3eclarin.pmd R.P. Felix and L.A . Eclarin 1 SCIENCE DILIMAN (JULY-DECEMBER 2014) 26:2, 1-20 Transitive Perfect Colorings of 2-Uniform Til ings Rene P. Fel ix University of the Philippines Diliman Lawrence A. Eclarin* Mariano Marcos State University _______________ *Corresponding Author ISSN 0115-7809 Print / ISSN 2012-0818 Online ABSTRACT In this work, a method to determine the nontrivial colorings of perfect a n d t r a n s i t i v e 2 - u n i f o r m t i l i n g s i s p r e s e n t e d . T h i s m e t h o d h a s b e e n applied to determine all nontrivial transitive perfect colorings of 2-uniform tilings that use the least number of colors. In addition, the equivalence of the colorings obtained was also ascertained. Keywords: Perfect colorings, 2-uniform tilings, equivalent colorings INTRODUCTION Numerous tilings of the plane by regular polygons have long been known, such as the regular tilings 36, 44, and 63 as well as the 8 semi-regular tilings 3.122, 4.6.12, 4.82, 3.4.6.4, 3.6.3.6, 34.6, 32.4.3.4, and 33.42, as illustrated by Grünbaum and Shepard (1987). These tilings are also known as Archimedean tilings. Given any pair of vertices of the tiling, the Archimedean tilings would exhibit symmetry (translation, rotation, reflection, or glide reflection) that sends one vertex to the other. That is, the vertices of an Archimedean tiling form one transitivity class under the action of the symmetry group of the tiling. For this reason, the arrangement of polygons about a vertex is the same for every vertex of an Archimedean tiling. For example, 3.4.6.4 means that a vertex is surrounded in cyclic order by a triangle (3-gon), a square (4-gon), a hexagon (6-gon) and a square. On the other hand, 63 is just 6.6.6, meaning a vertex is surrounded by three hexagons. Lesser known are the 2-uniform tilings, which are edge to edge tilings by regular polygons and where vertices of the tiling form two transitivity classes. These Transitive Perfect Colorings of 2-Uniform T ilings 2 tilings have 20 types, as shown in Figure 1. The enumeration of these tilings is attributed to Krötenheerdt (1969). Each of the 2-uniform tilings is described through the vertex types of the two transitivity classes. For example, (32.4.3.4; 3.4.6.4) describes a 2-uniform tiling where the vertices are of types 32.4.3.4, and 3.4.6.4. Figure 1. The twenty 2-uniform tilings in the Euclidean plane. R.P. Felix and L.A . Eclarin 3 In this study, the colorings of 2-uniform tilings, which fall under the theory of color s y m m e t r y, we r e co n s i d e r ed . T h e b a s i c p r o b l e m i n co l o r s y m m e t r y i s t h e classif ication of symmetrically colored symmetrical patterns. The work of Schwarzenberger (1984) provides a compendium of results on color symmetry, spanning decades of works. Senechal (1988) discussed results of interest and posed some problems on color symmetry. The paper of Rapanut (1988) provided useful results on subgroups of the seventeen plane crystallographic groups. In his paper, Roth (1993) determined that the minimum number n of colors that suff ice to color any multipattern with an associated symmetry group is 2 < n < 25. More recent works on coloring symmetrical patterns in the case of hyperbolic plane patterns have been done by De Las Peñas, Felix, and Laigo (2006). Frettlöh (2008) listed possible values for perfect k-colorings of some hyperbolic regular and Laves tilings. Felix and Loquias (2008) worked on semiperfect colorings. Precise perfect colorings were studied by Santos and Felix (2011). A study on transitive perfect colorings on semi-regular tilings was done by Gentuya (2013). PRELIMINARIES Let X be a set of objects in the plane and G the symmetry group of X. A coloring of X (using n colors ) is a surjective or onto