Symmetry of Structures That Can Be Approximated by Chains of Regular Tetrahedra
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TALLOGRAPHIC SYMMETRY
Symmetry of Structures That Can Be Approximated by Chains of Regular Tetrahedra A. L. Talisa,* and A. L. Rabinovichb a Nesmeyanov b Institute
Institute of Organoelement Compounds, Russian Academy of Sciences, Moscow, 119991 Russia of Biology, Karelian Research Centre, Russian Academy of Sciences, Petrozavodsk, 185910 Russia *e-mail: [email protected] Received September 17, 2018; revised December 4, 2018; accepted December 4, 2018
Abstract—The noncrystallographic symmetries of chains of regular tetrahedra are determined by mapping the system of algebraic geometry and topology designs to the structural level. It has been shown that the basic structural unit of such a chain is a tetrablock: a seven-vertex linear aggregation over faces of four regular tetrahedra, which is implemented in linear (right- and left-handed) and planar versions. The symmetry groups of linear and planar tetrablocks are isomorphic, respectively, to the projective special linear group PSL(2, 7) of order 168 and the projective general linear group PGL(2, 7) of order 336. A class of structures formed by an assembly of tetrablocks having no common tetrahedra is introduced. Examples of tetrablock assembly over common face, leading to a Boerdijk–Coxeter helix, an α helix, and a helix used as one of collagen models are presented. DOI: 10.1134/S106377451903026X
INTRODUCTION Space group is a group of rigid discrete motions of three-dimensional Euclidean space Е3. Therefore, the invariance of Е3 with respect to a space group leads to stability of a crystal structure in the same way as, for example, the invariance of Е3 relative to translations leads to the momentum conservation law [1]. In the general case, symmetry regularities of a discrete (in particular, crystalline) structure can be determined from the requirement for its correspondence to certain properties of Е3 space, which are not reduced to only the space invariance relative to a space group. There are other (along with space groups) consequences of the zero curvature of space E3, which make it possible to determine possible symmetries of its discrete substructures. According to [2, p. 99] “there is an osculating Euclidean metric to the given (Riemannian [2]) metric along the length of a given line”. One of the consequences of this theorem is that [2, p. 101] “an observer who moves only along the given line and is content to restrict his measurements to an immediate neighbourhood of this line, … would not be able to see that he is outside the Euclidean space for as long as he neglects the infinitesimals of an order greater than the first”. Let this curve (line) coincide with the framework of a real linear chain molecule. If we formulate a theorem similar to [2, p. 99] for this chain, its applicability will be limited by the frames of some substructure, which must be revealed; then one may suggest that mapping this substructure (ordered, crystalline,
or noncrystalline structure) from a non-Euclidean space to space Е3 will make it possible to find its symmetry. Obviously,
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