Penrose Tiling Research Articles

Multicomponent induced localized coupling in Penrose tiling for electromagnetic wave absorption and multiband compatibility

• Design quasiperiodic absorbers with long-range order but no periodicity. • Explored the relationship between electromagnetic parameters and frequency by structural design and component selection in depth. • The effective absorption bandwidth of Penrose absorber can realize 9.32 GHz in 2 mm thickness. • The multilayer metamaterials perfectly unify radar-infrared compatible capability. In the electronic information and intelligent era, microwave absorption materials have been focused on the issue of electromagnetic (EM) pollution, but high-performance absorbers with thin thickness are highly desirable. The positive action between structural design and component selection can be regarded as a function to optimize the microwave loss capacity, which contributes to understanding the response mechanisms from the micro and macro. Therefore, we designed the absorber with quasiperiodic structures of Penrose tiling that has finely engineered the location of maximum absorption peak, the spherical carbonyl iron in thin rhombus, and flakey carbonyl iron in fat rhombus, the effective absorption bandwidth (Reflection Loss ≤ –10 dB) was realized from 6.46 GHz to 15.78 GHz in the 2 mm thickness. The unique arrangement of Penrose tiling can respond alternatingly to EM waves at large scales, and multicomponent coupling is distinctly in the neighboring unit cells. In addition, spraying aluminum pigment as the upper layer, the infrared emissivity is less than 0.35 to meet the complex application environment. This work opens up novel ideas on quasiperiodic structure combined multicomponent and injects infinite vitality to provide prospects for broadband responsive and high-efficiency EM absorbers.

Coxeter pairs, Ammann patterns, and Penrose-like tilings

We identify a precise geometric relationship between: (i) certain natural pairs of irreducible reflection groups (``Coxeter pairs"); (ii) self-similar quasicrystalline patterns formed by superposing sets of 1D quasi-periodically-spaced lines, planes or hyper-planes (``Ammann patterns"); and (iii) the tilings dual to these patterns (``Penrose-like tilings"). We use this relationship to obtain all irreducible Ammann patterns and their dual Penrose-like tilings, along with their key properties in a simple, systematic and unified way, expanding the number of known examples from four to infinity. For each symmetry, we identify the minimal Ammann patterns (those composed of the fewest 1d quasiperiodic sets) and construct the associated Penrose-like tilings: 11 in 2D, 9 in 3D and one in 4D. These include the original Penrose tiling, the four other previously known Penrose-like tilings, and sixteen that are new. We also complete the enumeration of the quasicrystallographic space groups corresponding to the irreducible non-crystallographic reflection groups, by showing that there is a unique such space group in 4D (with nothing beyond 4D).

Self-similar one-dimensional quasilattices

We study 1D quasilattices, especially self-similar ones that can be used to generate two-, three- and higher-dimensional quasicrystalline tessellations that have matching rules and invertible self-similar substitution rules (also known as inflation rules) analogous to the rules for generating Penrose tilings. The lattice positions can be expressed in a closed-form expression we call {\it floor form}: $x_{n}=S(n-\alpha)+(L-S)\lfloor \kappa(n-\beta)\rfloor$, where $L >S>0$ and $0<\kappa<1$ is an irrational number. We describe two equivalent geometric constructions of these quasilattices and show how they can be subdivided into various types of equivalence classes: (i) {\it lattice equivalent}, where any two quasilattices in the same lattice equivalence class may be derived from one another by a local decoration/gluing rule; (ii) {\it self-similar}, a proper subset of lattice equivalent where, in addition, the two quasilattices are locally isomorphic; and (iii) {\it self-same}, a proper subset of self-similar where, in addition, the two quasilattices are globally isomorphic (i.e.) identical up to rescaling). For all three types of equivalence class, we obtain the explicit transformation law between the floor form expression for two quasilattices in the same class. We tabulate (in Table I and Figure 5) the ten special self-similar 1D quasilattices relevant for constructing Ammann patterns and Penrose-like tilings in two dimensions and higher, and we explicitly construct and catalog the corresponding self-same quasilattices.

Rectangle-triangle soft-matter quasicrystals with hexagonal symmetry.

