Polychromatic symmetry

Polychromatic symmetry is a colour symmetry which interchanges three or more colours in a symmetrical pattern. It is a natural extension of dichromatic symmetry. The coloured symmetry groups are derived by adding to the position coordinates (x and y in two dimensions, x, y and z in three dimensions) an extra coordinate, k, which takes three or more possible values (colours).

An example of an application of polychromatic symmetry is crystals of substances containing molecules or ions in triplet states, that is with an electronic spin of magnitude 1, should sometimes have structures in which the spins of these groups have projections of + 1, 0 and -1 onto local magnetic fields. If these three cases are present with equal frequency in an orderly array, then the magnetic space group of such a crystal should be three-coloured.

Example

The group p3 has three different rotation centres of order three (120°), but no reflections or glide reflections.

There are two distinct ways of colouring the p3 pattern with three colours: p3[3]1 and p3[3]2 where the figure in square brackets indicates the number of colours, and the subscript distinguishes between multiple cases of coloured patterns.

Taking a single motif in the pattern p3[3]1 it has a symmetry operation 3', consisting of a rotation by 120° and a cyclical permutation of the three colours white, green and red as shown in the animation.

This pattern p3[3]1 has the same colour symmetry as M. C. Escher's Hexagonal tessellation with animals: study of regular division of the plane with reptiles (1939). Escher reused the design in his 1943 lithograph Reptiles and it was also used as the cover art of Mott the Hoople’s debut album.