function from X to . The coloring results in a partition of X where two elements x and y in X are assigned the same color ci if and only if they are elements of the same set Pi . We may therefore treat a coloring as a partition P of X. If for every , we say that the partition P is G-invariant and that the associated coloring is perfect. We also say that each induces a permutation of the colors . A special class of perfect colorings of X is the class of transitive perfect colorings of X . A perfect coloring of X is transitive if G acts transitively on the set of colors , i.e. , if ci and cj are any two colors in there is an element in G that sends ci to cj . Thus, not only is each symmetry in G associated with a unique permutation of the colors of the colored set, but given any two colors, the pattern formed by elements of one color is congruent to the pattern formed by elements of the other color. Hence, the colored pattern may be thought of as a disjoint union of colored subpatterns that are congruent to each other. 1, 2,…,   1, 2,…,  1, 2,…,      ∈   ∈   1, 2,…,   1, 2,…,   1, 2,…,   Transitive Perfect Colorings of 2-Uniform T ilings 4 Consider two colorings of the same set X and the corresponding colored patterns arising from the two colorings. The two colorings are said to be equivalent if one of the colored patterns may be obtained from the other colored pattern by (1) a bijection from the set of colors used in the f irst coloring to the set of colors used in the second coloring, (2) a symmetry in the symmetry group G of X, or (3) a combination of (1) and (2). This def inition of equivalence is adapted from Roth (1982). The concepts are illustrated using the colored patterns in Figure 2. In Figure 2(a), X = {1, 2, 3, 4, 5, 6, 7, 8} is a set of eight points with symmetry group G = = {e, a, a2, a3, b, ab, a 2b, a3b} D 4 where a is a 90o-counterclockwise rotation about the center of the conf iguration and b is a mirror reflection about the horizontal line passing through the center of a. Figure 2(b) exhibits a transitive perfect coloring of X. The coloring corresponds to the partition P = {{1, 2}, {3, 4}, {5, 6}, {7, 8}}, where the points in {1, 2}, {3, 4}, {5, 6}, and {7, 8} are colored red (R), blue (B), green (G), and yellow (Y), respectively. The 90o rotation a results in the permutation (RYGB), whereas the reflection b results in the permutation (BY). Given any two colors in the set { R, Y, G, B}, there is a symmetry in G that sends one color to the other color. The coloring of X in Figure 2(c) is not perfect. The only elements of G that induce a permutation of the colors are e, a 2, b, and a2b. Figure 2(d) and 〈 , 〉 ≅ 4  ′  〈 , 〉 ≅ 4  Figure 2. The set X consisting of eight distinct points with symmetry group showing (a) the mirror elements of G, (b) a transitive and perfect coloring, (c) a non-transitive and non-perfect coloring, (d)–(e) equivalent colorings, and (f ) the set of eight points with symmetry group with a perfect but non-transitive coloring. ≅  R.P. Felix and L.A . Eclarin 5 Figure 2(e) exhibit equivalent colorings of X. Using the bijection redblue and green  yellow and then applying the 90o rotation a we obtain the colored pattern in Figure 2(e) from the colored pattern in Figure 2(d). In Figure 2(f ), we illustrate a coloring of a set X’ which is perfect but not transitive. The symmetry group of X’ is also G = D 4 . Let x denote the upper right hand corner point and the point immediately below as shown in the f igure. The G-orbit of x refers to the set and consists of the images of x under the elements of G. The set of corner points of X’ is the G-orbit of x whereas the G-orbit of y is the remaining set of points in X’. The rotation a induces the permutations (RB) (GY) and the reflection b induces the permutation (GY). Hence, the coloring is perfect. The coloring is not transitive because there is no symmetry in G that will send the color yellow to blue. COLORING FRAMEWORK In this study, nontrivial transitive perfect colorings of 2-uniform tilings were considered. The approach of Felix (2011) where a coloring of a set is treated as a partition of the set was used. We made use of the theorem described below. In the theorem, denotes the set and is called the stabilizer in G of x. Theorem. Let X be a set and let G be a group acting transitively on X . 