $\Omega$, and for translationally finite tilings of dimension 2 or less the K-theory is isomorphic to the direct sum of cohomology groups. In this paper we give a prescription for calculating the noncommutative analog of the cohomology of $\Omega/G$, that is, the K-theory of the crossed product of the tiling C*-algebra by $G$. We also provide a table with some calculated K-groups for many common examples, including the Penrose and pinwheel tilings.

A Fractal Version of the Pinwheel Tiling

T he pinwheel tilings are a remarkable class of tilings of the plane, and our main goal in this article is to introduce a fractal version of them. We will begin by describing how to construct the pinwheel tilings themselves and by discussing some of the properties that have generated so much interest. After that we will develop the fractal version and discuss some of its properties. Finding this fractile version was an inherently interesting problem, and the solution we found is unusual in the tiling literature. Like the well-known Penrose tilings [5], pinwheel tilings are generated by an ‘‘inflate-and-subdivide rule’’ (see Figure 1). Tilings generated by inflate-and-subdivide rules form a class of tilings that have a considerable amount of global structure called self-similarity. Self-similar tilings are usually nonperiodic but still exhibit a form of ‘‘long-range order’’ that makes their study particularly fruitful. Unlike Penrose tilings and most known examples of self-similar tilings, tiles in any pinwheel tiling appear in infinitely many different orientations. The pinwheel tilings were the first example of this sort and as such presented both new challenges and intriguing properties. Many examples of self-similar tilings are made of fractiles: tileswith fractal boundaries. Fractiles arise in the foundational work [7] for constructing a self-similar tiling for a given inflation factor. Two fractile versions of thePenrose tilings are introduced in [3]. Additionally, the procedure used in [8] may result in self-similar tilings made up of fractiles. This made it reasonable to expect that the pinwheel tilings might have a fractal variant, but did not provide a template for finding it. The technique for finding fractiles in both [3] and [8] is similar. One begins with an inflate-and-subdivide rule for which the edges of each inflated tile do not quite match up with the edges of the tiles that replace it (the Penrose kite and dart are anexample). The edgesof each tile are redrawnusing the edges of the tiles that replace it. These new edges are revised iteratively by the subdivision rule ad infinitum. The final result is a set of fractiles that are redrawings of the original tiles, but now they inflate-and-subdivide perfectly. Our technique is completely different.We found a fractal that runs through the interior of apinwheel triangle andbehavesnicely under the inflate-and-subdivide rule. The fact that this fractal extends to become the boundary of fractiles follows naturally (with some work) from the pinwheel inflation, but there was no way to know how many types of fractiles to expect. It was only by creating the images by computer that we were able to generate enough information to answer that question.

Some comments on pinwheel tilings and their diffraction

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 the pinwheel diffraction consists of sharp rings and, possibly, a continuous component with circular symmetry. We consider some combinatorial properties of the tiles and their orientations, and a numerical approach to the diffraction of weighted pinwheel point sets.

PV Cohomology of the Pinwheel Tilings, their Integer Group of Coinvariants and Gap-Labeling

In this paper, we first remind how we can see the “hull” of the pinwheel tiling as an inverse limit of simplicial complexes (Anderson and Putnam in Ergod Th Dynam Sys 18:509–537, 1998) and we then adapt the PV cohomology introduced in Savinien and Bellissard (Ergod Th Dynam Sys 29:997–1031, 2009) to define it for pinwheel tilings. We then prove that this cohomology is isomorphic to the integer Čech cohomology of the quotient of the hull by S 1 which let us prove that the top integer Čech cohomology of the hull is in fact the integer group of coinvariants of the canonical transversal Ξ of the hull. The gap-labeling for pinwheel tilings is then proved and we end this article by an explicit computation of this gap-labeling, showing that \({\mu^t \left( C(\Xi,\mathbb {Z}) \right) = \frac{1}{264}\mathbb {Z} \left [ \frac{1}{5}\right ]}\).

An Algorithm for Two‐Dimensional Mesh Generation Based on the Pinwheel Tiling

We propose a new two‐dimensional meshing algorithm called PINW able to generate meshes that accurately approximate the distance between any two domain points by paths composed only of cell edges. This technique is based on an extension of pinwheel tilings proposed by Radin and Conway (see [C. Radin, Ann.of Math. (2), 139 (1994), pp. 661–702]). We prove that the algorithm produces triangles of bounded aspect ratio. This kind of mesh would be useful in cohesive interface finite element modeling when the crack propagation path is an outcome of a simulation process.

The isoperimetric problem for pinwheel tilings

In aperiodic “pinwheel” tilings of the plane there exist unions of tiles with ratio (area)/(perimeter)2 arbitrarily close to that of a circle. Such approximate circles can be constructed with arbitrary center and any sufficiently large radius. The existence of such circles follows from the metric on pinwheel space being almost Euclidean at large distances; ifP andQ are points separated by large Euclidean distanceR, then the shortest path along tile edges fromP toQ has lengthR+o(R).

Space tilings and substitutions

We generalize the study of symbolic dynamical systems of finite type and ℤ2 action, and the associated use of symbolic substitution dynamical systems, to dynamical systems with ℝ2 action. The new systems are associated with tilings of the plane. We generalize the classical technique of the matrix of a substitution to include the geometrical information needed to study tilings, and we utilize rotation invariance to eliminate discrete spectrum. As an example we prove that the pinwheel tilings have no discrete spectrum.

The Pinwheel Tilings of the Plane

prototiles) in Euclidean n-space, En, for n > 2. The prototiles are usually required to be rather nice topologically, at least homeomorphs of the closed unit ball. One then makes arbitrarily many congruent copies, called tiles, of these prototiles, and considers all ways (called tilings) that such tiles may provide a simultaneous covering and packing of En; a tiling is thus an unordered collection of tiles for which the union is all of En, but such that the interiors of each pair of tiles do not intersect. Wang's original problem was to determine if it was possible to have a finite set of prototiles, with associated tilings of space, for which all the tilings were nonperiodic. (A tiling is nonperiodic if it is not invariant under any single simultaneous translation of its tiles.) This was settled in the affirmative in the thesis of Wang's student Robert Berger [1].

Order versus symmetry in pinwheel tilings of the plane

This article calculates the Fourier transform of a quasicrystalline solid, the pinwheel tiling, proposed by Conway and Radin. The solid is isotropic, but the Fourier transform has large fluctuations as a consequence of correlations.