The colour symmetry of phase patterns formed by the interference of plane waves

Three plane waves in two dimensions or four plane waves in three dimensions interfere to produce patterns of amplitude and phase. If there is a rational relation between the wavevectors, both patterns repeat on a lattice, but the unit cell for amplitude (modulus of the complex wavefunction) is smaller than that for phase. Although the symmetry of the amplitude pattern is describable by the usual space groups, this is not so for the phase pattern, which requires colours to characterize it. An example is given that needs five colours, but, in general, the number needed is unlimited.