Perfect colorings of multipatterns in the plane

Abstract

In this paper we consider the problem of finding nontrivial perfect colorings of planar multipatterns. By a ‘multipattern’ we mean a symmetric planar figure, with symmetry group G, having several classes of motifs each transitive under the action of G. A perfect or symmetric coloring, then, is an assignment of colors to the motifs such that each symmetry operation of the group induces a unique permutation of the colors; 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 possible 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 multipatterns have nontrivial perfect colorings with the following exceptions: there are no nontrivial perfect colorings if some motif is fixed by all gϵG, or if G is the dihedral group dn with n=2 r , r⩾1 and motifs lie on all reflection axes. For wallpaper groups, 2⩽ N( G)⩽25. As a consequence it follows that every periodic k-isohedral tiling of the plane has a perfect coloring using at most 25 colors.