PERFECT COLORINGS OF MULTIPATTERNS IN THE PLANE

In this paper we consider the problem of finding nontrivial perfect co lorings Of planar multipatterns. By a 'multipattern' we mean a symmetr ic planar figure, with symmetry group G, having several classes of mot ifs each transitive under the action of G. A perfect or symmetric colo ring, then, is an assignment of colors to the motifs such that each sy mmetry operation of the group induces a unique permutation of the colo rs; we will here assume further that the action of G on the colors is transitive. If G is a wallpaper or frieze group, this is always possib le and there is a number N(G), the coloring number of G, which is the minimum number (> 1) of colors which suffice to color any multipattern with symmetry group G. In the case of finite groups, all multipattern s have nontrivial perfect colorings with the following exceptions: the re are no nontrivial perfect colorings if some motif is fixed by all g is-an-element-of G, or if G is the dihedral group dn with n = 2r, r g reater-than-or-equal-to 1 and motifs lie on all reflection axes. For w allpaper groups, 2 less-than-or-equal-to N (G) less-than-or-equal-to 2 5. As a consequence it follows that every periodic k-isohedral tiling of the plane has a perfect coloring using at most 25 colors.