Effect of rotational symmetry on colored lattices
Abstract
A colored lattice Lc has a geometrical lattice L. A subgroup lattice Lâ² of L and each of its cosets consist of like-colored points, each coset having a different color. The index of Lâ² in L is given by Î, the determinant of the matrix (tjk) that converts L into Lâ². This is the order of the factor group {L/Lâ²}, and is also the number n of colors present. The crystal systemsâi.e., the combinations of rotational symmetry axesâof Lc, L, and Lâ² are all the same, but L and Lâ² 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 (tjk) and the formulas for Î. In only 7 of the 39 types can the number of colors be arbitrary; in most types certain numbers of colors are impossible.