Penrose Tiling Research Articles

Embedding Quasicrystals in a Periodic Cell: Dynamics in Quasiperiodic Structures

We introduce a construction to "periodize" a quasiperiodic lattice of obstacles, i.e., embed it into a unit cell in a higher-dimensional space, reversing the projection method used to form quasilattices. This gives an algorithm for simulating dynamics, as well as a natural notion of uniform distribution, in quasiperiodic structures. It also shows the generic existence of channels, where particles travel without colliding, up to a critical obstacle radius, which we calculate for a Penrose tiling. As an application, we find superdiffusion in the presence of channels, and a subdiffusive regime when obstacles overlap.

Quasicrystals—The impact of N.G. de Bruijn

In this paper we put the work of Professor N.G. de Bruijn on quasicrystals in historical context. After briefly discussing what went before, we shall review de Bruijn’s work together with recent related theoretical and experimental developments. We conclude with a discussion of Yang–Baxter integrable models on Penrose tilings, for which essential use of de Bruijn’s work has been made.

Bond percolation on isoradial graphs: criticality and universality

In an investigation of percolation on isoradial graphs, we prove the criticality of canonical bond percolation on isoradial embeddings of planar graphs, thus extending celebrated earlier results for homogeneous and inhomogeneous square, triangular, and other lattices. This is achieved via the star–triangle transformation, by transporting the box-crossing property across the family of isoradial graphs. As a consequence, we obtain the universality of these models at the critical point, in the sense that the one-arm and \(2j\)-alternating-arm critical exponents (and therefore also the connectivity and volume exponents) are constant across the family of such percolation processes. The isoradial graphs in question are those that satisfy certain weak conditions on their embedding and on their track system. This class of graphs includes, for example, isoradial embeddings of periodic graphs, and graphs derived from rhombic Penrose tilings.

Functions of substitution tilings as a Jacobian

In this paper we show that the function defined by a primitive, star shaped substitution tiling of the plane, can be realized as a Jacobian of a biLipschitz homeomorphism of R 2 . In particular it holds for any Penrose tiling.

Cubulating rhombus groups

We study CAT(0) 2-complexes all of whose polygons are Euclidean rhombi. We describe walls in these complexes, and examine the dual CAT(0) cube complex. Our viewpoint relates Sageev’s dual cube complexes to de Bruijn’s approach to the Penrose tilings

Symmetry of quasicrystals

The definition of an aperiodic crystal (quasicrystal) as a solid that is characterized by the forbidden symmetry suggests the existence of an unsolved problem, because, in a mutually exclusive manner, it appeals to the fundamental theorem of classical crystallography. Using the Penrose tiling as an example, we have investigated the symmetry properties of aperiodic tilings for the purpose to establish the allowed symmetry groups of quasicrystals. The filling of the Euclidean space according to an aperiodic law is considered as the action of an infinite number of group elements on a fundamental domain in the non-Euclidean space. It is concluded that all locally equivalent tilings have a common “parent” structure and, consequently, the same symmetry group. An idealized object, namely, an infinitely refined tiling, is introduced. It is shown that the symmetry operations of this object are operations of the similarity (rotational homothety). A positive answer is given to the question about a possible composition of operations of the similarity with different singular points. It is demonstrated that the transformations of orientation-preserving aperiodic crystals are isomorphic to a discrete subgroup of the Mobius group PSL(2, ℂ); i.e., they can be realized as discrete subgroups of the full group of motions in the Lobachevsky space. The problem of classification of the allowed types of aperiodic tilings is reduced to the procedure of enumeration of the aforementioned discrete subgroups.

The combinatorics of N.G. de Bruijn

In memoriam: N.G. de Bruijn.In this article we present a survey of his papers on combinatorics. The section titles show its variety. 1.Common systems of representatives2.De Bruijn cycles3.The De Bruijn–Erdös theorem from incidence geometry4.Bases for integers5.The BEST theorem6.The De Bruijn–Erdös theorem from graph theory7.Factorizations of finite groups8.Rooted trees in the plane9.Permutations of a given shape10.Covering of graphs by dimers11.Counting (Pólya’s fundamental theorem, Color designs)12.Penrose tilings

