Enumerating and identifying semiperfect colorings of symmetrical patterns

Abstract

If G is the symmetry group of an uncolored pattern then a coloring of the pattern is semiperfect if the associated color group H is a subgroup of G of index 2. We give results on how to identify and enumerate all inequivalent semiperfect colorings of certain patterns. This is achieved by treating a coloring as a partition fhJ i Y i : i 2 I; h 2 Hg of G, where H is a subgroup of index 2 in G, J i H for i 2 I, and Y ¼ [ i2I Y i is a complete set of right coset representatives of H in G. We also give a one-to-one correspondence between inequivalent semiperfect colorings whose associated color groups are conjugate subgroups with respect to the normalizer of G in the group of isometries of R n .

Symmetric Perfect and Symmetric Semiperfect Colorings of Groups

“…If x ∈ X , the G-orbit of x is the set Gx = {gx : g ∈ G}, while the stabilizer of x ∈ X is given by Stab G (x) = {g ∈ G : gx = x}. Throughout this paper, we observe the coloring setting considered in [9]: G acts transitively on X , and for all x ∈ X , Stab G (x) = {e}. With this, we obtain a one-to-one correspondence between G and X given by g ↔ gx.…”

“…These eventually led to a coloring framework in [8] that resulted in a method to come up with perfect colorings of any symmetrical object. On the matter of semiperfect colorings, Felix and Loquias [9] gave results on how to obtain all semiperfect colorings of any symmetrical pattern.…”

Color groups arising from index-nsubgroups of symmetry groups

“…On the other hand, a method for identifying the color groups arising from index-2 subgroups of symmetry groups was outlined in De Las Peñ as et al (2011). This method was then applied to tilings of the hyperbolic plane to obtain various colorings, some of which are perfect, and others with associated color groups of index 2 in the symmetry group (called semi-perfect colorings; Felix & Loquias, 2008).…”

Semi-perfect colourings of hyperbolic tilings

“…In previous works, the study of perfect and semi-perfect colourings of symmetrical patterns [3][4][5][6][7] was carried out assuming certain considerations. For example, in [5], it was assumed that G acts transitively on the set S of tiles of a given tiling to be coloured and for all s 2 S, the stabiliser of s in G is {e}.…”

“…For example, in [5], it was assumed that G acts transitively on the set S of tiles of a given tiling to be coloured and for all s 2 S, the stabiliser of s in G is {e}. In this work, we present a method that will allow for the construction and enumeration of semiperfect colourings of symmetrical tilings where the stabiliser of a tile in G could be non-trivial.…”