Penrose Tiling Research Articles

Average unit cell for the Generalized Penrose Tiling

The Generalized Penrose Tiling (GPT) can be considered a promising alternative for Penrose Tiling (PT) as a model for decagonal quasicrystal refinement procedure, particularly in the statistical approach (also called the Average Unit Cell (AUC) approach) [1]. The statistical method using PT has been successfully applied to the structure optimization of various decagonal phases [2]. The application of the AUC concept to the GPT will be presented. In the higher dimensional (nD) approach, PT is obtained by projecting a 5D hypercubic lattice through a window consisting of four pentagons, called the atomic surfaces (ASs), in the perpendicular space. The vertices of these pentagons together with two additional points form a rhombicosahedron. The GPT is created by projecting the 5D hypercubic lattice through a window consisting of five polygons, generated by shifting the ASs along the body diagonal of the rhombicosahedron. Three of the previously pentagonal ASs will become decagon, one will remain pentagonal and one more pentagon will be created (for PT it is a single point). The structure of GPT will depend on the shift parameter, its building units are still thick and thin rhombuses, but the matching rules and the tiling changes. In the AUC concept the probability distribution for rhombuses of PT can be obtained as an oblique projection of the ASs on the physical space. This holds true also for the GPT. The derivation of the AUC distribution for a given type of rhomb in a given orientation of an arbitrarily chosen GPT will be presented. In the PT, these distributions are triangular, whereas in the case of the GPT they are triangular (originating from the pentagonal AS) or hexagonal (originating from the decagonal AS). The AUC of GPT for shift parameters 0.2 and 0.5 has been calculated. The derivation of the analytical formula for structure factor using AUC formalism, for the decorated GPT is made similarly to the calculation for the PT [3].

The phase problem for quasicrystals in reciprocal space

The phase problem is very well-known in crystallography and is particularly important for structure solution of quasicrystals. Structure solution ( initial phasing of the diffraction pattern) is the first step of atomic structure determination against the diffraction data. Many tools for solving the phase problem in crystallography were developed over the years. Besides the pioneer Patherson function method or direct methods, also the low density elimination method and, more recently, maximum entropy method or charge flipping algorithm are widely used. We propose another way of phase recovery, directly from the diffraction pattern. It has been shown, that diffraction patterns of aperiodic structures (quasicrystals and modulated structures) consists of periodic series of peaks [1,2]. The peaks are, of course, aperiodically distributed in reciprocal space. However, the envelopes of such peaks are strictly periodic. For Fibonacci sequence the shape of envelopes is given by a sinx/x – type function. In the centrosymmetric case all peaks belonging to a given series have the same phase (0 or π). Moreover, once we rescale peak positions to obtain the reduced envelope of structure factor (intensity), we can easily find the shape of the average unit cell, by applying the inverse Fourier transform. The functionality of such approach has been shown in [3] for d-AlNiCo quasicrystal. In this paper we show a few examples of model structures with recovered phases. Decorated Fibonacci sequence, Penrose tiling and Ammann tiling are used as model structures for 1D, 2D, and 3D quasicrystals respectively. We compare the results with conventionally used charge flipping method.

Aperiodic crystals and beyond

Following Dan Shechtman's discovery of quasicrystals in 1982, the realm of crystallography has been extended to include structures that lack translational periodicity. While periodic crystals can be modelled as decorations of lattices, aperiodic crystals require more general discrete structures such as point sets or tilings in space for the description of their structure. For a mathematical introduction to the field of aperiodic order, we refer to the recent monograph [1]. Interesting examples are obtained by projections from higher-dimensional lattices, leading to model sets which have pure Bragg diffraction, though the Bragg peaks are, in general, dense in space. All symmetries that have been experimentally observed in quasicrystals can be reproduced in this way, and some of the resulting structures are standard examples of tilings that are frequently used in quasicrystal modelling, such as the famous Penrose tiling. But there exists a plethora of ordered structures beyond cut and project sets, some of which have even weirder properties. After a general introduction to aperiodically ordered structures, a couple of examples of such systems are briefly described, offering a glimpse at the largely unexplored world of order beyond (aperiodic) crystals.

