9.28. How many symmetries (i.e., movements that translate a polyhedron into itself) are there for each of the regular polyhedra?

## § 5. Different Definitions of Regular Polyhedra
9.28. Any face of a regular polyhedron can be translated into any other by a motion. If the faces of the polyhedron are $n$-sided, there are exactly $2n$ self-coincidences that preserve one of the faces: $n$ rotations and $n$ symmetries relative to planes. Therefore, the number of self-coincidences (including the identity transformation) is $2 n \Gamma$.

The number of self-coincidences of a tetrahedron is 24, of a cube and octahedron - 48, of a dodecahedron and icosahedron - 120.

Note. Similar reasoning can show that the number of self-coincidences of a regular polyhedron is twice the product of the number of its vertices and the number of faces of its polyhedral angles.