Symmetries with Multiple High-Order Axes

12 SYMMETRIES WITH MULTIPLE HIGH-ORDER AXES 

To implement the above plan we consider the regular polyhedra, sometimes called the Platonic solids, of which there are five. By a regular polyhedron we mean a polyhedron 

By equivalent we mean, as usual, interchangeable by symmetry operations. The five regular polyhedra are depicted and their essential characteristics listed in Table 3.2. 

Our first task is to show that the five Platonic solids do, in fact, represent all the possibilities. This is quite easy to do. 

In order to construct a polyhedron, three or more of the desired faces must meet at a point so as to produce a closed, pyramidal (not planar) arrangement. Using equilateral triangles, we have the following possibilities  

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