Burnside's Lemma - Example Application

Example Application

The number of rotationally distinct colourings of the faces of a cube using three colours can be determined from this formula as follows.

Let X be the set of 36 possible face colour combinations that can be applied to a cube in one particular orientation, and let the rotation group G of the cube act on X in the natural manner. Then two elements of X belong to the same orbit precisely when one is simply a rotation of the other. The number of rotationally distinct colourings is thus the same as the number of orbits and can be found by counting the sizes of the fixed sets for the 24 elements of G.

• one identity element which leaves all 36 elements of X unchanged
• six 90-degree face rotations, each of which leaves 33 of the elements of X unchanged
• three 180-degree face rotations, each of which leaves 34 of the elements of X unchanged
• eight 120-degree vertex rotations, each of which leaves 32 of the elements of X unchanged
• six 180-degree edge rotations, each of which leaves 33 of the elements of X unchanged

A detailed examination of these automorphisms may be found here.

The average fix size is thus

Hence there are 57 rotationally distinct colourings of the faces of a cube in three colours. In general, the number of rotationally distinct colorings of the faces of a cube in n colors is given by