Symmetries of the cube

We first consider the symmetries of a cube. After talking (rather a lot) about this important concept, we will go back to Jennifer's puzzle from Chapter 8 to see how it casts light on the relation of the symmetries of a cube to the symmetries of a regular octahedron and those of a regular tetrahedron.

We picture the cube occupying a certain part of space; by a symmetry we mean the effect of a rotation of the cube about its center that brings it into a position occupying the same original part of space. Thus, for example, we may rotate the cube through an angle of π/2 about an axis passing through the midpoints of two opposite faces; this is a symmetry of the cube. It is plain that

1. if we follow one symmetry by another, the composite effect is again a symmetry,

2. if we reverse a symmetry we again get a symmetry, and

3. the “zero” rotation, that is, the “rotation” that holds every point fixed, is trivially a symmetry.

These three facts allow us to talk of the group of symmetries of the cube (or, more generally, of the group of symmetries of any polyhedron). Notice that a symmetry of a cube is completely determined when we describe the position of the points of the cube after the rotation – it is thus sufficient to describe the destinations of each vertex.