Polyhedra with noncrystallographic symmetry as the orbits of crystallographic point symmetry groups
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TALLOGRAPHIC SYMMETRY
Polyhedra with Noncrystallographic Symmetry As the Orbits of Crystallographic Point Symmetry Groups T. I. Ovsetsina and E. V. Chuprunov Lobachevsky State University of Nizhny Novgorod, pr. Gagarina 23, Nizhny Novgorod, 603950 Russia e-mail: [email protected] Received April 16, 2015
Abstract—Polyhedra with noncrystallographic symmetry are analyzed as the orbits of crystallographic point symmetry groups on a set of smooth or structured (“hatched”) planes. Polyhedra with symmetrically equivalent faces, obtained using crystallographic point groups but having noncrystallographic symmetry, and polyhedra, the symmetry group T of which is crystallographic but can be implemented only on the assumption of a noncrystallographic character of the internal structure of polyhedron, are studied. The results of the analysis for all 32 point symmetry groups are listed in table. DOI: 10.1134/S1063774515060255
Simple crystal forms are traditionally obtained by the multiplication of a smooth or structured (hatched) initial face by all symmetry operations of crystallographic point symmetry group G [1]. The symmetry of the polyhedron obtained can either coincide with group G or be described by some its supergroup T ⊃ G. The cases where groups T and G belong to the same system (are subgroups of the same holohedral point group) were described in detail in [2]. In some cases the symmetry of the polyhedron obtained can be described by a noncrystallographic point group. In particular, a polyhedron with smooth faces, formed by the multiplication of symmetry operations of group C6v of the initial face, which makes equal angles with the symmetry planes of this group, is invariant with respect to group C12v [3]. Figure 1a presents a gnomostereographic projection corresponding to this simple form. The polyhedra for which groups T and G are crystallographic but belong to different systems are also of interest. Under certain conditions, the orthorhombic tetrahedron obtained by the multiplication of a smooth initial face by symmetry operations of group D2 can be invariant with respect to cubic symmetry group Тd (Fig. 1b). This may occur if the corresponding crystal space is characterized by exact equality (which is unlikely) of the elementary translation lengths (unit-cell edges) or on the assumption that this polyhedron does not have a lattice structure. In this paper we describe polyhedra with symmetrically equivalent faces, obtained based on crystallographic point groups but having noncrystallographic symmetry, and polyhedra the symmetry group T of which is crystallographic but can be implemented only
on the assumption of noncrystallographic character of the polyhedron internal structure. One way to enumerate and describe these simple forms with sufficient rigor is to present them as orbits of point symmetry groups on a set of smooth or structured (hatched) planes. This approach was used in [2] to describe simple forms of crystals. Let us recall briefly the main concepts of the theory of orbits of groups [4, 5]. A set A o
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