Girih 타일링을 이용한 초등수학영재 프로그램 개발 및 적용 연구

The purpose of this study is to develop a new program for elementary math-gifted students by using `Girih Tililng` and apply it to the elementary students to improve their math-ability. Girih Tililng is well known for `the secrets of mathematics hidden in Mosque decoration` with lots of recent attention from the world. The process of this study is as follows; (1) Reference research has been done for various tiling theories and the theories have been utilized for making this study applicable. (2) The characteristic features of Mosque tiles and their basic structures have been analyzed. After logical examination of the patterns, their mathematic attributes have been found out. (3) After development of Girih tiling program, the program has been applied to math-gifted students and the program has been modified and complemented. This program which has been developed for math-gifted students is called `Exploring the Secrets of Girih Hidden in Mosque Patterns`. The program was based on the Renzulli`s three-part in-depth learning. The first part of the in-depth learning activity, as a research stage, is designed to examine Islamic patterns in various ways and get the gifted students to understand and have them motivated to learn the concept of the tiling, understanding the characteristics of Islamic patterns, investigating Islamic design, and experiencing the Girih tiles. The second part of the in-depth learning activity, as a discovery stage, is focused on investigating the mathematical features of the Girih tile, comparing Girih tiled patterns with non-Girih tiled ones, investigating the mathematical characteristics of the five Girih tiles, and filling out the blank of Islamic patterns. The third part of the in-depth learning activity, as an inquiry or a creative stage, is planned to show the students` mathematical creativity by thinking over different types of Girih tiling, making the students` own tile patterns, presenting artifacts and reflecting over production process. This program was applied to 6 students who were enrolled in an unified(math and science) gifted class of D elementary school in Daejeon. After analyzing the results produced by its application, the program was modified and complemented repeatedly. It is expected that this program and its materials used in this study will guide a direction of how to develop methodical materials for math-gifted education in elementary schools. This program is originally developed for gifted education in elementary schools, but for further study, it is hoped that this study and the program will be also utilized in the field of math-gifted or unified gifted education in secondary schools in connection with `Penrose Tiling` or material of `quasi-crystal`.

Fractal structures of regular pentagonal stars in Penrose tiling

Fractal structures of regular pentagonal stars are proposed. Some of these structures are detected in Penrose tiling. The fractals are based on generalized stars, whose vertices generate the centers of smaller size stars. There are two possible orientations of generalized stars interrelated by the operation of inversion in a point. The generalized stars to initial star linear sizes ratios are equal to integer degrees of golden mean.

Surface scaling properties of decagonal quasicrystals under nonequilibrium vertical growth

Restricted solid-on-solid (RSOS) growth models are studied on two different decagonal quasicrystal lattices, namely the Penrose tiling lattice and the random tiling lattice. There exist two types of growth blocks-fat and skinny tiles-which may have different sticking probabilities. We found that the RSOS growths on both lattices belong to the Kardar-Parisi-Zhang universality class when they have the same sticking probabilities in spite of the lack of periodicity in the substrates. However, when they have tile-type dependent sticking probabilities, the RSOS models on two lattices may produce different scaling behaviors. Growth on Penrose tiling shows that the roughness exponent is around 0.4 while that on random tiling is around 0.49. Our observation may provide an effective way to investigate the bulk structures of decagonal quasicrystals.

Structural transformations in quasicrystals induced by higher dimensional lattice transitions

We study the structural transformations induced, via the cut-and-project method, in quasicrystals and tilings by lattice transitions in higher dimensions, with a focus on transition paths preserving at least some symmetry in intermediate lattices. We discuss the effect of such transformations on planar aperiodic Penrose tilings, and on three-dimensional aperiodic Ammann tilings with icosahedral symmetry. We find that locally the transformations in the aperiodic structures occur through the mechanisms of tile splitting, tile flipping and tile merger, and we investigate the origin of these local transformation mechanisms within the projection framework.

Statistical mechanics of two dimensional tilings

Reduced dimensionality in two dimensions is a topic of current interest. We use model systems to investigate the statistical mechanics of ideal networks. The tilings have possible applications such as the 2D locations of pore sites in nanoporous arrays (quantum dots), in the 2D hexagonal structure of graphene, and as adsorbates on quasicrystalline crystal surfaces. We calculate the statistical mechanics of these networks, such as the partition function, free energy, entropy, and enthalpy. The plots of these functions versus the number of links in the finite networks result in power law regression. We also determine the degree distribution, which is a combination of power law and rational function behavior. In the large-scale limit, the degree of these 2D networks approaches 3, 4, and 6, in agreement with the degree of the regular tilings. In comparison, a Penrose tiling has a degree also equal to about 4.

Trees, Ultrametrics, and Noncommutative Geometry

Noncommutative geometry is used to study the local geometry of ultrametric spaces and the geometry of trees at infinity. Connes’s example of the noncommutative space of Penrose tilings is interpreted as a non-Hausdorff orbit space of a compact, ultrametric space under the action of its local isometry group. This is generalized to compact, locally rigid, ultrametric spaces. The local isometry types and the local similarity types in those spaces can be analyzed using groupoid C∗-algebras. The concept of a locally rigid action of a countable group Γ on a compact, ultrametric space by local similarities is introduced. It is proved that there is a faithful unitary representation of Γ into the germ groupoid C∗-algebra of the action. The prototypical example is the standard action of Thompson’s group V on the ultrametric Cantor set. In this case, the C∗-algebra is the Cuntz algebra O2 and representations originally due to Birget and Nekrashevych are recovered. End spaces of trees are sources of ultrametric spaces. Some connections are made between locally rigid, ultrametric spaces and a concept in the theory of tree lattices of Bass and Lubotzky.

