Chiral Polyhedra Derived from Coxeter Diagrams and Quaternions

There are two chiral Archimedean polyhedra, the snub cube and snub dodecahedron together with their dual Catalan solids, pentagonal icositetrahedron and pentagonal hexacontahedron. In this paper we construct the chiral polyhedra and their dual solids in a systematic way. We use the proper rotational subgroups of the Coxeter groups and to derive the orbits representing the solids of interest. They lead to the polyhedra tetrahedron, icosahedron, snub cube, and snub dodecahedron respectively. We prove that the tetrahedron and icosahedron can be transformed to their mirror images by the proper rotational octahedral group so they are not classified in the class of chiral polyhedra. It is noted that vertices of the snub cube and snub dodecahedron can be derived from the vectors, which are linear combinations of the simple roots, by the actions of the proper rotation groupsand respectively. Their duals are constructed as the unions of three orbits of the groups of concern. We also construct the polyhedra, quasiregular in general, by combining chiral polyhedra with their mirror images. As a by-product we obtain the pyritohedral group as the subgroup the Coxeter group and discuss the constructions of pyritohedrons. We employ a method which describes the Coxeter groups and their orbits in terms of quaternions.

Snub polyhedra and organic growth

This paper describes a new application of polyhedral theory to the growth of the outer sheath of certain viruses. Such structures are often modular, consisting of one or two types of units arranged in a symmetric pattern. In particular, the polyoma virus has a structure apparently related to the snub dodecahedron. Here, we consider the problem of how such patterns might grow in time, starting from a given number N of randomly placed circles on the surface of a sphere. The circles are first jostled by random perturbations, then their radii are enlarged, then they are jostled again, and so on. This ‘yin–yang’ method of growth can result in some very close packings. When N =12, the closest packing corresponds to the snub tetrahedron, and when N =24 the closest packing corresponds to the snub cube. However, when N =60 the closest packing does not correspond to the snub dodecahedron but to a less-symmetric arrangement. Special attention is given to the structure of the human polyoma virus, for which N =72. It is shown that the yin–yang procedure successfully assembles the observed structure provided that the 72 circles are pre-assembled in clusters of six. Each cluster consists of five circles arranged symmetrically around a sixth at the centre, as in a flower with five petals. This has implications for the assembly of the capsomeres in a polyoma virus.

Archimedean polyhedra as the basis of tetrahedrally-coordinated frameworks

SummaryThe linkage of Archimedean polyhedra has been studied to provide trial models for tetrahedrally-coordinated structures such as zeolites. The cuboctahedron, rhombicuboctahedron, snub cube, icosidodecahedron, rhombicosidodecahedron, and snub dodecahedron cannot be used because four edges meet at a corner. The truncated tetrahedron, truncated cube, and truncated dodecahedron have triangular faces, and are unlikely to occur in silicate frameworks because of the instability of 3-rings. The truncated icosahedron and truncated icosidodecahedron have fivefold symmetry and cannot form four-connected frameworks with lattice symmetry. The truncated octahedron and truncated cuboctahedron may be linked either directly, in combination, or with square, hexagonal, or octagonal prisms. There are nine structures of which four are represented by sodalite, Linde A, faujasite, and ZK-5. The other five have the following properties: truncated octahedra linked H(H-S),Fd3m, a17·5 Å; truncated octahedra linked H′(H-S) and (H-H),P63/mmc, a12·4,c20·5 Å; truncated octahedra linked H′(H-S) and (H-H),P63/mmc, a17·5,c28·5 Å; truncated cuboctahedra linked O′(S-S),Im3m, a15·1 Å; truncated octahedra linked to truncated cuboctahedra H′(S-S),Fm3m, a31·1 Å. The first symbol specifies the linking unit (H hexagon, H′ hexagonal prism, O′ octagonal prism) while the symbols in brackets specify the type of faces opposing across the contact (S square).