Geometry, orbits and decorations

It is important to understand the notion of an orbit of structure, for an object exhibiting a particular point symmetry, in order to operate the GT Calculator. 

how these orbits fit into the general geometry of polygons and polyhedra  

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The examination of coordinate transformations as local contractions and expansions of decorations about the poles of the principal rotational axes on the unit sphere for objects of Oh symmetry leads to intermediate geometries corresponding to particular Archimedean polyhedra related to the cube. In a similar manner, partial contractions and expansions of the decorations of the regular orbit of Ih point symmetry, i.e. the vertices of the great rhombicosidodecahedron, leads to the remaining polyhedra within the icosahedral family of Archimedean structures and orbits of Ih. 

Myriads of structures of isopoly and heteropolyacid anion cages are found as minerals with many different metal cations completing the chemical structures. The basic tetrahedral or near tetrahedral local geometry of, for example, component SiO " units can be formed between neighbouring shells of decorations of cubic vertices. Three examples, which emphasize the orbit-by-orbit perspective are shown in Figures 2.28 to 2.30. 

The observation that the permutation character on a set of vertices and the a-character for a set of ns orbitals localized on these vertices are the same is illustrated in Figure 3.1, using as in Chapter 2, the example of the ammonia molecule, but, in this case, considering the permutations of the hydrogen Is atomic orbitals about the equilateral triangular geometry of an O3 orbit decorated in this manner. 

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