Aperiodic (quasicrystalline) tilings, such as Penrose's tiling, can be built up from, e.g., kites and darts, squares and equilateral triangles, rhombi- or shield-shaped tiles, and can have a variety of different symmetries. However, almost all quasicrystals occurring in soft matter are of the dodecagonal type. Here we investigate a class of aperiodic tilings with hexagonal symmetry that are based on rectangles and two types of equilateral triangles. We show how to design soft-matter systems of particles interacting via pair potentials containing two length scales that form aperiodic stable states with two different examples of rectangle-triangle tilings. One of these is the bronze-mean tiling, while the other is a generalization. Our work points to how more general (beyond dodecagonal) quasicrystals can be designed in soft matter.

Continuum limits of discrete isoperimetric problems and Wulff shapes in lattices and quasicrystal tilings

We prove discrete-to-continuum convergence of interaction energies defined on lattices in the Euclidean space (with interactions beyond nearest neighbours) to a crystalline perimeter, and we discuss the possible Wulff shapes obtainable in this way. Exploiting the “multigrid construction” of quasiperiodic tilings (which is an extension of De Bruijn’s “pentagrid” construction of Penrose tilings) we adapt the same techniques to also find the macroscopical homogenized perimeter when we microscopically rescale a given quasiperiodic tiling.

Wide-Scan and Energy-Saving Phased Arrays by Exploiting Penrose Tiling Subarrays

A partitioning scheme based on Penrose-inspired subarrays is proposed for wide-scan planar phased arrays. The regular triangular lattice adopted for the arrangement of the antenna elements is organized into irregularly shaped tiles that exploit the properties offered by the Penrose tessellation. The novel array architecture ensures a remarkable reduction of the transmit/receive modules without harming the scan angle range and peak sidelobe level thanks to the optimization based on the Pareto front that is able to tackle conflicting objectives. The performance of the proposed approach is assessed in different scenarios and proves the robustness and flexibility of the new tiling architecture. Moreover, a thorough analysis is carried out to highlight the unique feature of the partition scheme of offering a noteworthy reduction in energy consumption.

Exploiting Anyonic Behavior of Quasicrystals for Topological Quantum Computing

The concrete realization of topological quantum computing using low-dimensional quasiparticles, known as anyons, remains one of the important challenges of quantum computing. A topological quantum computing platform promises to deliver more robust qubits with additional hardware-level protection against errors that could lead to the desired large-scale quantum computation. We propose quasicrystal materials as such a natural platform and show that they exhibit anyonic behavior that can be used for topological quantum computing. Different from anyons, quasicrystals are already implemented in laboratories. In particular, we study the correspondence between the fusion Hilbert spaces of the simplest non-abelian anyon, the Fibonacci anyons, and the tiling spaces of the one-dimensional Fibonacci chain and the two-dimensional Penrose tiling quasicrystals. A concrete encoding on these tiling spaces of topological quantum information processing is also presented by making use of inflation and deflation of such tiling spaces. While we outline the theoretical basis for such a platform, details on the physical implementation remain open.

Intrinsic vortex pinning in superconducting quasicrystals

We numerically show that a vortex pinning occurs in superconducting quasicrystal without impurities and defects. This vortex pinning is intrinsic since the superconducting order parameter in quasicrystal is always inhomogeneous due to the lack of the translational symmetry. We propose that experiments influenced by vortex pinning effects can detect the atomic-scale inhomogeneous superconducting order parameter in quasicrystals. We develop a numerical method to solve the Bogoliubov-de Gennes equations and gap equations in large systems, which is based on the localized-Krylov subspace and a sparse modeling technique. Two two-dimensional quasicrystals, the Penrose and Amman-Beenker tiling, are considered.

The density and minimal gap of visible points in some planar quasicrystals

It has previously been observed that the limiting gap distribution of the directions to visible points of planar quasicrystals may vanish near zero, that is, there exist planar quasicrystals with a positive limiting minimal normalised gap between the angles of visible points. The exact values of these limiting minimal normalised gaps have not been determined. In this paper we give explicit formulas for the densities of visible points for planar quasicrystals from several families, which include the Ammann–Beenker point set and the vertex sets of some rhombic Penrose tilings. Combining these results with a known characterisation of the limiting minimal gap in terms of a probability measure on an associated homogeneous space of quasicrystals, we give explicit values of the limiting minimal normalised gap between the angles of visible points for several families of planar quasicrystals, in particular, for the Ammann–Beenker point set and for the vertex sets of some rhombic Penrose tilings. We also compare our results with numerical observations.

