Quasiperiodic tilings: a generalized grid-projection method

Physica A: Statistical Mechanics and its Applications, 1992

We generate a quasiperiodic, dodecagonally symmetric tiling of the plane by squares and equilateral triangles embedded in a higher-dimensional periodic structure. Starting from a 4D lattice frequently used for the embedding of dodecagonal structures, we iteratively construct an acceptance domain (AD) for a quasiperiodic dodecagonal point set which proves to be the vertex set of a square-triangle tiling. It turns out that our procedure leads to fractally bounded ADS but leaves enough freedom to generate several different local isomorphism classes.

Acta crystallographica. Section A, Foundations and advances, 2015

A group-theoretical discussion on the hypercubic lattice described by the affine Coxeter-Weyl group Wa(Bn) is presented. When the lattice is projected onto the Coxeter plane it is noted that the maximal dihedral subgroup Dh of W(Bn) with h = 2n representing the Coxeter number describes the h-fold symmetric aperiodic tilings. Higher-dimensional cubic lattices are explicitly constructed for n = 4, 5, 6. Their rank-3 Coxeter subgroups and maximal dihedral subgroups are identified. It is explicitly shown that when their Voronoi cells are decomposed under the respective rank-3 subgroups W(A3), W(H2) × W(A1) and W(H3) one obtains the rhombic dodecahedron, rhombic icosahedron and rhombic triacontahedron, respectively. Projection of the lattice B4 onto the Coxeter plane represents a model for quasicrystal structure with eightfold symmetry. The B5 lattice is used to describe both fivefold and tenfold symmetries. The lattice B6 can describe aperiodic tilings with 12-fold symmetry as well as a...

Discrete & Computational Geometry, 2016

In this paper we prove the existence of quasiperiodic rhombic substitution tilings with 2n-fold rotational symmetry, for any n. The tilings are edge-to-edge and use ⌊ n 2 ⌋ rhombic prototiles with unit length sides. We explicitly describe the substitution rule for the edges of the rhombuses, and prove the existence of the corresponding tile substitutions by proving that the interior can be tiled consistently with the given edge substitutions.

Physical Review B, 2003

We analyze the spreading of wavepackets 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 constrast, 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. PACS numbers: 61.44.Br, 71.23.Ft It is now well established that quasiperiodic order has a strong influence on the quantum dynamics of wavepackets. Indeed, the nature of the eigenstates in quasiperiodic systems, which are neither spatially extended (as in periodic systems) nor localized (as in disordered systems) but critical, is often responsible for a sub-ballistic motion. Although most of the studies about this anomalous diffusion concern one-dimensional systems such as the Fibonacci or the Harper chain, there has also been a great interest for the, more physical, higher-dimensional ones . However, the parameters that determine the characteristics of the long time dynamics, such as the diffusion exponent β, remains misunderstood.

2005

Arxiv preprint arXiv:1101.0223, 2010

We consider here square tilings of the plane. By extending the formalism introduced in [3] we build a correspondence between plane maps endowed with an harmonic vector and square tilings satisfying a condition of regularity. In the case of periodic plane square tiling the relevant space of harmonic vectors is actually isomorphic to the first homology group of a torus. So, periodic plane square tilings are described by two parameters and the set of parameters is split into angular sectors.

Le Journal de Physique Colloques

Resume-Les proprietes generates des stuctures quasi-pbriodiques obtenues par la methode de projection sont presentees en vue d'un modhle structural des cristaux aperiodiques du type AI-Mn. On 6tudie particulihrement les environnements locaux, les projections & 2 dimensions de ces structures et leur transformee de Fourier. Absract-The general properties of quasiperiodic structures obtained by the projection method are given in view of a structural model of aperiodic crystals of type AI-Mn. The local environments, the 2-dimensional projections and the Fourier transform of these structures are particularly examined. 1 INTRODUCTION This talk deals with some specific features of quasiperiodic tilings or patterns obtained by the projection method. Non periodic tilings were first introduced in connection with problems of mathematical logics but the invention by R. Penrose of his well known aperiodic tilings of the plane initiated the study of the geometrical properties of these patterns with the pionnering works of R. Penrose [I], J. Conway (see [3]) and N. G. de Bruijn [2] (see the beautiful survey by M. Gardner [3]). The possible implications of these structures for solid state physics were mentioned by A. L. Mackay [4] and Mosseri and Sadoc 151. A 3-dimensional generalization of the Penrose tilings (using two rhombohedra instead of two rhombs) was devised by R. Ammann and theoretically described by P. Kramer and R. Neri [6] and P. Kramer [q using the de Bruijn's framework. The spectral properties of the Penrose patterns were first observed by A. L. Mackay, using optical methods. The presence of sharp Bragg peaks in the diffraction patterns is now identified [8,9,10,11] as the expression of quasiperiodicity : the Fourier transform of the measure defined as the sum of Dirac deltas at all vertices of the tiling, is a sum of weighted Dirac measures carried by a 2-module. The experimental discovery of icosahedral symmetry together with some kind of long range spatial order in quenched AI-Mn alloys by D. Shechtman et al. [12] aroused a great interest among theorists, as well on the structural problem , as on stability questions which will not be discussed here.

arXiv (Cornell University), 2021

We show that diffraction features of 1D quasicrystals can be retrieved from a single topological quantity, the Čech cohomology group, Ȟ1 ∼ = Z 2 , which encodes all relevant combinatorial information of tilings. We present a constructive way to calculate Ȟ1 for a large variety of aperiodic tilings. By means of two winding numbers, we compare the diffraction features contained in Ȟ1 to the gap labeling theorem, another topological tool used to label spectral gaps in the integrated density of states. In the light of this topological description, we discuss similarities and differences between families of aperiodic tilings, and the resilience of topological features against perturbations.

Journal de Physique I, 1997

Physical Review B, 2002

We investigate the properties of electronic states in two and three-dimensional quasiperiodic structures : the generalized Rauzy tilings. Exact diagonalizations, limited to clusters with a few thousands sites, suggest that eigenstates are critical and more extended at the band edges than at the band center. These trends are clearly confirmed when we compute the spreading of energy-filtered wavepackets, using a new algorithm which allows to treat systems of about one million sites. The present approach to quantum dynamics, which gives also access to the low frequency conductivity, opens new perspectives in the analyzis of two and three-dimensional models.