Construction of quasiperiodic patterns in the Moroccan ornamental art

The similarity between the structure of Islamic decorative patterns and quasicrystal, aroused the interest of several crystallographers. They analyzed these patterns, by different approaches, various kinds of ornamental quasiperiodic patterns encountered in the Morocco and Alhambra (Andalusia) as well as in the eastern of Islamic world. In this work, we are interesting in the quasiperiodic patterns found in several Moroccan historical buildings constructed in the 14th century. We first describe the Zellige panels (fine mosaics) decorating the Madrasas (schools) Attarine and Bou Inania in Fez in term of Penrose tiling, to confirm that both panels have quasiperiodic structure (Makovicky et al, 1998). The panel Madrasas Attarine appears as a finite part of this quasiperiodic pattern (Figure 1-c). As already mentioned by several authors, we can notice the similarity of the decagonal pattern with the diffraction pattern of the quasicrystal Al Mn (Schechtman et al, 1984) (Figure 1-a and c). The multigrid method developed by De Bruijn (1981) and reformulated by Gratias (2002) to obtain a quasiperiodic paving, is used to construct known quasiperiodic patterns from periodic patterns extracted from the Madrasas Bou Inania and Ben Youssef (Marrakech). At last, we propose a method of construction of heptagonal, enneagonal, tetradecagonal and octadecagonal quasiperiodic patterns, not encountered in the Moroccan ornamental art. They are built from tiling (skeleton) generated by the multigrid method and decorated by motifs obtained by the craftsmen method.

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.

Direct and Reciprocal Space Properties of the Generalized Penrose Tiling

Direct and Reciprocal Space Properties of the Generalized Penrose Tiling

Electrons in deterministic quasicrystalline potentials and hidden conserved quantities

We propose an ansatz for the wave function of a non-interacting quantum particle in a deterministic quasicrystalline potential. It is applicable to both continuous and discrete models and includes Sutherlandʼs hierarchical wave function as a special case. The ansatz is parameterized by a first cohomology class of the hull of the structure. The structure of the ansatz and the values of its parameters are preserved by the time evolution. Numerical results suggest that the ground states of the standard vertex models on Ammann–Beenker and Penrose tilings belong to this class of functions. This property remains valid for the models perturbed within their mutual local derivability class, e.g. by adding links along diagonals of rhombi. The convergence of the numerical simulations in a finite patch of the tiling critically depends on the boundary conditions, and can be significantly improved when the choice of the latter respects the structure of the ansatz.

The space of Penrose tilings and the noncommutative curve with homogeneous coordinate ring $k\langle x,y\rangle /(y^2)$

It is shown that the noncommutative algebraic curve with homogeneous coordinate ring \mathbb{C}\langle x,y\rangle/(y^2) is a noncommutative algebraic-geometric analogue of the space of Penrose tilings of the plane. Individual tilings determine “points” on the noncommutative curve and the tilings coincide under isometry if and only if the skyscraper sheaves of the corresponding points are isomorphic. The category of quasi-coherent sheaves on the curve is equivalent to the category of modules over a von Neumann regular ring that is a direct limit of finite dimensional semisimple algebras. The norm closure of this von Neumann regular ring is the AF-algebra that Connes associates to the space of Penrose tilings. There is an algebraic analogue of the fact that every isometry-invariant subset of tilings is dense in the set of all Penrose tilings.