Real Space Structure Factor for Different Quasicrystals

AbstractThe statistical approach to the description of aperiodic structures using the concept of the so called Average Unit Cell (AUC) is presented. The use of this method is shown for 1D, 2D, and 3D structures. We start with the presentation of the basic ideas for 1D simple sinusoidal modulation (1q) and the Fibonacci chain (a 1D model of a quasicrystalline structure). Up to now the AUC concept has been most thoroughly studied for 2D decagonal quasicrystals. The idea of structure factor derivation for the decorated Penrose tiling, a quasilattice for the description of decagonal quasicrystals, is presented. Two examples of structure refinement of decagonal phases from the Al‐Ni‐Co system are discussed. The potential of the cluster description of decagonal structures is also presented. The AUC construction and structure factor derivation is shown for the 3D Amman–Kramer–Neri tiling, which is a model quasilattice for the description of icosahedral phases. The AUC concept can also be extended to structures with singular‐continuous Fourier spectrum. An example for the 1D Thue–Morse sequence is studied.

Quasicrystals: Sections of Hyperspace

Since the moment Dan Shechtman discovered a material with an ordered but aperiodic (“quasiperiodic”) structure, which was believed to be impossible at the time, he began a long battle to convince his colleagues that his data were real. His efforts were finally rewarded with a Nobel Prize. The picture shows the projected electron density distribution function of decagonal Al-Co-Ni with Penrose tiling as an example of a quasicrystal.

Crystal and electronic structure study of AgPd 3Se

The AgPd 3Se compound was synthesised from individual elements by solid-state chemical reactions and structurally characterized by powder X-ray diffraction data. AgPd 3Se displays cubic symmetry, space group Pa3̄, unit cell parameter a=8.6289(1) Å and Z=8. Double-Friauf polyhedra (DFP), defined by Pd and Se atoms, form the basic structural building block of the AgPd 3Se crystal structure. The Ag atoms occupy the centres of DFPs forming an Ag–Ag dimer (2.792(2) Å). The packing of DFPs forms two kinds of interpenetrating networks that show similar features as three-dimensional Penrose tiles. AgPd 3Se is isostructural with CaAu 3Ga. The electric resistivity as well as the electronic structure calculation suggests metallic behaviour.

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.

INTERFACIAL ENERGIES ON PENROSE LATTICES

In this paper we prove a homogenization theorem for interfacial discrete energies defined on an a-periodic Penrose tiling in ℝ2. A general result on the homogenization of surface energies cannot be directly adapted to this case; the existence of the limit interfacial energy is therefore proved by showing some refined "quasi-periodic" properties of the tilings.

Geometry of non-Hausdorff spaces and its significance for physics

Hausdorff relation, topologically identifying points in a given space, belongs to elementary tools of modern mathematics. We show that if subtle enough mathematical methods are used to analyze this relation, the conclusions may be far-reaching and illuminating. Examples of situations in which the Hausdorff relation is of the total type, i.e., when it identifies all points of the considered space, are the space of Penrose tilings and space-times of some cosmological models with strong curvature singularities. With every Hausdorff relation a groupoid can be associated, and a convolutive algebra defined on it allows one to analyze the space that otherwise would remain intractable. The regular representation of this algebra in a bundle of Hilbert spaces leads to a von Neumann algebra of random operators. In this way, a probabilistic description (in a generalized sense) naturally takes over when the concept of point looses its meaning. In this situation counterparts of the position and momentum operators can be defined, and they satisfy a commutation relation which, in the suitable limiting case, reproduces the Heisenberg indeterminacy relation. It should be emphasized that this is neither an additional assumption nor an effect of a quantization process, but simply the consequence of a purely geometric analysis.

Rigidity and mapping class group for abstract tiling spaces

AbstractWe study abstract self-affine tiling actions, which are an intrinsically defined class of minimal expansive actions of ℝdon a compact space. They include the translation actions on the compact spaces associated to aperiodic repetitive tilings or Delone sets in ℝd. In the self-similar case, we show that the existence of a homeomorphism between tiling spaces implies conjugacy of the actions up to a linear rescaling. We also introduce the general linear group of an (abstract) tiling, prove its discreteness, and show that it is naturally isomorphic with the (pointed) mapping class group of the tiling space. To illustrate our theory, we compute the mapping class group for a five-fold symmetric Penrose tiling.