Ma. Louise Antonette N. De Las Peñas

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.

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.

Using Dynamic Tools to Develop an Understanding of the Fundamental Ideas of Calculus.

Although dynamic geometry software has been extensively used for teaching calculus concepts, few studies have documented how these dynamic tools may be used for teaching the rigorous foundations of the calculus. In this paper, we describe lesson sequences utilizing dynamic tools for teaching the epsilon-delta definition of the limit and the fundamental theorem of calculus. The lessons were designed on the basis of observed student difficulties and the existing scholarly literature. We show how a combination of dynamic tools and guide questions allows students to construct their understanding of these calculus ideas.

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.

Colorings of single-wall carbon nanotubes

Abstract In this work, we describe a process to classify and characterize nanotubes with several types of atoms by constructing colorings associated with single-wall carbon nanotubes. We also illustrate how colored single-wall carbon nanotubes can be used to model rolled-up versions of particular two-dimensional patterns such as carbon-boron nitride ternary graphite-like monolayers.

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.

Colored patterns and the subgroups of their symmetries effecting color permutations

Abstract In this paper, we study colorings corresponding to the partition of the form of the symmetry group G of an uncolored pattern where Ji, H, K are subgroups of G such that K ≤ Ji ≤ H ≤ NG(K) and is a complete set of right coset representatives of H in G. In particular, we consider those colorings obtained when Y is partitioned into one set, two sets or singletons and determines the subgroup H* consisting of elements of G effecting color permutations.