The icosahedral symmetry group H3 of order 120 and its dihedral subgroup H2 of order 10 are used for exact geometric construction of polytopes that are known to exist in nature. The branching rule for the H3 orbit of the fullerene C60 to the subgroup H2 yields a union of eight orbits of H2: four of them are regular pentagons and four are regular decagons. By inserting into the branching rule one, two, three or n additional decagonal orbits of H2, one builds the polytopes C70, C80, C90 and nanotubes in general. A minute difference should be taken into account depending on whether an even or odd number of H2 decagons are inserted. Vertices of all the structures are given in exact coordinates relative to a non-orthogonal basis naturally appropriate for the icosahedral group, as well as relative to an orthonormal basis. Twisted fullerenes are defined. Their surface consists of 12 regular pentagons and 20 hexagons that have three and three edges of equal length. There is an uncountable number of different twisted fullerenes, all with precise icosahedral symmetry. Two examples of the twisted C60 are described.
[Show abstract][Hide abstract] ABSTRACT: The principle of affine symmetry is applied here to the nested fullerene cages (carbon onions) that arise in the context of carbon chemistry. Previous work on affine extensions of the icosahedral group has revealed a new organizational principle in virus structure and assembly. This group-theoretic framework is adapted here to the physical requirements dictated by carbon chemistry, and it is shown that mathematical models for carbon onions can be derived within this affine symmetry approach. This suggests the applicability of affine symmetry in a wider context in nature, as well as offering a novel perspective on the geometric principles underpinning carbon chemistry.
[Show abstract][Hide abstract] ABSTRACT: This paper considers Platonic solids/polytopes in the real Euclidean space
R^n of dimension 3 <= n < infinity. The Platonic solids/polytopes are described
together with their faces of dimensions 0 <= d <= n-1. Dual pairs of Platonic
polytopes are considered in parallel. The underlying fi?nite Coxeter groups are
those of simple Lie algebras of types An, Bn, Cn, F4 and of
non-crystallographic Coxeter groups H3, H4. Our method consists in recursively
decorating the appropriate Coxeter-Dynkin diagram. Each recursion step provides
the essential information about faces of a speci?c dimension. If, at each
recursion step, all of the faces are in the same Coxeter group orbit, i.e. are
identical, the solid is called Platonic.
Acta Crystallographica Section A 07/2014; 70(4):358-363. DOI:10.1107/S205327331400638X
[Show abstract][Hide abstract] ABSTRACT: Exact icosahedral symmetry of C60 is viewed as the union of 12 orbits of the symmetric subgroup of order 6 of the icosahedral group of order 120. Here, this subgroup is denoted by A2 because it is isomorphic to the Weyl group of the simple Lie algebra A2. Eight of the A2 orbits are hexagons and four are triangles. Only two of the hexagons appear as part of the C60 surface shell. The orbits form a stack of parallel layers centered on the axis of C60 passing through the centers of two opposite hexagons on the surface of C60. By inserting into the middle of the stack two A2 orbits of six points each and two A2 orbits of three points each, one can match the structure of C78. Repeating the insertion, one gets C96; multiple such insertions generate nanotubes of any desired length. Five different polytopes with 78 carbon-like vertices are described; only two of them can be augmented to nanotubes.
Acta Crystallographica Section A: Foundations and Advances 11/2014; 70(6). DOI:10.1107/S2053273314017215
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