The computation of overlap coincidence in Taylor--Socolar substitution tiling

Abstract. Recently Taylor and Socolar introduced an aperiodic mono-tile. The associated tilingcan be viewed as a substitution tiling. We use the substitution rule for this tiling and applythe algorithm of [1] to check overlap coincidence. It turns out that the tiling has overlap coinci-dence. So the tiling dynamics has pure point spectrum and we can conclude that this tiling has aquasicrystalline structure.Keywords: Mono-tile tiling, Quasicrystals, Pure point diffraction/dynamical spectrum, Self-affinetilings, Overlap coincidence. 1. IntroductionAperiodic tiles are the set of prototiles which tile the space with their isomorphic images byEuclidean motions (composition of translations, rotations and reflections) but only in non-periodicway. There have been many examples of aperiodic tiles and study on them [2, 4, 6, 11, 13, 24, 25, 28,29]. Two of well-known examples of aperiodic tiles with simple prototiles up to Euclidean motionsare Penrose tiles and Ammann tiles which are uncovered in the mid 70’s. These sets consist of twoprototiles and it has been the smallest number of prototiles which form aperiodic tiles until recently.Since then, people have been interested in finding a single prototile for an aperiodic tile. This problemis coined as ‘Mono-tile’ problem or ‘Einstein’ problem (one stone in German). It had taken quitesome time before Taylor and Socolar announced in 2010 the existence of an aperiodic mono-tile.Their tile is a hexagonal tile with colored decorations and matching rules which can be embeddedonto a single tile using shape only. Penrose had found earlier a mono-tile using matching rules whichis the reformulation of (1+ǫ+ǫ

Color theorems, chiral domain topology, and magnetic properties of Fe(x)TaS2.

Common mathematical theories can have profound applications in understanding real materials. The intrinsic connection between aperiodic orders observed in the Fibonacci sequence, Penrose tiling, and quasicrystals is a well-known example. Another example is the self-similarity in fractals and dendrites. From transmission electron microscopy experiments, we found that FexTaS2 crystals with x = 1/4 and 1/3 exhibit complicated antiphase and chiral domain structures related to ordering of intercalated Fe ions with 2a × 2a and √3a × √3a superstructures, respectively. These complex domain patterns are found to be deeply related with the four color theorem, stating that four colors are sufficient to identify the countries on a planar map with proper coloring and its variations for two-step proper coloring. Furthermore, the domain topology is closely relevant to their magnetic properties. Our discovery unveils the importance of understanding the global topology of domain configurations in functional materials.

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.

Universal characteristics of fractal fluctuations in prime number distribution

The frequency of occurrence of prime numbers at unit number spacing intervals exhibits self-similar fractal fluctuations concomitant with inverse power law form for power spectrum generic to dynamical systems in nature such as fluid flows, stock market fluctuations and population dynamics. The physics of long-range correlations exhibited by fractals is not yet identified. A recently developed general systems theory visualizes the eddy continuum underlying fractals to result from the growth of large eddies as the integrated mean of enclosed small scale eddies, thereby generating a hierarchy of eddy circulations or an inter-connected network with associated long-range correlations. The model predictions are as follows: (1) The probability distribution and power spectrum of fractals follow the same inverse power law which is a function of the golden mean. The predicted inverse power law distribution is very close to the statistical normal distribution for fluctuations within two standard deviations from the mean of the distribution. (2) Fractals signify quantum-like chaos since variance spectrum represents probability density distribution, a characteristic of quantum systems such as electron or photon. (3) Fractal fluctuations of frequency distribution of prime numbers signify spontaneous organization of underlying continuum number field into the ordered pattern of the quasiperiodic Penrose tiling pattern. The model predictions are in agreement with the probability distributions and power spectra for different sets of frequency of occurrence of prime numbers at unit number interval for successive 1000 numbers. Prime numbers in the first 10 million numbers were used for the study.

Free Path Lengths in Quasicrystals

Previous studies of kinetic transport in the Lorentz gas have been limited to cases where the scatterers are distributed at random (e.g. at the points of a spatial Poisson process) or at the vertices of a Euclidean lattice. In the present paper we investigate quasicrystalline scatterer configurations, which are non-periodic, yet strongly correlated. A famous example is the vertex set of the Penrose tiling. Our main result proves the existence of a limit distribution of the free path length, which answers a question of Wennberg. The limit distribution is characterised by a certain random variable on the space of higher dimensional lattices, and is distinctly different from the exponential distribution observed for random scatterer configurations. The key ingredients in the proofs are equidistribution theorems on homogeneous spaces, which follow from Ratner's measure classification.

