The three-colored three-dimensional space groups

Abstract

This article contains a table of the groups of combined geometrical and color-permutational symmetry operations that leave a certain kind of three-colored, three-dimensionally periodic object apparently unchanged. The asymmetric units - 'motifs' - of the object are all either geometrically congruent to, or are mirror images of, one another. Each motif has a 'color' representing a scalar quality of some kind, and three different colors of motifs are assumed to occur in the object. Two types of three-colored space groups exist: type I in which all of the geometrical lattice translations leave all the motifs unchanged in color, and type II in which at least one of the geometrical lattice translations requires a permutation of the three colors in order to restore the original appearance. There are 88 three-colored space groups of type I, and 341 of type II. Type I can belong only to the trigonal, hexagonal and cubic systems; type II can belong to any system, except the cubic. A notation for the three-colored, three-dimensional space groups is proposed. It is based on similar principles to those used in the article on colored point groups by Harker [Acta Cryst. (1976), A32, 133-139].