The decagonal AlCuRh quasicrystal modelled with five atomic surfaces - high-temperature X-ray diffraction data analysis.

Five datasets of high-temperature X-ray diffraction performed upon the decagonal phase of AlCuRh are used to derive the temperature-related structural changes. Two sets of atomic structure refinements are conducted, with four and five atomic surfaces, respectively. The fifth atomic surface emerges as a consequence either of the transition to a tiling with different local isomorphism than the Penrose tiling or of the structure being phason disordered. The correlation of the occupancy of the fifth atomic surface with the thermal dependence of the lattice constants is 80%. In addition, the structural refinement with five atomic surfaces allowed a better agreement with the experimental data to be obtained, compared with the model with four atomic surfaces.

Doped Mott insulator on a Penrose tiling

We study the effect of carrier doping to the Mott insulator on the Penrose tiling, aiming at clarifying the interplay between quasiperiodicity and strong electron correlations. We numerically solve the Hubbard model on the Penrose-tiling structure within a real-space dynamical mean-field theory, which can deal with a singular self-energy necessary to describe the Mott insulator and spatial inhomogeneity. We find that the strong correlation effect produces a charge distribution unreachable by a static mean-field approximation. In a small doping region, the spectrum shows a site-dependent gap just above the Fermi energy, which is generated by a singularly large self-energy emergent from the Mott physics and regarded as a real-space counterpart of the momentum-dependent pseudogap observed in a square-lattice Hubbard model.

Envelope Function Analysis of Quasicrystals

Quasicrystals have attracted a growing interest in material science because of their unique properties and applications. Proper determination of the atomic structure is important in designing a useful application of these materials, for which a difficult phase problem of the structure factor must be solved. Diffraction patterns of quasicrystals consist of a periodic series of peaks, which can be reduced to a single envelope. Knowing the distribution of the diffraction image into series, it is possible to recover information about the phase of the structure factor without using time-consuming iterative methods. By the inverse Fourier transform, the structure factor can be obtained (enclosed in the shape of the average unit cell, or atomic surface) directly from the diffraction patterns. The method based on envelope function analysis was discussed in detail for a model 1D (Fibonacci chain) and 2D (Penrose tiling) quasicrystal. First attempts to apply this technique to a real Al-Cu-Rh decagonal quasicrystal were also made.

Photoinduced dynamics of a quasicrystalline excitonic insulator

We study the photoinduced dynamics of the excitonic insulator in the two-orbital Hubbard model on the Penrose tiling by means of the time-dependent real-space mean-field approximation. We show that, with a single-cycle electric-field pulse, the bulk (spatially averaged) excitonic order parameter decreases in the Bardeen-Cooper-Schrieffer (BCS) regime, while it increases in the Bose-Einstein condensate (BEC) regime. This behavior is in common with the excitonic insulator in the square lattice reported previously. To explore the dynamics peculiar to the Penrose tiling, we examine the coordination number dependence of observables and analyze the perpendicular space. In the BEC regime, characteristic oscillations of the electron number at each site are induced by the pulse, which are not observed in normal crystals with single coordination number. On the other hand, the dynamics in the BCS regime is characterized by drastic change in the spatial pattern of the excitonic order parameter. Our results imply that potentially interesting nonequilibrium physics is caused by rich local structural patterns involved in quasicrystals.

Photoinduced Phase Transition in Two-Band model on Penrose Tiling

We study the effects of the photo irradiation on the band insulating state in the two-band Hubbard model on the Penrose tiling. Examining the time- and site-dependent physical quantities, we find that the excitionic state is dynamically induced with site-dependent order parameters. It is also clarified that, in the excitonic state induced by the photo irradiation, local oscillatory behavior appears in the electron number as well as in the order parameter, which should be characteristic of the quasiperiodic lattice.

