Effect of rotational symmetry on colored lattices

A colored lattice L(c) has a geometrical lattice L. A subgroup lattice L' of L and each of its cosets consist of like-colored points, each coset having a different color. The index of L' in L is given by Delta, the determinant of the matrix (t(jk)) that converts L into L'. This is the order of the factor group {L/L'}, and is also the number n of colors present. The crystal systems-i.e., the combinations of rotational symmetry axes-of L(c), L, and L' are all the same, but L and L' may have different centerings; this results in 39 combinations of centerings, called here the 39 symmetry types of colored lattices. These types are tabulated here, together with the special forms taken by (t(jk)) and the formulas for Delta. In only 7 of the 39 types can the number of colors be arbitrary; in most types certain numbers of colors are impossible.