1 . If is a coloring of X for which G permutes the colors, then for e v e r y , t h e r e e x i s t s s u c h t h a t a n d t h e c o l o r i n g is described by the partition 2. Let and such that and . Then is a coloring of X with n colors for which G permutes the colors. (See Evidente, 2012 for the proof ). Remark: The above theorem determines all perfect colorings of X on the assumption that G acts transitively on the set X. The partition P above corresponds to a coloring of X that is perfect and transitive.   1  ∈       : ∈ .  ∈       ∶ ∞  : ∈   ≅  ∈ :   : ∈   Transitive Perfect Colorings of 2-Uniform T ilings 6 Based on the theorem, a procedure for arriving at nontrivial transitive perfect colorings of a 2-uniform tiling where the number of colors used is minimal is described. METHOD FOR DETERMINING TRANSITIVE PERFECT COLORINGS OF A 2-UNIFORM TILING USING THE LEAST NUMBER OF COLORS 1. Given a 2-uniform tiling, let G denote the symmetry group of the tiling. This group G is a plane crystallographic group. 2. The set X of tiles of the tiling is partitioned into a f inite number of G-orbits X i , i = 1, 2, ..., m. 3. For each G-orbit , X i , i = 1, 2, ..., m, obtain a G-orbit representative . 4. Obtain , i = 1, 2, ..., m. This group is a finite group, which is cyclic or dihedral . 5. Look for a proper subgroup of G of least index such that for each i = 1, 2, ..., m, a conjugate of is contained in . Assume there is a subgroup that was obtained in 5. 6. Obtain the -orbits of tiles of the tiling. 7. For each i = 1, 2, ..., m, choose a tile such that or equivalently . 8. Form the set and denote by the set ; i.e. , is the union of the -orbits i = 1, 2, ..., m. 9. Let be a complete set of left coset representatives of in G. 10. The partition describes a nontrivial transitive perfect coloring of the tiling using n colors . The coloring is given by the assignment , i = 1, 2, ..., n; i.e. , the tiles in are all colored (assigned the color) . ∈     ≅   ≅         ∈       1, 2,…,     1 ∪ 2 ∪ …  …∪       ,  1, 2,…,     , 1,2,…,    1, 2,…,   ⟶          R.P. Felix and L.A . Eclarin 7 For the basic ideas involved in formulating the procedure, some explanations are provided below. First, G-orbits X i was considered to be independently colored. Based on the theorem, the coloring or partition where and was used. Note that if , there exists such that and thus ; i.e. , and are conjugate. The number of colors corresponding to the partition is given by . Since using the least number of colors in the coloring is preferred, the same set of colors was used in coloring the G -orbits , and thus we take for i = 1, 2, ..., m for some where . If instead of , was used to represent the G-orbit X i then should be obtained. However, if for some . is required. All of the transitive perfect colorings of 2-uniform tilings were looked into, with the least number of colors (which were f inite) and in all cases, a subgroup was found. Otherwise, if no proper subgroup of G was found, then the option will be that = G, and the coloring will be trivial. The procedure used to arrive at the results for four of the twenty 2-uniform tilings, namely, the tilings (3 2.4.3.4; 3.4.6.4), (3 3.42; 32.4.3.4) 1 , (3 3.4 2; 