Quasiperiodic Tilings Research Articles

Remarks on 2-dimensional quasiperiodic tilings with rotational symmetries

We construct a sequentially compact space of patches. By using this construction, we analyze certain symmetries of tilings obtained by substitution rules.

Quasiperiodic plane tilings based on stepped surfaces

Static and dynamic characteristics of layerwise growth in two-dimensional quasiperiodic Ito-Ohtsuki tilings are studied. These tilings are the projections of three-dimensional stepped surfaces. It is proved that these tilings have hexagonal self-similar growth with bounded radius of neighborhood. A formula is given for the averaged coordination number. Deviations of coordination numbers from its average are quasiperiodic. Ito-Ohtsuki tiling can be decomposed into one-dimensional sector layers. These sector layers are one-dimensional quasiperiodic tilings with properties like Ito-Ohtsuki tilings.

Quasi-periods of layer-by-layer growth of Rauzy tiling

The dynamic characteristics of shaping of a layer-by-layer growth polyhedron in the 2D quasi-periodic Rauzy tiling have been investigated. Quasi-periodicity of deviations ρi(n) of sectoral growth rates from their means has been found in a computer experiment and rigorously mathematically substantiated. It is shown that sectoral deviations ρi(n) have exact upper and low boundaries, which are determined by induced maps L′i, isomorphous to the rotation of a unit circle by an angle βi. Quasi-periods and amplitudes of deviations ρi(n) are calculated from partial quotients of the expansion of βi in a continued fraction as well as from the denominators of the partial convergents determined by these expansion.

Complexity function and forcing in the 2D quasi-periodic Rauzy tiling

The quantitative characteristics of the long-range translational order in the 2D quasi-periodic Rauzy tiling (complexity function and forcing depth) have been investigated. It is proved that the complexity function c(n) is equal to the number of figures in the n-corona grown from a seed composed of three figures of different types. The complexity function c(n) is found to be additive. A relationship between the jumps in the maximum forcing depth and large incomplete particular expansions in a chain fraction of irrational angles of rotation of a unit circle, which determine the growth of geodetic chains, is established.

Layer-by-layer growth of quasi-periodic Rauzy tiling

Some characteristics of the 2D quasi-periodic Rauzy tiling, which allows parametrization and is obtained from finite torus tilings, have been investigated. The main characteristics of the layer-by-layer growth of this tiling were investigated theoretically and in a computer experiment. The polygonal character of self-similar growth with a limited radius of deviation from the limiting polygon is demonstrated and formulas for both sectoral and global growth rates are obtained.

A dodecagonal quasiperiodic tiling with a fractal window

A dodecagonal quasiperiodic tiling is generated by a substitution rule (SR) for four kinds of tiles, namely, a triangle, a square, a trigonal hexagon and a dodecagon. The scaling factor of the SR is equal to . The same tiling (exactly, quasilattice) is obtained with the projection method from a four-dimensional dodecagonal lattice and the relevant window has a fractal boundary. A set equation for the window is presented. It is emphasized that investigating quasiperiodic tilings of this type is a less explored but rich field in the crystallography of quasicrystals.

On the combinatorial characterization of quasicrystals

In various fields of materials science, many interesting two-dimensional (2D) and three-dimensional (3D) structures (fullerenes, nanotubes, frothes, metal foams, polycrystals and, notably, various quasicrystals) can be considered as finite or infinite cellular systems. For a cell T of a given 2D cellular system, we write n ( T ) to denote the number of sides of T , and m ( T ) to denote the average number of sides of the neighbours of T . In the 3D case, we replace sides by faces in the definition. D. Weaire first observed, for trivalent random tilings of the plane, that 〈 n ⋅ m 〉 = 〈 n 2 〉 , where 〈 ⋅ 〉 stands for the expected value. Following his discovery, the Weaire sum rule has been proved for various tilings of a sphere or a torus, and for periodic tilings of the plane or space. In this paper we extend the Weaire sum rule to quasiperiodic tilings of the plane or space. Actually, the method of this paper yields the Weaire sum rule for tilings of any compact surface or three-manifolds as well.

MAGNETIC ORDERING IN QUASICRYSTALS

An overview of the theoretical advances in description of the magnetic ordering and its stability in two-dimensional quasiperiodic tilings with strongly localized magnetic moments is presented. It is demonstrated that combination of the magnetic frustration and the quasiperiodic order of atoms leads to noncollinear ground states. An experimental and theoretical evidence for the possibility of coexistence of stable, magnetically ordered subtilings with highly frustrated, glass-like phases in a single sample is given.

