The Rotations of a Cube

A symmetry of a cube is a permutation of its eight corners that sends edges to edges; in other words if a pair of corners are joined by an edge, then each symmetry must send this pair to another pair that is also joined by an edge. A cube has 48 symmetries, 24 of which can be realised physically on a solid cube; the other 24 effectively turn the cube inside out. To see why there are 24 physical symmetries, notice that a cube has six faces any of which can be moved to the bottom, and this face can then be rotated into four different positions: 6 × 4 = 24. This is explained in my book Symmetry and the Monster, on page 8, and several readers have asked whether each of the 24 positions can be reached by doing a single rotation. The answer is yes, and here is how this works.

First one should understand that each rotation must leave the cube occupying the same space before and after. This means that without distinguishing the faces of the cube from one another by colours, or numbers, or some other markings, we cannot tell whether a rotation has been done. Each possible rotation has an axis — think of it as a spindle going through the centre of the cube. There are three possibilities for this spindle. Either it goes through the centres of two opposite faces, or through the centres of two opposite edges, or through two opposite corners.

A cube has six faces, so there are three pairs of opposite faces. For a spindle going through the centres of two opposite faces there are two possible 90° rotations, one going one way, one the other. This gives six 90° rotations. There is also a 180° rotation when the spindle goes through the centres of two opposite faces. This gives three further rotations.

A cube also has twelve edges, so there are six pairs of opposite edges. For a spindle going through the centres of two opposite edges, the only possibility is a rotation of 180˚. This gives six rotations, one for each pair of opposite edges.

Finally a cube also has eight corners, so there are four pairs of opposite corners. For a spindle going through two opposite corners, there are two possible rotations, both of 120°, one going one way, one the other. This means there are eight 120° rotations, two for each pair of opposite corners.

We have counted 6+3+6+8 different rotations. Add the ‘rotation’ by 0°, which does nothing, and we have a group of 24 rotations. This is the rotation group of the cube.