32.4.3.4) 2 , and (36; 32.4.3.4) is illustrated as follows. The 2-Uniform T il ing (32.4.3.4; 3.4.6.4) Consider the 2-uniform tiling (32.4.3.4; 3.4.6.4) given in Figure 1. If we let G denote the symmetry group of the tiling, then , where are two linearly independent translations, a is a six-fold rotation centered at a hexagonal tile, and r is a reflection with symmetry axis passing through the center of a. These are shown in Figure 3 together with G-orbits of tiles where tiles of the tiling belonging to the same G-orbit have the same color. There are four G-orbits of tiles of the tiling: X 1 , the set of hexagons; X 2 , the set of squares; X 3 , the set of triangles whose sides are sides of squares, and; X 4 , the set of triangles that share one side with another triangle. Without loss of generality, tiles and can be chosen, as given in Figure 3. The stabilizer of the hexagonal tile 1 in X 1 , generated by the six-fold rotation a and the reflection r, is isomorphic to D 6 . For the square tile 2 in X 2 , the stabilizer is isomorphic to D 1 generated by the reflection r. The triangular : ∈   ∈     , ∈   ∈     1     : ∈   :   1, 2,…,   1, 2,…,   ,  1,2,…,         ,  1,2,…,   ∈   ∈     1    ∈           〈 , , , 〉 ≅ 6   ,   1 ∈ 1, 2 ∈ 2, 3 ∈ 3,  4 ∈ 4  Transitive Perfect Colorings of 2-Uniform T ilings 8 tile 3 in X 3 has a stabilizer generated by a three-fold rotation about its center and the reflection r. The subgroup is isomorphic to D 3 . The stabilizer of the triangular tile 4 is generated by a reflection r and is isomorphic to D 1 . · ·  Figure 3. The (32.4.3.4; 3.4.6.4) tiling showing the generators and the distinct G-orbits with the tiles , and .1 ∈ 1, 2 ∈ 2, 3 ∈ 3  4 ∈ 4  To f ind the subgroup of smallest index in G that yields a nontrivial transitive perfect coloring of the (32.4.3.4; 3.4.6.4) tiling, a plane crystallographic group H that contains subgroups of type D 6 , D 3 , and D 1 must be identif ied. Among the subgroups of G isomorphic to H is the subgroup , which is needed in this instance. The results of Rapanut (1988) indicate that . Moreover, the subgroups of G of type are of index n 2 or 3n 2, where n is a natural number. This gives the possible indices 1, 3, 4, 9, and so on. Since the least possible index n is being determined such that the coloring is nontrivial, the subgroup of index 3 is f irst considered. Figure 4 shows a unit cell corresponding to the subgroup . For simplicity, a unit cell of the tilings will be looked into. Schattschneider (1978) can be referred to for the unit cells corresponding to the 17 plane crystallographic groups and the symbols used to denote centers of rotations. An inspection of the unit cell shows that no 3-fold rotation of stabilizes a triangle in X 3 hence another low index subgroup must be considered. If we let , a subgroup of index 4 in G, each G-orbit splits into 2 or more -orbits. As shown in Figure 5, there are two -orbits of hexagons, three -orbits of squares, two -orbits of triangles in and three -orbits of triangles in X 4 .     ≅ 6   6   0 〈 2 1, 1 2, , 〉   0  0  〈 2, 2, , 〉            R.P. Felix and L.A . Eclarin 9 Figure 4. The (3 2.4.3.4; 3.4.6.4) tiling with the stabilizers in shown in a unit cell (shaded region). 0  Figure 5. The -orbits of the tiles of the tiling in (a) X 1 , (b) X 2 , (c) X 3 , and (d) X 4 .  Next, the set T = {t 1 , t 2 , t 3 , t 4 } is formed, where and . Any blue hexagonal tile in X 1 can be chosen as t 1 , any grey or green square tile in X 2 can be chosen as t 2 , any purple triangular tile in X 3 can be chosen as t 3 , and any orange or blue triangular tile in X 4 can be chosen as t 4 . Note that in the -orbits of tiles of the tiling in each X i , . Form the partition, Assigning distinct colors to each results in a transitive perfect coloring of the tiling using only four colors. Considering all possible combinations of tiles for T results in exactly four inequivalent transitive perfect 4-colorings of the (3 2.4.3.4; 3.4.6.4) tiling, as given in Figure 6(a)–(d). ∈           , , , .  