Layer-by-layer growth of vertex graph of Penrose tiling
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RY OF CRYSTAL STRUCTURES
Layer-by-Layer Growth of Vertex Graph of Penrose Tiling A. V. Shutov and A. V. Maleev* Vladimir State University, Vladimir, 600000 Russia *e-mail: [email protected] Received May 10, 2016
Abstract—The growth form for the vertex graph of Penrose tiling is found to be a regular decagon. The lower and upper bounds for this form, coinciding with it, are strictly proven. A fractal character of layer-by-layer growth is revealed for some subgraphs of the vertex graph of Penrose tiling. DOI: 10.1134/S1063774517050194
INTRODUCTION The shaping of crystalline and quasicrystalline states of matter is an important problem of crystallography. An approach to the modeling of crystal formation, based on successive attachment of coordination encirclements in a periodic tiling of space into polyhedra or in a periodic packing of coordination polyhedra to some chosen finite set of polyhedra (to a seed), was proposed in [1]. This approach was generalized in [2, 3] to arbitrary adjacency graphs, and the concept of the form of layer-by-layer growth was strictly defined. One can prove [4] that the growth form always exists for periodic structures; it is a convex centrosymmetric polyhedron (polygon in the two-dimensional case), which can easily be calculated. The effect of polygonal growth was also found for a number of 1periodic graphs [5] on plane. For two-dimensional quasi-periodic Ito–Otsuki tilings, constructed based on stepped surfaces, it was shown [6] that the form of layer-by-layer growth is a convex centrosymmetric hexagon, whose vertices are determined by the parameters a, b, c , specifying these tilings. A study of the layer-by-layer growth of two-dimensional quasi-periodic Rauzy tilings made it possible not only to establish the growth form as a centrosymmetric octagon [7, 8] but also find the growth rate quasi-periods [9]. A polygonal layer-by-layer growth form was also found for a series of self-similar quasi-periodic tilings constructed on a complex plane using β-expansions, corresponding to some cubic irrationalities [10, 11]. It was shown in [12] that in the case of some graphs with elements of chance, the growth form is a combination of linear and nonlinear (hypothetically elliptical) portions. In this paper, we report the results of studying the layer-by-layer growth of vertex graph of Penrose tilings.
PROBLEM OF LAYER-BY-LAYER GROWTH OF VERTEX GRAPH OF PENROSE TILING Baake et al. [13–16] proposed an approach where a set of Penrose tiling vertices is presented in the form of a model set ΛW = {π1(x) : x ∈ L , π 2 (x) ∈ W }, where L is an integer four-dimensional lattice Z 4 , whose points have coordinates (h, j, k, l ), where h, j, k, l run all possible integer values, π1 is the projection of lattice L onto a complex plane C (physical space in which tiling vertices are built), and π2 is the projection of lattice L onto the direct product of another complex plane and fifth-order cyclic group: A = C × C 5 (phase or 2πi
parameter space). If ζ = e 5 is a complex fifth root of unity, projections π1 an
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