This paper studies colorings of patterns with multiple orbits, particularly those colorings where the orbits share colors. The main problem is determining when such colorings become perfect. This problem is attacked by characterizing all perfect colorings of patterns through the construction of sufficient and necessary conditions for a coloring to be perfect. These results are then applied on symmetrical objects to construct both perfect and non‐perfect colorings.

In this paper, orbit colorings of a lattice Λ were considered, extending the idea of obtaining lattice colorings based on cosets of a sublattice. Given a subgroup S of finite index in the symmetry group G of the lattice Λ, if T(S) is the subgroup of translations in G and L is the sublattice of Λ determined by T(S), then a homomorphism f from N = N G (T(S)) to the subgroup S C of permutations of the cosets of L in Λ is defined. This homomorphism was used to obtain S-orbit colorings of Λ and the corresponding color group H and color fixing group K. Moreover, S ker f ≤ K ≤ H ≤ N.

Let G be a group. A k-coloring of G is a surjection λ:G→{1,2,…,k}. Equivalently, a k-coloring λ of G is a partition P={P1,P2,…,Pk} of G into k subsets. If gP=P for all g in G, we say that λ is perfect. If hP=P only for all h∈H≤G such that [G:H]=2, then λ is semiperfect. If there is an element g∈G such that λ(x)=λ(gx−1g) for all x∈G, then λ is said to be symmetric. In this research, we relate the notion of symmetric colorings with perfect and semiperfect colorings. Specifically, we identify which perfect and semiperfect colorings are symmetric in relation to the subgroups of G that contain the squares of elements in G, in H, and in G∖H. We also show examples of colored planar patterns that represent symmetric perfect and symmetric semiperfect colorings of some groups.

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