Transitive Perfect Colorings of 2-Uniform T ilings 10 Figure 6. The four possible partitions for the (32.4.3.4; 3.4.6.4) tiling and their corresponding inequivalent transitive perfect 4-colorings. 〈 , , R.P. Felix and L.A . Eclarin 11 〈 , , , 〉 ≅ 4   In each of the 4-colorings generated, color permutations are given in Table 1 (where the colors red, green, blue, and yellow are denoted by R, G, B, and Y, respectively). Table 1. The color permutations correspond ing to generators of G Generator Color Permutation u (RG) (BY) v (RB (GY) a (BYG) r (BG) The 2-Uniform T il ings and (33.42.32.4.3.4) 1 and (33.42.32.4.3.4) 2 Consider the 2-uniform tiling (33.42.32.4.3.4) 1 with symmetry group , where u and v are two linearly independent translations, a is a 4-fold rotation and r is a reflection with symmetry axis not passing the center of rotation a, as shown in Figure 7. There are four G-orbits of tiles of the tiling: (1) the squares that share no side with other squares, (2) the squares that share one side with another square, (3) the triangles that share exactly one side with a square, and (4) the triangles that share two sides with squares. The stabilizers in G for a tile in each G-orbit are isomorphic to C 4 , D 1 , D 1 , and C 1 , respectively. The proper subgroup of G that contains them must be isomorphic to . The subgroup of least possible index n 2 where n is a natural number is , as generated by GAP and . 〈 , , , 〉 ≅ 4     4     〈 3, 3, , 〉  ∶ 9  Figure 7. The (33.42;32.4.3.4) 1 tiling with the generators u, v, a, and r. Transitive Perfect Colorings of 2-Uniform T ilings 12 Form where for i = 1, 2, 3, 4. Observe that in Figure 8(a) any yellow square tile can be chosen as t 1 , in (b) t 2 can be chosen from any of the square tiles of colors red, pink, or peach, in (c) t 3 can be chosen from any of the triangular tiles of the tiling of colors orange, purple, or green, and in (d) t 4 can be any of the nine colored triangular tiles of the tiling. From all the possible choices of tiles for inequivalent nontrivial transitive perfect colorings of the (33.42.32.4.3.4) 1 tiling is obtained. It may be checked that these colorings are inequivalent. One such transitive perfect coloring of the (33.42.32.4.3.4) 1 tiling using nine distinct colors is shown in Figure 9. The 2-uniform tiling (33.42.32.4.3.4) 2 has the same vertex types 33.42 and 32.4.3.4 but its symmetry group , where u and v are two linearly independent translations, p is a glide reflection, and q is a glide reflection with glide axis perpendicular to the glide axis of p, as shown in Figure 10.  1, 2, 3, 4   〈 , , , 〉 ≅   1 ⋅ 3 ⋅ 3 ⋅ 9 81 1, 2, 3, 4 ,  Figure 8. The -orbits of square and triangular tiles t i with .    R.P. Felix and L.A . Eclarin 13 Figure 9. A transitive perfect 9-coloring of the (33.42; 32.4.3.4) 1 tiling. There are three G-orbits of tiles of the tiling: X1 consisting of squares, X2 consisting of triangles that share two sides with squares, X3 and consisting of the triangles not included in X2. The stabilizers of the tiles of the tiling in each G-orbit are all isomorphic to C1. This is contained in any subgroup of G. Thus, we only need subgroups of least possible index greater than 1. Using GAP, three subgroups of index 2 are obtained, and these are given in Table 2. Figure 10. The (33.42; 32.4.3.4) 2 tiling with its generators and tiles belonging to the three G-orbits. Transitive Perfect Colorings of 2-Uniform T ilings 14 〈 , 2〉 〈 , 2〉          ∪ ,  ,     ,   ,   ,     (a)  1, (b)  2, (c)  3 where  〈 , 2〉  If , each G-orbit splits into two -orbits, as shown in Figure 11. Form T = {t 1 , t 2 , t 3 }, where t i is in X i for each i = 1, 2, 3 and . In Figure 11(a), t 1 can be any yellow square tile or grey square tile. In (b), t 2 can be any orange tile or purple tile, and in (c), t 3 can be any pink tile or blue tile. These give eight possible combinations for the set T. The subgroup is of index 2 in G and we have where h is the 2-fold rotation in G whose center is shown in Figure 11. If a set of f i xed tiles for and the half-turn h in G are considered, the partition could be obtained. Assigning the color red to and the color green to results in a transitive perfect 2-coloring of the (33.42; 3 2.4.3.4) 2 tiling. Nevertheless, it should be noted that 〈 2, 1 , 1 1〉  Subgroup Index Symmetry Group 2 pg 2 pg 2 p2 〈 , 2〉  〈 2, 〉  Table 2. Subgroups of of index 2〈 , , , 〉    , , 2 , .  That is, the symmetry maps the partition to the partition . Hence, the coloring described by is equivalent to the coloring described by . ∈   ,   Figure 11. -orbits of tiles of the tiling in and a half-turn h in G.   1, 2, 3   R.P. Felix and L.A . Eclarin 15   〈 , 2〉  In turn, this reduces the possible number of nontrivial transitive perfect colorings to four instead of eight, as presented in Figures 12(a)-(h). The coloring in (a) is equivalent to (b), (c) is equivalent to (d), (e) is equivalent to (f ), and (g) is equivalent to (h). Figure 12. The eight transitive perfect 2-colorings of (33.42; 32.4.3.4) 2 when . Transitive Perfect Colorings of 2-Uniform T ilings 16 Similarly, when we let , each G-orbit of tiles of the tiling splits into 2 -orbits and results into four inequivalent transitive perfect 2-colorings, as seen in Figure 13. If we let , each G-orbit of tiles of the tiling also splits into two -orbits. The resulting 2-colorings are shown in Figure 14. In all, there are 12 inequivalent transitive perfect 2-colorings of the (33.42; 32.4.3.4) 2 tiling. 〈 2, 〉 〈 2, 〉  〈 2, 1 , 1 1〉 〈 2, 1 , 1 1〉    Figure 14. The four inequivalent transitive perfect 2-colorings of (33.42; 32.4.3.4) 2 when .〈 2, 1 , 1 1〉  Figure 13. The four inequivalent transitive perfect 2-colorings of (33.42; 32.4.3.4) 2 when .〈 2, 〉    R.P. Felix and L.A . Eclarin 17 The 2-Uniform T il ing (36; 32.4.3.4) Using the method discussed, f ive inequivalent transitive perfect 25-colorings of (36; 32.4.3.4) were obtained. The least number of colors that can be used to color the tiling is 25. One such coloring is shown in Figure 15 where tiles of the same number are assigned the same color. Figure 16 indicates the remaining four transitive perfect 25-colorings of (36; 32.4.3.4). Applying the method for finding nontrivial transitive perfect colorings to all 2-uniform tilings, results were obtained, as summarized in Table 3. The patterns for all of the inequivalent colorings in each of the twenty 2-uniform tilings were illustrated. The results show that if n is the least number of colors needed in coloring a 2-uniform tiling in such a way that it is nontrivial, transitive, and perfect then . This r e s u l t i s ex p ected b a s ed o n t h e w o r k of Ro t h ( 1 9 9 3 ) . Applying the method to 3-uniform tilings, i.e. , tilings by regular polygons where the vertices of the tiling form three transitivity classes, is also of interest. Additional insights may be acquired from looking at transitive perfect colorings of 3-uniform tilings. The complete list of drawings for the 61 3-uniform tilings are found in Chavey (1989). 