About crystal lattices and quasilattices in Euclidean space
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TALLOGRAPHIC SYMMETRY
About Crystal Lattices and Quasilattices in Euclidean Space A. S. Prokhoda Oles Honchar Dnepropetrovsk National University, Dnepropetrovsk, 49050 Ukraine e-mail: [email protected] Received February 10, 2016
Abstract—Definitions are given, based on which algorithms have been developed for constructing computer models of two-dimensional quasilattices and the corresponding quasiperiodic tilings in plane, the point symmetry groups of which are dihedral groups Dm (m = 5, 7, 8, 9, 10, 12, 14, 18), and the translation subgroups are free Abelian groups of the fourth or sixth rank. The angles at the tile vertices in the constructed tilings are calculated. DOI: 10.1134/S1063774517040174
INTRODUCTION According to the definition given by the Commission on Aperiodic Crystals of the International Union of Crystallographers, a crystal is any solid having an essentially discrete diffraction diagram. Quasicrystals are solids that are characterized by a symmetry forbidden in classical crystallography and the presence of long-range order and have at the same time a discrete diffraction pattern. It follows from these definitions that quasicrystals form a subclass of the class of crystals. The crystals having rotational symmetry axes of only the first, second, third, fourth, or sixth orders form another subclass. Quasicrystals may also have forbidden symmetry axes, for example, the fivefold axis. The long-range order characterizing quasicrystals differs from the long-range order the “conventional” crystals possess. Since there are different types of long-range order, we define below this characteristic. Definition 1. A crystal structure in space has a longrange order if, independent of the location of some point in space, there is an algorithm making it possible to determine (with a finite number of steps) if a crystal structure element (atom, molecule) is located or is not located at this point. From the group-theoretical point of view, the presence of long-range order indicates that the symmetry group of crystal should contain an infinite subgroup, which acts in space and determines this order. In the case of conventional crystals, this subgroup is a subgroup of parallel translations, which is a free Abelian group with a rank (i.e., the maximum number of independent elements) equal to the space dimension. The lattice in an n-dimensional space is defined as a set of points with integer affine coordinates. In other words, a lattice in n-dimensional space is a free Abelian group
of rank n. In the case of physical Euclidean space Е n (n = 2, 3), a lattice cannot have rotational axes of fivefold symmetry and symmetries of order higher than six. However, quasicrystals have some of the aforementioned axes. Therefore, depending on the type of the infinite subgroup the crystal symmetry group possess, a particular theory of quasicrystals will work. Several such theories have been developed to date. Two theories (the equivalence of which, as far as we know, has not been proven) are most popular. The first theory, develo
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