Symmetry of Structures That Can Be Approximated by Chains of Regular Tetrahedra

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 o...

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Bibliographic Details
Published in:Crystallography reports 2019-05, Vol.64 (3), p.367-375
Main Authors: Talis, A. L., Rabinovich, A. L.
Format: Article
Language:English
Subjects:
Assembly
Chains
Crystallographic Symmetry
Crystallography and Scattering Methods
Group theory
Mapping
Physics
Physics and Astronomy
Symmetry
Tetrahedra
Topology
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Summary: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.
ISSN:1063-7745
1562-689X
DOI:10.1134/S106377451903026X