Moroccan ornamental quasiperiodic patterns constructed by the multigrid method

The similarity between the structure of Islamic decorative patterns and quasicrystals has aroused the interest of several crystallographers. Many of these patterns have been analysed by different approaches, including various kinds of ornamental quasiperiodic patterns encountered in Morocco and the Alhambra (Andalusia), as well as those in the eastern Islamic world. In the present work, the interest is in the quasiperiodic patterns found in several Moroccan historical buildings constructed in the 14th century. First, thezelligepanels (fine mosaics) decorating the Madrasas (schools) Attarine and Bou Inania in Fez are described in terms of Penrose tiling, to confirm that both panels have a quasiperiodic structure. The multigrid method developed by De Bruijn [Proc. K. Ned. Akad. Wet. Ser. A Math. Sci.(1981),43, 39–66] and reformulated by Gratias [Tangente(2002),85, 34–36] to obtain a quasiperiodic paving is then used to construct known quasiperiodic patterns from periodic patterns extracted from the Madrasas Bou Inania and Ben Youssef (Marrakech). Finally, a method of construction of heptagonal, enneagonal, tetradecagonal and octadecagonal quasiperiodic patterns, not encountered in Moroccan ornamental art, is proposed. They are built from tilings (skeletons) generated by the multigrid method and decorated by motifs obtained by craftsmen.

Alignment verification for electron beam lithography

Alignment between lithography layers is essential for device fabrication. A minor defect in a single marker can lead to incorrect alignment and this can be the source of wafer reworks. In this paper we show that this can be prevented by using extra alignment markers to check the alignment during patterning, rather than inspecting vernier patterns after the exposure is completed. Accurate vernier patterns can often only be read after pattern transfer has been carried out. We also show that by using a Penrose tile as a marker it is possible to locate the marker to about 1nm without fully exposing the resist. This means that the marker can be reused with full accuracy, thus improving the layer to layer alignment accuracy. Lithography tool noise limits the process.

Quasicrystalline tilings with nematic colloidal platelets

Complex nematic fluids have the remarkable capability for self-assembling regular colloidal structures of various symmetries and dimensionality according to their micromolecular orientational order. Colloidal chains, clusters, and crystals were demonstrated recently, exhibiting soft-matter functionalities of robust binding, spontaneous chiral symmetry breaking, entanglement, shape-driven and topological driven assembly, and even memory imprinting. However, no quasicrystalline structures were found. Here, we show with numerical modeling that quasicrystalline colloidal lattices can be achieved in the form of original Penrose P1 tiling by using pentagonal colloidal platelets in layers of nematic liquid crystals. The tilings are energetically stabilized with binding energies up to 2500 kBT for micrometer-sized platelets and further allow for hierarchical substitution tiling, i.e., hierarchical pentagulation. Quasicrystalline structures are constructed bottom-up by assembling the boat, rhombus, and star maximum density clusters, thus avoiding other (nonquasicrystalline) stable or metastable configurations of platelets. Central to our design of the quasicrystalline tilings is the symmetry breaking imposed by the platelet shape and the surface anchoring conditions at the colloidal platelets, which are misaligning and asymmetric over two perpendicular mirror planes. Finally, the design of the quasicrystalline tilings as platelets in nematic liquid crystals is inherently capable of a continuous variety of length scales of the tiling, ranging over three orders of magnitude in the typical length (from ~ 10 nm to ~ 10 μm), which could allow for the design of quasicrystalline photonics at multiple frequency ranges.

