METHODS OF PERFECT COLORINGR. PérezGómez ^{1}andCeferino Ruiz ^{2}
Keywords: Mosaic, Symmetry, Group, Tiling, Crystallographic Groups. 1 IntroductionAll the plane periodic mosaics may be classified by means of the determination of their symmetry group, having 17 different classes as a result [1], although from an artistic point of view their number may be infinite. However, if we bear in mind the colors, the number of groups which determine the symmetry of the mosaic is greater [10]. These groups are called chromatic groups. R.L.E. Schwarzenberger [8] contributed certain clarifications to the work of classifying chromatic compositions combining two monochromatic groups. This paper presents two methods of coloring monochromatic plane and
periodic mosaics by means of which color may be distributed making use
of the symmetry group associated to the monochromatic mosaic and its subgroups.
The main result is the Theorem where it is proved
that, when the subgroup used for the two colorings is normal, the coloring
result obtained is independent from the method used.
2 First Method of mosaic coloringLet G be the crystallographic group associated to a periodic tiling
T, and G_{1} one of its crystallographic subgroups
of index k. In these conditions, the union of basic, suitable,
k
tiles for G define a basic tile for G_{1}.
Method 1 (M1)With k colors available, a basic tile for G_{1} is colored by assigning a color to each of the k basic tiles of the tiling T (which has been primarily determined). The coloration is then extended over the whole tiling by the action of group G_{1} on the basic tile. This form of coloring has the definition of the color stains, which we will deal with in a forthcoming article [3], and of the classes g_{i}G_{1} and G_{1}g_{i}, with i = 1,2,¼,k, to the left and to the right respectively. If G_{1} is a normal subgroup in G, both classes coincide. In another case we will see that the colorings which can be obtained are less fine such as will be shown in the following example. Example 1 Let G be the group of the type p4m associated to a pattern of squares which tiles the plane. Taking the translations T, and the reflection s_{L} from the G group represented in Figure 1, we form T È{s_{L}} and we take this as a set of generators of the subgroup G_{1} which proves to be of the cm type. G_{1} is not normal in G, as if we take a rotation of G, for example, r_{C}_{,p/2}, we have that:
Figure 1: Basic tile for G. Figure 2: 1st Basic tile for G_{1}.
Figure 3: Colored by the action of G_{1} on the 1st basic tile. Proposition 1 Let G_{1} be a crystallographic subgroup of G of index k. Let K^{G} be a basic tile for G and K^{G}^{1} a basic tile for G_{1} obtained by uniting k tiles g_{i}K^{G}, i = 1,2,¼,k, with g_{i}Î G and g_{i}G_{1}¹ g_{j}G_{1} if g_{i} ¹ g_{j}. The resulting coloring when applying M1 to the tiling T_{G} = {gK^{G}; g Î G} from K^{G}^{1} does not depend on K^{G}^{1}. Proof: Let K^{G}^{1} and L^{G}^{1} be two basic different G_{1} tiles, obtained by uniting g_{i}K^{G}, i = 1,2,¼,k and g_{i}ÎG, and [`g]_{s}K^{G}, s = 1,2,¼,k and [`g]_{s}ÎG, respectively. For each g_{i}K^{G} a g^{1}Î G exists such that g^{1}(g_{i}K^{G}) ÍL^{G}^{1}, therefore a unique [`g]_{s} exists such that g^{1}(g_{i}K^{G}) = [`g]_{s}K^{G}. Later g^{1}g_{i} = [`g]_{s} and g^{1} = [`g]_{s}g^{1}_{i}. The permutation of colors which takes the color from g_{i}K^{G} in that of [`g]_{s}K^{G} permutes the T_{G }tile colors, colored by M1 starting from K^{G}^{1}, in the colors obtained from L^{G}^{1}. In fact, any tile obtained starting from g_{i}K^{G} by the action of G_{1} will have the type h^{1}(g_{i}K^{G}). As h^{1} Î G_{1}, m = h^{1}(g^{1})^{1} ÎG_{1} and h^{1} = mg^{1}. Then, h^{1}(g_{i}K^{G}) = mg^{1}(g_{i}K^{G}) = m(g^{1}g_{i}K^{G}) = m([`g]_{s}K^{G}). With the tiles g_{i}K^{G} and [`g]_{s}K^{G} having the same color as h^{1}(g_{i}K^{G}) and m([`g]_{s}K^{G}), the permutation which takes the color from g_{i}K^{G} to that of [`g]_{s}K^{G} also takes that of h^{1}(g_{i}K^{G}) to that of m([`g]_{s}K^{G}). Figure 4: 2nd basic tile. Coloring by G_{1} If we change subgroup, logically, other results are obtained if the G_{1} index is maintained in G like the basic G tile, see Figure 5 in which a subgroup of the p1m1 type has been used. Figure 5. Definition 1 Let a periodic tiling be of R^{2}, in which each tile is colored with a color from among k possible colors. According to Senechal [9] we will say that kcoloring is perfect or compatible if each symmetry of the periodical mosaic discoloring induces a permutation of the k colors. This type of coloring makes all the tiles which have the same color (for example, red) be transformed by means of a symmetry of the discolored periodical mosaic in tiles of the same color (for example, blue); symmetries of the discolored periodical mosaic do not exist which may mix colors in the meaning of transforming, for example, some red tiles in blue and others, also red, in green. Proposition 2 The coloring done by M1 is perfect if and only if G_{1} is normal in G. Proof: Let us call G_{i} the set of elements of G which stabilize the color i :
(Note: By means of g(i) = j we want to express that all basic tiles for G colored in the i color passes by means of the element g to another of the j color). It is evident that G_{1} ÍG_{i}. Let us see if the other inclusion is fulfilled. Let K_{i} be an i colored tile. g_{1}ÎG_{1} and K exist, and they are unique. Taking gÎG_{i} we have a K_{i}¢ tile, K_{i}¢ = gK_{i} = gg_{1}K, of icolor. Then gg_{1}ÎG_{1} from where gÎG_{1}. Now, let K_{i} be a basic i colored tile for G, K_{i} = g_{1}K. We will obtain another basic tile for G of j color applying any g of G to K_{i }: gK_{i} = gg_{1}K. On the other hand, g_{1}¢ÎG_{1} and K_{j}, will exist and they are unique, first basic tile for G of j color, such that:
Let K_{i}¢ be another basic tile for G of color i:
3 Second Method of mosaic coloringMethod 2 (M2)The symmetries of G_{1} act upon a chasen basic tile, for G, which is painted with a determined color (for example, yellow) and all its images because of the action of G_{1} will have its the same color. Each one of the classes on the left hand side of G_{1} in G will be identified with one of the available k colors and the symmetries of such a class are utilized to transform the original tile (in our example, black) into tiles which have the color assigned to such a class. In this way, the tiling remains painted with k colors. Example 2 If M2 is applied to the basic tile
for G of Figure 1, and we take the same
generators for G_{1}  that is, s_{L}
and T the resulting coloring is the same that of Figure
4.
