Rene P. Felix

Colorings of hyperbolic plane crystallographic patterns

Abstract 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 G a hyperbolic plane crystallographic group, this question can be addressed by applying a general framework for coloring symmetrical patterns and using right coset colorings as a tool to study the subgroup structure of G. In this paper, we present colored patterns that emerge from the hyperbolic 3 · 4 · 3 · 4 · 3 · 3 tiling where all the symmetries of the uncolored tiling permute the colors of the patterns.

On index 2 subgroups of hyperbolic symmetry groups

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.

Color groups associated with square and hexagonal lattices

In this paper, we obtain the list of all color groups associated with a coloring of a square or hexagonal lattice Λ arising from sublattices of Λ. The results are obtained using a canonical representation of a sublattice L of Λ by a 2 × 2 matrix and no restriction on whether L is compatible or not with Λ is imposed.

On color groups of Bravais colorings of planar modules with quasicrystallographic symmetries

Abstract In this work we study the color symmetries pertaining to colorings of Mn = Z[ξ], where ξ = exp (2πi/n) for n ∈ {5,8,12} which yield standard symmetries of quasicrystals. The first part of the paper treats Mn as a four dimensional lattice Λ with symmetry group G and a result is provided on sublattices of Λ 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 Mn using an approach involving ideals.

Semi-perfect colourings of hyperbolic tilings

If G is the symmetry group of an uncoloured tiling, then a colouring of the tiling is semi-perfect if the associated colour group is a subgroup of G of index 2. Results are presented that show how to identify and construct semi-perfect colourings of symmetrical tilings. Semi-perfectly coloured tilings that emerge from the hyperbolic semi-regular tiling 8·10·16 are reported.

On subgroups of hyperbolic tetrahedral Coxeter groups

Abstract In this work we address the problem on the determination of the subgroup structure of crystallographic groups in hyperbolic space by deriving the low index subgroups of hyperbolic tetrahedral Coxeter groups and tetrahedral Kleinian groups. This paper continues the work giv en in [5, 6] on the subgroups of triangle groups.

On subgroups of crystallographic Coxeter groups.

A framework is presented based on color symmetry theory that will facilitate the determination of the subgroup structure of a crystallographic Coxeter group. It is shown that the method may be extended to characterize torsion-free subgroups. The approach is to treat these groups as groups of symmetries of tessellations in space by fundamental polyhedra.

Perfect precise colorings of plane regular tilings

Abstract Let (pq) denote the tessellation of the plane by regular p-gons meeting q at a vertex. In this paper, we consider the problem of coloring (pq) using q colors such that all the q colors appear at every vertex, and that every symmetry (or every direct symmetry) of the tiling (pq) induces a permutation on the set of q colors. Such a coloring is called a perfect precise coloring.

Enumeration of index 3 and 4 subgroups of hyperbolic triangle symmetry groups

Abstract This paper explores the area of crystallography on the hyperbolic plane, in particular the study of the subgroup structure of hyperbolic symmetry groups. In this work, the index 3 and 4 subgroups of triangle groups are derived, using color symmetry theory. This paper continues the work started in [7] on index 2 subgroups of hyperbolic symmetry groups.

Symmetries and color symmetries of a family of tilings with a singular point.

Tilings with a singular point are obtained by applying conformal maps on regular tilings of the Euclidean plane and their symmetries are determined. The resulting tilings are then symmetrically colored by applying the same conformal maps on colorings of regular tilings arising from sublattice colorings of the centers of the tiles. In addition, conditions are determined in order that the coloring of a tiling with singularity that is obtained in this manner is perfect.