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8, in which, two bond lengths can be identified. A continuous range of equisymmetric structures is permitted. Within that range, the other Archimedean polyhedra based on cubic geometry appear, in turn, as the local sets of vertices about the principal rotational axes are allowed to coalesce. 9 shows how contractions of local sets of 8, 6 and 4 vertices onto the poles of C4, C3 and C2 axes, respectively, recover the octahedron, cube and cuboctahedron and thereby identify the 06, 08 and O12 orbits of Oh symmetry.

Space group 227 (Fd3m O 7) describes the diamond crystal structure and we see from this information that the structure is non-symmorphic and that while the factor group is of order 48, not all of these symmetry operations are common to the point symmetry group Oh. 5 lists the 'point' symmetry operations of O 7, in which the non-primitive translation (89 89 1) accompanying some rotations is required to mimic some of the operations of the Oh point group. 5 reveals that the 24 point symmetry operations of the Td point group are present amongst the 48 operations required to generate the isomorphic group to Oh.

18 (a) The regular orbit of T h symmetry as an elliptical projection and (b) in perspective displaying the structure as a modified small rhombicuboctahedron, in which there is no reflection symmetry across face diagonals of the inscribing cube. 19b-d, the local sets of 10, 6 and 4 vertices of the great rhombicosidodecahedron identified about a representative pole position on a face and, then, in the second column of diagrams, as fully decorated elliptical projections of the 120-vertex cage. There are 6 five-fold rotational symmetry elements in an object of Ih point symmetry.

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