2 25  Figure 15. A transitive perfect 25-coloring of the (3 6; 32.4.3.4) tiling. Transitive Perfect Colorings of 2-Uniform T ilings 18 Figure 16. The other four motifs of a 25-coloring of the (36; 32.4.3.4) tiling. Table 3. The 2-uniform til ings with their correspond ing symmetry groups and the least number of colors needed for generating transitive perfect colorings (36; 34.6) 1 p6 4 64 (36; 34.6) 2 p6m 3 1 (36; 33.42) 1 cmm 2 8 (36; 33.42) 2 pmm 2 4 (36; 32.4.3.4) p6m 25 5 (36; 32.4.12) p6m 3 1 (36; 32.62) p6m 2 2 (34.6; 32.62) cmm 2 2 (33.42; 32.4.3.4) 1 p4g 9 81 (33.42; 32.4.3.4) 2 pgg 2 12 (33.42; 3.4.6.4) p6m 4 4 (33.42; 44) 1 cmm 2 2 (33.42; 44) 2 cmm 2 4 (33.4.3.4; 3.4.6.4) p6m 4 4 (32.62; 3.6.3.6) pmm 2 2 (3.4.3.12; 3.122 ) p4m 9 3 (3.42.6; 3.4.6.4) p6m 25 25 (3.42.6; 3.6.3.6) 1 pmm 3 3 (3.42.6; 3.6.3.6) 2 cmm 2 2 (3.4.6.4; 4.6.12) p6m 25 4 2-Uniform Til ings Symmetry Group Least Number n of Colors Number of Inequivalent n-Colorings R.P. Felix and L.A . Eclarin 19 ACKNOWLEDGMENTS We wish to thank the Commission on Higher Education for the f inancial support provided in the preparation of this research through the Off ice of the Vice-Chancellor for Research and Development of the University of the Philippines Diliman. REFERENCES Chavey D. 1989. Tilings by regular polygons–II: A catalog of tilings. Computers Math. Applic. 17(1-3): 147-165. Conway J. , Burgiel H. , Goodman-Strauss C. 2008. The symmetries of things. Wellesley, MA: A.K. Peters, Ltd. De Las Peñas M. , Felix R. , Laigo G. 2006. Colorings of hyperbolic plane crystallographic patterns. Z. Kristallogr. 221: 665-672. Evidente I. 2012. Colorings of regular tilings with a singular center (Unpublished doctoral dissertation). University of the Philippines, Diliman. Felix R. Color symmetry. Available from http://www.crystallography.fr/mathcryst/pdf/ Manila/Felix.pdf. Accessed on 3 November 2011. F e l i x R . , Lo q u i a s M . 2 0 0 8 . E n u m e r a t i n g a n d i d e n t i f y i n g s e m i p e r fec t co l o r i n g s of symmetrical patterns. Z. Kristallogr. 223(8): 483-491. Frettlöh D. 2008. Counting perfect colourings of plane regular tilings. Z. Kristallogr. 223: 773-776. The GAP Group. 2014. GAP – Groups, algorithms, and programming ( Version 4.7.4) [Software]. Available from http: //www.gap-system.org. Gentuya J. 2013. Transitive perfect colorings of semi-regular tilings (Unpublished master’s thesis). University of the Philippines, Diliman. Grunbaüm B. , Shephard G.C. 1977. Perfect colorings of transitive tilings and patterns in the plane. Discrete Math. 20: 235-247. Grunbaüm B. , Shephard G.C. 1987. T ilings and Patterns. New York: W. H. Freeman and Company. K r ö t e n h e e r d t O . 1 9 6 9 . D i e h o m o g e n e m o s a i k e n - t e r o r d n u n g i n d e r e u k l i d i s c h e n ebene. I, Wiss. Z. Mar tin-Luther-Univ. Halle-Wittenberg. Math.-natur. Reihe 18: 273-290. Rapanut T. 1988. Subgroups, conjugate subgroups, and -color groups of the seventeen plane crystallographic groups (Unpublished doctoral dissertation). University of the Philippines, Diliman. Roth R. 1982. Color symmetry and group theory. Discrete Math. 38: 273-296. Transitive Perfect Colorings of 2-Uniform T ilings 20 Roth R. 1993. Perfect colorings of multipatterns in the plane. Discrete Math. 122: 269-286. Santos R. , Felix R. 2011. Perfect precise colorings of plane regular tilings. Z. Kristallogr. 226: 726-730. Schattschneider D. 1978. The plane symmetry groups: their recognition and notation. Amer. Math. Monthly 85(6): 439-450. Schwarzenberger R. 1984. Colour Symmetry. Bull. London Math. Soc. 16: 209-240. Senechal M. 1988. Color symmetry. Comput. Math. Appl. 16(5-8): 545-553. _____________ Lawrence A. Eclarin is an Assistant Professor at the Department of Mathematics, Mariano Marcos State University. Rene P. Fel ix is a Professor at the Institute of Mathematics, University of the Philippines Diliman. He is a member of the Commission on Mathematical and Theoretical Crystallography, International Union of Crystalography, 2008 - 2014.