Tetrahedron symmetry groups

Figure 13.17 Molecular structure of some sulfides of arsenic, stressing the relationship to the AS4 tetrahedron (point group symmetry in parentheses).

The point groups Td,Oh, and / are the respective symmetry group of Tetrahedron, Cube, and Icosahedron the point groups T, O, and I are their respective normal subgroup of rotations. The point group 7 is generated by T and the central symmetry inversion of the centre of the Isobarycenter of the Tetrahedron. 

These are groups which contain more than one threefold or higher axis. We will limit our consideration to the symmetry groups which describe the Platonic solids Td for the regular tetrahedron, Oh for the cube and regular octahedron, I/, for the regular dodecahedron and icosahedron, and JCh for the sphere. Some molecules in the cubic groups are shown below ... 

The dihedral group is the symmetry group for an n-sided regular polygon. The cubic group includes the octahedral group that represents the symmetries of the octahedron and the tetrahedral group that represents the symmetries of the tetrahedron. 

Tetrahedral sites Each tetrahedral or d-site is surrounded by 4 fi-sites to form a tetrahedron. There are 24 d-sites in each unit cell. Each d-site has the point symmetry group of 4(84). 

If there remains any doubt that it is the symmetry of the frame and not of the inscribed tetrahedron that determines the number of isomers, it can be dispelled by considering example 28. The four points to which the bivalent ligands a, b,. are attached form a tetrahedron with Om symmetry, slightly compressed from the regular one. This inscribed tetrahedron would allow only half the number of stereoisomers that are actually permitted by the frame which belongs to point group S, (5) if a = b = c = d H2. 

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