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On symmetry breaking of dual polyhedra of non‐crystallographic group H 3
Author(s) -
Myronova Mariia
Publication year - 2021
Publication title -
acta crystallographica section a
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.742
H-Index - 83
ISSN - 2053-2733
DOI - 10.1107/s2053273321002254
Subject(s) - polytope , polyhedron , dodecahedron , combinatorics , icosahedral symmetry , physics , lambda , crystallography , symmetry group , symmetry (geometry) , group (periodic table) , geometry , mathematics , chemistry , quantum mechanics
The study of the polyhedra described in this paper is relevant to the icosahedral symmetry in the assembly of various spherical molecules, biomolecules and viruses. A symmetry‐breaking mechanism is applied to the family of polytopes constructed for each type of dominant point λ. Here a polytope is considered as a dual of a polytope obtained from the action of the Coxeter group H 3 on a single point . The H 3 symmetry is reduced to the symmetry of its two‐dimensional subgroups H 2 , A 1 × A 1 and A 2 that are used to examine the geometric structure of polytopes. The latter is presented as a stack of parallel circular/polygonal orbits known as the `pancake' structure of a polytope. Inserting more orbits into an orbit decomposition results in the extension of the structure into various nanotubes. Moreover, since a polytope may contain the orbits obtained by the action of H 3 on the seed points ( a , 0, 0), (0, b , 0) and (0, 0, c ) within its structure, the stellations of flat‐faced polytopes are constructed whenever the radii of such orbits are appropriately scaled. Finally, since the fullerene C 20 has the dodecahedral structure of , the construction of the smallest fullerenes C 24 , C 26 , C 28 , C 30 together with the nanotubes C 20+6 N , C 20+10 N is presented.