The Pinwheel Tilings of the Plane

Radin, Charles

doi:10.2307/2118575

Cited by 148 publications

(133 citation statements)

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“…An example of a system that is statistically isotropic is a tiling of identical triangles [19] called the 'Pinwheel' tiling ( Fig 6) . The Pinwheel tiling can be constructed by recursive substitution rules, so patterns of size repeat (modulo rotation) every D ∼ in a perfect tiling, as for the 1D sequences discussed above.…”

Section: Quasiperiodicmentioning

confidence: 99%

Order in glassy systems

Kurchan¹,

Levine²

2010

J. Phys. A: Math. Theor.

140

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A directly measurable correlation length may be defined for systems having a two-step relaxation, based on the geometric properties of density profile that remains after averaging out the fast motion. We argue that the length diverges if and when the slow timescale diverges, whatever the microscopic mechanism at the origin of the slowing down. Measuring the length amounts to determining explicitly the complexity from the observed particle configurations. One may compute in the same way the Renyi complexities Kq, their relative behavior for different q characterizes the mechanism underlying the transition. In particular, the 'Random First Order' scenario predicts that in the glass phase Kq = 0 for q > x, and Kq > 0 for q < x, with x the Parisi parameter. The hypothesis of a nonequilibrium effective temperature may also be directly tested directly from configurations.

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Section: Quasiperiodicmentioning

confidence: 99%

Order in glassy systems

Kurchan¹,

Levine²

2010

J. Phys. A: Math. Theor.

140

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“…A prominent one is the Conway-Radin pinwheel tiling. 106 This tiling is based on a single triangular prototile (of edge lengths 1, 2 and √ 5), with an inflation rule of linear inflation multiplier √ 5, so each re-scaled triangle (which is planar) is dissected into five congruent copies. Figure 35 shows a photograph of a patch of the tiling, which has been used as a theme for Melbourne's Federation Square development.…”

Section: Discussionmentioning

confidence: 99%

Mathematical diffraction of aperiodic structures

Baake

Grimm

2012

Chem. Soc. Rev.

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Kinematic diffraction is well suited for a mathematical approach via measures, which has substantially been developed since the discovery of quasicrystals. The need for further insight emerged from the question of which distributions of matter, beyond perfect crystals, lead to pure point diffraction, hence to sharp Bragg peaks only. More recently, it has become apparent that one also has to study continuous diffraction in more detail, with a careful analysis of the different types of diffuse scattering involved. In this review, we summarise some key results, with particular emphasis on non-periodic structures. We choose an exposition on the basis of characteristic examples, while we refer to the existing literature for proofs and further details. IntroductionDiffraction techniques have dominated the structure analysis of solids for the last century, ever since von Laue and Bragg employed X-ray diffraction to determine the atomic structure of crystalline materials. Despite the availability of direct imaging techniques such as electron and atomic force microscopy, diffraction by X-rays, electrons and neutrons continues to be the method of choice to detect order in the atomic arrangements of a substance; see Cowley's book 35 and references therein for background.In its full generality, the diffraction of a beam of X-rays, electrons or neutrons from a macroscopic piece of solid is a complicated physical process. It is the presence of inelastic and multiple scattering, prevalent particularly in electron diffraction, which makes it essentially impossible to arrive at a complete mathematical description of the process. Here, we restrict to kinematic diffraction in the far-field or Fraunhofer limit. In this case, powerful tools of harmonic analysis are available to attack the direct problem of calculating the (kinematic) diffraction pattern of a given structure.In contrast, the inverse problem of determining a structure from its diffraction intensities is extremely involved. A diffraction pattern rarely determines a structure uniquely, as there can be homometric structures sharing the same autocorrelation (and hence the same diffraction). 9,54,57,103 We are far away from a complete understanding of the homometry classes of structures, in particular if the diffraction spectrum contains continuous components. At present, a picture is emerging, based on the analysis of explicit examples, which highlight how large the homometry classes may be.Originally, much of the effort concentrated on the pure point part of diffraction, also called the Bragg diffraction, for † Part of a themed issue on Quasicrystals in honour of the 2011 Nobel Prize in Chemistry winner, Professor Dan Shechtman.

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“…, it can be subdivided into five pinwheel triangles of the original size (see Figure 1). This is known as the pinwheel inflate-and-subdivide rule, or more simply as the pinwheel substitution rule [9]. (Readers who wish to make drawings for yourselves: notice that all of the images in this paper are oriented with a standard triangle at the origin and the origin marked.)…”

mentioning

confidence: 99%

“…Another surprising property of pinwheel tilings is that the hierarchical structure mandated by the inflate-and-subdivide rule can be enforced by local constraints called matching rules [9], decorations on the edges of tiles that specify how they are allowed to meet up. Although many famous tilings, for instance the Penrose tilings, were known to come equipped with matching rules that force the hierarchical structure, this was the first example for which the matching rules also enforced infinite rotations.…”

mentioning

confidence: 99%

“…Although many famous tilings, for instance the Penrose tilings, were known to come equipped with matching rules that force the hierarchical structure, this was the first example for which the matching rules also enforced infinite rotations. In [9], a new set of triangles is constructed by making numerous copies of the pinwheel triangles, each with markings on their edges that specify how they are allowed to meet.…”

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confidence: 99%

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A Fractal Version of the Pinwheel Tiling

Frank

Whittaker

2011

Math Intelligencer

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Dedicated to the inspiration of Benoit Mandelbrot.The pinwheel tilings are a remarkable class of tilings of the plane, and our main goal in this paper 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 fruit- The technique for finding fractiles in both [3] and [8] is similar. One begins with an inflate-andsubdivide 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 an example). The edges of each tile are redrawn using the edges of the tiles that replace it. These new edges are revised iteratively by the The authors would like to thank Dirk Frettlöh for helpful discussions about the aorta and Edmund Harriss for pointing out our theorem on rotations in the aorta. The second author is partially supported by ARC grant 228-37-1021, Australia.

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The Pinwheel Tilings of the Plane

Cited by 148 publications

References 9 publications

Order in glassy systems

Order in glassy systems

Mathematical diffraction of aperiodic structures

A Fractal Version of the Pinwheel Tiling

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