Structure of an Al–Cu–Co Decagonal Quasicrystal Studied by <i>C</i>s-Corrected STEM

The structure of an Al­Cu­Co decagonal quasicrystal (DQC) with two quasiperiodic planes in an annealed Al64Cu22Co14 alloy has been studied by spherical aberration (Cs)-corrected scanning transmission electron microscopy (STEM) with high-angle annular detector dark-field (HAADF) and annular bright-field (ABF) techniques. The observed HAADF-STEM images taken with the incident beam parallel to the periodic axis clearly represent individual transition-metal (TM) Cu/Co atoms and mixed sites (MSs) of Al and TM atoms as separated bright dots, and consequently arrangements of TM atoms and MSs on the two quasiperiodic planes can be directly determined. The TM atoms on the two quasiperiodic planes are arranged in pentagonal tiling with an edge-length of 0.76nm, and also the TM atoms and MSs are located at vertices of Penrose tiling with an edge-length of 0.25nm, and so they are arranged with a bond orientational order (BOO). Pentagonal frames with definite directions in the pentagonal tiling of TM atoms are also arranged in ¸ 2 -inflated pentagonal tiling with an edge-length of 2nm. Arrangements of TM atoms derived from the observed HAADF-STEM image are placed in ideal pentagonal Penrose tiling with an edge-length of 2nm, which is generated by the projection of five-dimensional (5D) hyper-cubic lattice, and are projected on occupation domains (ODs) in the perpendicular space. The arrangement of Al atoms and TM atoms and MSs in a well-symmetric region is interpreted from the observed HAADF- and ABFSTEM images. [doi:10.2320/matertrans.M2014008]

Penrose nanotiles: design of the thin and thick rhomb molecules to self-assemble into a quasicrystal

New organic molecules are designed, with synthesis in mind, to fit the shape of the thin and thick rhombs of the Penrose tiling and self-assemble by hydrogen bonding into nearly flat sheets (that may stack face-to-face) following the matching rules.

Dragon Curves Revisited

It has happened several times in recent history that a mathematical discovery of great beauty and importance was originally published in a journal that would not likely to be read by many a working mathematician. One famous example is the Penrose tilings [15]. Surely, Penrose tilings and the theory of quasicrystals is now a major area of research (see, e.g., [17]), and not only in mathematics but also in physics and chemistry, as witnessed by the 2011 Nobel Prize awarded to D. Shechtman for “the discovery of quasicrystals” in 1982. It is a pleasure to mention that this magazine played a role in popularizing Penrose tilings [16]. The topic of this column is another mathematical object of comparable beauty, the Dragon curves, whose theory was created by Chandler Davis and Donald Knuth [4]. The original articles are not easily available (they are reprinted in [10], along with previously unpublished addendum). Mathematical Intelligencer wrote about Dragon curves more than 30 years ago [5, 6, 7]. In spite of the existence of a Wikipedia article on the subject and in spite of their appearance in M. Crichton’s popular novel “Jurassic Park”, Dragon curves are not sufficiently well known to contemporary mathematicians, especially the younger ones who missed the original excitement of 40+ years ago. The goal of this article is to bring Dragon curves to spotlight again and to pay tribute to Chandler Davis, a co-author of an elegant theory that explains the striking features of these curves. This article is merely an invitation to the subject; the reader should not expect a thorough survey of the results or proofs.

Electronic Structure and Transport in Approximants of the Penrose Tiling

We present numerical calculations of electronic structure and transport in Penrose approximants. The electronic structure of perfect approximants shows a spiky density of states and a tendency to localization that is more pronounced in the middle of the band. Near the band edges the behavior is more similar to that of free electrons. These calculations of band structure and in particular the band scaling suggest an anomalous quantum diffusion when compared to normal ballistic crystals. This is confirmed by a numerical calculation of quantum diffusion which shows a crossover from normal ballistic propagation at long times to anomalous, possibly insulator-like, behavior at short times. The time scale t*(E) for this crossover is computed for several approximants and is detailed. The consequences for electronic conductivity are discussed in the context of the relaxation time approximation. The metallic like or non metallic like behavior of the conductivity is dictated by the comparison between the scattering time due to defects and the time scale t*(E).