Interplay between magnetic and spatial order in quasicrystals

The stable magnetization configurations of antiferromagnets on quasiperiodic tilings are investigated theoretically. The exchange coupling is assumed to decrease exponentially with the distance between magnetic moments. It is demonstrated that the combination of geometric frustration and the quasiperiodic order of atoms leads to complicated non-collinear ground states. The structure can be divided into subtilings of different energies. The symmetry of the subtilings depends on the quasiperiodic order of magnetic moments. The subtilings are spatially ordered. However, the magnetic ordering of the subtilings in general does not correspond to their spatial arrangements. While subtilings of low energy are magnetically ordered, those of high energy can be completely disordered due to local magnetic frustration.

Ground state of a two-dimensional quasiperiodic quantum antiferromagnet

We consider the spin 1/2 Heisenberg antiferromagnet on a two dimensional quasiperiodic tiling. The T=0 state is long range ordered, with an inhomogeneous distribution of the staggered moment. An approximate real space renormalization scheme first developed for the square lattice is generalized and used to obtain information on the ground state energy as well as the distribution of local correlations in the quasicrystal.

Cauchy Problem for Integrable Discrete Equations on Quad-Graphs

Initial value problems for the integrable discrete equations on quad-graphs are investigated. A geometric criterion of the well-posedness of such a problem is found. The effects of the interaction of the solutions with the localized defects in the regular square lattice are discussed for the discrete potential KdV and linear wave equations. The examples of kinks and solitons on various quad-graphs, including quasiperiodic tilings, are presented.

Cauchy problem for integrable discrete equations on quad-graphs

Initial value problems for the integrable discrete equations on quad-graphs are investigated. We give a geometric criterion of when such a problem is well-posed. In the basic example of the discrete KdV equation an effective integration scheme based on the matrix factorization problem is proposed and the interaction of the solutions with the localized defects in the regular square lattice are discussed in details. The examples of kinks and solitons on various quad-graphs, including quasiperiodic tilings, are presented.

Noncollinear magnetic order in quasicrystals.

Based on Monte Carlo simulations, the stable magnetization configurations of an antiferromagnet on a quasiperiodic tiling are derived theoretically. The exchange coupling is assumed to decrease exponentially with the distance between magnetic moments. It is demonstrated that the superposition of geometric frustration with the quasiperiodic ordering leads to a three-dimensional noncollinear antiferromagnetic spin structure. The structure can be divided into several ordered interpenetrating magnetic supertilings of different energy and characteristic wave vector. The number and the symmetry of subtilings depend on the quasiperiodic ordering of atoms.

Atomic structure of anAl–Co–Nidecagonal quasicrystalline surface

We have analyzed the structure and composition of the first layer of an ${\text{Al}}_{72}{\text{Co}}_{16}{\text{Ni}}_{12}$ tenfold surface by means of scanning tunneling microscopy (STM), ion scattering spectroscopy (ISS), and Auger electron spectroscopy (AES). High-resolution STM images reveal local structures that have decagonal symmetry in addition to the usual pentagonal symmetry of the surface. This quasicrystal surface resembles a random tiling instead of an ideal quasiperiodic tiling. After annealing at $1100\phantom{\rule{0.3em}{0ex}}\mathrm{K}$, the total surface atomic density found by ISS is $(9\ifmmode\pm\else\textpm\fi{}1)\ifmmode\times\else\texttimes\fi{}{10}^{14}\phantom{\rule{0.3em}{0ex}}{\text{cm}}^{\ensuremath{-}2}$. The surface densities of Al and TM (transition metal, i.e., Co and Ni) are determined as $(8\ifmmode\pm\else\textpm\fi{}1)\ifmmode\times\else\texttimes\fi{}{10}^{14}\phantom{\rule{0.3em}{0ex}}{\text{cm}}^{\ensuremath{-}2}$ and $(1.0\ifmmode\pm\else\textpm\fi{}0.2)\ifmmode\times\else\texttimes\fi{}{10}^{14}\phantom{\rule{0.3em}{0ex}}{\text{cm}}^{\ensuremath{-}2}$, respectively from ISS, indicating a similar density of Al and much lower density of the TM atoms in the surface layer than in a truncated bulk. The Al surface atomic density agrees well with the number of corrugation maxima in the STM images. A model of the arrangement of the Al atoms in the top layer is presented. Scanning tunneling spectroscopy (STS) is performed to study the local electronic structure. The STS spectrum at the corrugation maxima is similar to that at the corrugation minima. A few $\ensuremath{\approx}0.12\phantom{\rule{0.3em}{0ex}}\text{nm}$ high protrusions in the STM images are attributed to local oxide clusters due to their STS spectra different from the corrugation maxima and through in situ STM observations during exposure to ${\mathrm{O}}_{2}$ gas at $2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}\phantom{\rule{0.3em}{0ex}}\text{Pa}$ at RT.

