Lattice Color Groups of Quasicrystals

It is well known that the points of a square lattice may be partitioned into two square subsets, related by a translation, but that the same is not true for triangular lattices. This allows a certain kind of anti-ferromagnetic order in tetragonal crystals which is not possible for hexagonal crystals. One often encounters similar situations in which a set of lattice points is to be partitioned into n “symmetry-related” subsets (to be properly defined below). If the lattice points correspond, for example, to atomic positions in a crystal then the different subsets may correspond to different chemical species or to n different orientations of a magnetic moment. One may also single out just one of the subsets to play a significant role such as in describing superlattice ordering. We shall address here the generalization of this question to quasiperiodic crystals. In doing so we shall introduce some aspects of the theory of color symmetry for periodic and quasiperiodic crystals. Please consult Ref. 1 for complete detail and a rigorous derivation of the results given here.