Spherical Symmetry and the Platonic Solids

Any plane through the center of a sphere is a reflection plane, and any axis through the center is a rotation axis, as well as a rotation-reflection axis. In addition, the sphere also is centrosymmetric, which means that the center is a point of inversion. The resulting infinite-dimensional symmetry group of the sphere is usually denoted 

Van t Hoff published his findings in 1874 in Utrecht. In the same year, Le Bel came to the same conclusion, based on the investigation of optical rotatory power. An Enghsh translation of the original papers of both chemists can be found in [1]. 

The symmetry elements are labeled by the indices x, y, and z, which refer to their orientation in the Cartesian coordinate system, e.g., indicates the C3 axis, which is the diagonal of the positive Cartesian directions. This notation emphasizes that symmetry elements are tied to the coordinate system and stay fixed in space. 

Icosahedral symmetry is less obvious and thus more intriguing than cubic symmetry. Whilst tetrahedral and octahedral molecules were already known before the turn of the nineteenth/twentieth century, the first structural study of an icosahedral molecule was the closo-dodecaborane, in 1960. More examples would 

Each of the cosets refers to an inscribed cube, exactly as the three Q cosets in Csu ammonia refer to the three equivalent hydrogen sites. Hence, in a dodecahedron there will be five inscribed cubes. Since the order of the 4 group is 120, which equals 5 , it is tempting to think of this group as isomorphic to Ss, permuting the five inscribed cubes. This, however, is not the case since 4 contains 10-fold 

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