Polyhedral truncations as eutactic transformations.

An eutactic star is a set of N vectors in Rn (N > n) that are projections of N orthogonal vectors in RN. First introduced in the context of regular polytopes, eutactic stars are particularly useful in the field of quasicrystals where a method to generate quasiperiodic tilings is by projecting higher-dimensional lattices. Here are defined the concepts of eutactic transformations (as mappings that preserve eutacticity) and of vector radiations (vectors that stem from the vectors of an eutactic star), which are used to describe and parameterize polyhedral truncations. The polyhedral truncations preserve eutacticity, a result of relevance to the faceting and habit-forming characteristics of quasicrystals.

Quasicrystallic Tilings and Projection Method

Basic notions related to quasiperiodic tilings and Delone sets in Eucledean space are discussed. It is shown how the cut and project method of constructing them is used to calculate their spectra. Special attention is paid to self-similar tilings and the way one can obtain one-dimensional substitutional tilings by the projection scheme. Bibliography: 18 titles.

Inflation and wavelets for the icosahedral Danzer tiling

The distribution of atoms in quasi-crystals lacks periodicity and displays point symmetry associated with non-crystallographic modules. Often it can be described by quasi-periodic tilings on built from a finite number of prototiles. The modules and the canonical tilings of five-fold and icosahedral point symmetry admit inflation symmetry. In the simplest case of stone inflation, any prototile when scaled by the golden section number ? can be packed from unscaled prototiles. Observables supported on for quasi-crystals require symmetry-adapted function spaces. We construct wavelet bases on for the icosahedral Danzer tiling. The stone inflation of the four Danzer prototiles is given explicitly in terms of Euclidean group operations acting on . By acting with the unitary representations inverse to these operations on the characteristic functions of the prototiles, we recursively provide a full orthogonal wavelet basis of . It incorporates the icosahedral and inflation symmetry.

Quasiperiodic tilings in a magnetic field

We study the electronic properties of a two-dimensional quasiperiodic tiling, the isometric generalized Rauzy tiling, embedded in a magnetic field. Its energy spectrum is computed in a tight-binding approach using the recursion method. As for the square lattice, a rich band structure is obtained when the magnetic flux per elementary plaquette is varied. Then, we study the quantum dynamics of wave packets and discuss the influence of the magnetic field on the diffusion and spectral exponents. We show that contrary to the zero field case where the propagation is faster in a periodic structure than in a quasiperiodic one, the diffusion exponent may be larger in the Rauzy tiling than in the square lattice for some incommensurate fluxes. Finally, we consider a quasiperiodic superconducting wire network with the same geometry and we determine the critical temperature as a function of the magnetic field. A non-trivial cusp-like structure arises that reveals the real importance of the quasiperiodic order.

Quantum dynamics in high codimension tilings: From quasiperiodicity to disorder

We analyze the spreading of wave packets in two-dimensional quasiperiodic and random tilings as a function of their codimension, i.e., of their topological complexity. In the quasiperiodic case, we show that the diffusion exponent that characterizes the propagation decreases when the codimension increases and goes to 1/2 in the high codimension limit. By contrast, the exponent for the random tilings is independent of their codimension and also equals 1/2. This shows that, in high codimension, the quasiperiodicity is irrelevant and that the topological disorder leads in every case, to a diffusive regime, at least in the time scale investigated here.

Self-avoiding walks and polygons on quasiperiodic tilings

We enumerate self-avoiding walks and polygons, counted by perimeter, on the quasiperiodic rhombic Penrose and Ammann-Beenker tilings, thereby considerably extending previous results. In contrast to similar problems on regular lattices, these numbers depend on the chosen start vertex. We compare different ways of counting and demonstrate that suitable averaging improves converge to the asymptotic regime. This leads to improved estimates for critical points and exponents, which support the conjecture that self-avoiding walks on quasiperiodic tilings belong to the same universality class as self-avoiding walks on the square lattice. For polygons, the obtained e numeration data does not allow to draw decisive conclusions about the exponent.