Subgroups of crystallographic groups play an important role in many branches of mathematics, physics and crystallography such as representation theory, the theories of phase transitions, manifolds and in the comparative study of crystal structures [14]. In this work, the index 2 subgroups of a huge family of crystallographic groups called triangle groups are derived using black and white tilings. The focus of the work will be in determining the index 2 subgroups of triangle groups in the hyperbolic plane.

Abstract. In this work we study the color symmetries pertaining to colorings of M n ¼ Z½x, where x ¼ exp ð2pi=nÞ for n 2 f5; 8; 12g which yield standard symmetries of quasicrystals. The first part of the paper treats M n as a four dimensional lattice L with symmetry group G and a result is provided on sublattices of L which are invariant under the point group of G. The second part of the paper characterizes the color symmetry groups and color fixing groups corresponding to Bravais colorings of M n using an approach involving ideals.

One of the main goals in the study of color symmetry is to classify colorings of symmetrical objects through their color groups. The term color group is taken to mean the subgroup of the symmetry group of the uncolored symmetrical object which induces a permutation of colors in the coloring. This work looks for methods of determining the color group of a colored symmetric object. It begins with an index n subgroup H of the symmetry group G of the uncolored object. It then considers H-invariant colorings of the object, so that the color group H(*) will be a subgroup of G containing H. In other words, H ≤ H(*) ≤ G. It proceeds to give necessary and sufficient conditions for the equality of H(*) and G. If H(*) ≠ G and n is prime, then H(*) = H. On the other hand, if H(*) ≠ G and n is not prime, methods are discussed to determine whether H(*) is G, H or some intermediate subgroup between H and G.

In color symmetry the basic problem has always been to classify symmetrically colored symmetrical patterns [13]. An important step in the study of color symmetry in the hyperbolic plane is the determination of a systematic approach in arriving at colored symmetrical hyperbolic patterns. For a given uncolored semi-regular tiling with symmetry group

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 .