Proof: We can see that the technique utilized guarantees that it is a perfect kcoloring. Let the K¢ and K¢¢ tiles be of the same color. g¢ and g¢¢ of G, exist, belonging to the same class, g¢G_{1} = g¢¢G_{1}, and the basic tile for G, K, to color in such a way that:
Let g be any element of G:
later gK¢ and gK¢¢ has the same color (that corresponding to the gg¢G_{1} class). As a consequence, g induces a permutation of colors. Proposition 4 The correspondence which is established on coloring according to M2 between the elements of G and the group of permutations of k colors, S_{k}, is a homomorphism of groups. Proof: Taking left hand side classes a homomorphism can be defined between G and the group of the permutations of k elements S_{k} with the composition. In effect, representing the quotient G/G_{1} by
The application p: g Î
G ® p(g), where
p(g)
Î
S_{k }is the matrix
is a homomorphism. In effect:
In this homomorphism, used in the previous demonstration, we can verify that Ker p Í Ç_{i = 1}^{k} G_{i} = G^{1}, G_{i} being the stabilizer of i color, that is, fixed an i, G_{i} = {gÎG; gh_{i}G_{1} = h_{i}G_{1}}. Each one of the subgroups G_{i} is a class of conjugation of G_{1}. Remark 1 A permutation of the K colors is associated to each element of G. That is, all symmetry operations have an element of the group of permutations of order k, S_{k} associated. Besides, as the product of symmetries is corresponded to the composition of permutations, we have seen that the nucleus of the homomorphism p: G ® S_{k} is formed by all the symmetries which leave all the colors invariant because it is associated to the permutation identity. The knowledge of G/Ker p is useful for being isomorphous to p(G), but it has the problem that knowing Ker p we know nothing about the permutation, p(g), which corresponds to any g. Therefore we have to substitute Ker p for a subgroup G_{1}, Ker p Í G_{1}, by means of which one unique color remains invariant. In the particular case in which G_{1} = Ker p, it will be necessarily normal and the classes will coincide on the right and on the left. Remark 2 In the demonstration of the Proposition 3 we have taken classes from the left. We make this obsevation because it is not the same to take classes from one side as from the other. Only in the case of G_{1} being normal will the results coincide; this happens in the bicolor symmetry when k = 2. Remark 3 One color is assigned to each class on the left, in this way the action of the elements of that class transform the assigned color to the G_{1} class in it. Therefore, we can write G as finite union classes on the left, G = G_{1}Èh_{1}G_{1}È¼Èh_{k}_{1}G_{1}. Remark 4 We believe we contribute a different method to obtain perfect kcolorings on those noted in Senechal [9] and before, giving an interpretation of their mathematical significance. Corollary 1 If a tiling is colored by the action of a G group and one of its subgroups G_{1} according to M2, the number of colors coincides with the index of G_{1} in G. 4 Relationship between the two MethodsTheorem 1 If the subgroup G_{1} is normal in G, the obtained colorings for M1 and M2 coincide.Proof: It is deduced from abovementioned results and that, in this case:
Definition 2 Let us call a chromatic group, with k colors, the trio (G,G_{1},f ) where
Remark 6 The subgroup G^{1} = Ç_{gÎG} gG_{1}g^{1}, complete intersection of the conjugation family of G_{1}, is a normal and crystallographic subgroup of G. Besides, G^{1} = Ker f. Remark 7 The homomorphism f: G ® S_{k} induces a monomorphism:
References:
Rafael PérezGómez Dpt. Matemática Aplicada, EU Arquitectura Técnica, Univ. Granada, 18071  Granada, Spain. email: rperez@ugr.es Ceferino Ruiz
1991 Mathematical Subject Classification. Primary 20H15; Secondary 52C20. ^{1)} Research partially supported by a Research Group in Applied Mathematics grant FQM0191 ^{2)} Research partially supported by a DGICYT grant PB970785 and Research Group in Geometry grant FQM0203