Quasiperiodic metamaterials with broadband absorption: Tailoring electromagnetic wave by Penrose tiling

Quasicrystals can enhance wave transmission and boost nonlinear phenomena due to rotational symmetry forbidden in conventional periodic crystals. As a projection of higher-dimensional quasicrystal lattice, quasiperiodic structures represent a novel platform that exhibits robust microwave-absorbing capacity in thin thickness and digs out the influence of structural configurations on electromagnetic wave absorption from local and global aspects. Due to the synergy effect of composites and structure, quasiperiodic metamaterials with Penrose tiling achieve 12.2 GHz in effective absorption bandwidth (reflection loss < −10 dB), from 5.78 GHz to 18.00 GHz with 2 mm thickness. It has obvious advantages in absorbing electromagnetic wave compared with corresponding periodic absorbers according to the characteristic of local anisotropy and higher order translational symmetry in Brillouin zone . By enhancing multiple scattering and nonlinear coupling effects, Penrose tiling provides new insights to design absorbers with outstanding electromagnetic performance and higher flexibility and tunability.

Interlaced wire medium with quasicrystal lattice

We propose a design of an interlaced wire medium with a quasicrystalline lattice based on fivefold rotation symmetry Penrose tiling. The transport properties of this structure are studied. We distinguish two transport regimes, namely, a propagation regime related to the low-frequency interval and localization regime in the high-frequency interval. While the former is observed in structures both with and without the translation symmetry property, the latter is exclusive for aperiodic structures only. We show that the localization regime is promising for many applications, including the engineering of effective multichannel devices for telecommunication and imaging systems.

Cohomology groups for spaces of 12-fold tilings

We consider tilings of the plane with 12-fold symmetry obtained by the cut and projection method. We compute their cohomology groups using the techniques introduced in [5]. To do this we completely describe the window, the orbits of lines under the group action and the orbits of 0-singularities. The complete family of generalized 12-fold tilings can be described using 2-parameters and it presents a surprisingly rich cohomological structure. To put this finding into perspective, one should compare our results with the cohomology of the generalized 5-fold tilings (more commonly known as generalized Penrose tilings). In this case the tilings form a 1-parameter family, which fits in simply one of two types of cohomology.

Informational Measure of Symmetry vs. Voronoi Entropy and Continuous Measure of Entropy of the Penrose Tiling. Part II of the “Voronoi Entropy vs. Continuous Measure of Symmetry of the Penrose Tiling”

The notion of the informational measure of symmetry is introduced according to: Hsym(G)=−∑i=1kP(Gi)lnP(Gi), where P(Gi) is the probability of appearance of the symmetry operation Gi within the given 2D pattern. Hsym(G) is interpreted as an averaged uncertainty in the presence of symmetry elements from the group G in the given pattern. The informational measure of symmetry of the “ideal” pattern built of identical equilateral triangles is established as Hsym(D3)= 1.792. The informational measure of symmetry of the random, completely disordered pattern is zero, Hsym=0. The informational measure of symmetry is calculated for the patterns generated by the P3 Penrose tessellation. The informational measure of symmetry does not correlate with either the Voronoi entropy of the studied patterns nor with the continuous measure of symmetry of the patterns. Quantification of the “ordering” in 2D patterns performed solely with the Voronoi entropy is misleading and erroneous.

Voronoi Entropy vs. Continuous Measure of Symmetry of the Penrose Tiling: Part I. Analysis of the Voronoi Diagrams

A continuous measure of symmetry and the Voronoi entropy of 2D patterns representing Voronoi diagrams emerging from the Penrose tiling were calculated. A given Penrose tiling gives rise to a diversity of the Voronoi diagrams when the centers, vertices, and the centers of the edges of the Penrose rhombs are taken as the seed points (or nuclei). Voronoi diagrams keep the initial symmetry group of the Penrose tiling. We demonstrate that the continuous symmetry measure and the Voronoi entropy of the studied sets of points, generated by the Penrose tiling, do not necessarily correlate. Voronoi diagrams emerging from the centers of the edges of the Penrose rhombs, considered nuclei, deny the hypothesis that the continuous measure of symmetry and the Voronoi entropy are always correlated. The Voronoi entropy of this kind of tiling built of asymmetric convex quadrangles equals zero, whereas the continuous measure of symmetry of this pattern is high. Voronoi diagrams generate new types of Penrose tiling, which are different from the classical Penrose tessellation.

Decagonal random tilings and their quasi-unit cell cluster coverings

Electron microscopy images of decagonal quasicrystals obtained recently have been shown to be related to cluster coverings with a Hexagon–Bow–Tie decagon as single structural unit. Most decagonal phases show more complex structural orderings than models based on deterministic tilings like the Penrose tiling. We analyze different types of decagonal random tilings and their coverings by a Hexagon–Bow–Tie decagon.