On the frequency module of the hull of a primitive substitution tiling.

Understanding the properties of tilings is of increasing relevance to the study of aperiodic tilings and tiling spaces. This work considers the statistical properties of the hull of a primitive substitution tiling, where the hull is the family of all substitution tilings with respect to the substitution. A method is presented on how to arrive at the frequency module of the hull of a primitive substitution tiling (the minimal {\bb Z}-module, where {\bb Z} is the set of integers) containing the absolute frequency of each of its patches.

Primitive substitution tilings with rotational symmetries.

This work introduces the idea of symmetry order, which describes the rotational symmetry types of tilings in the hull of a given substitution. Definitions are given of the substitutions σ and σ which give rise to aperiodic primitive substitution tilings with dense tile orientations and which are invariant under six- and sevenfold rotations, respectively; the derivation of the symmetry orders of their hulls is also presented.

Symmetry groups associated with tilings on a flat torus.

This work investigates symmetry and color symmetry properties of Kepler, Heesch and Laves tilings embedded on a flat torus and their geometric realizations as tilings on a round torus in Euclidean 3-space. The symmetry group of the tiling on the round torus is determined by analyzing relevant symmetries of the planar tiling that are transformed to axial symmetries of the three-dimensional tiling. The focus on studying tilings on a round torus is motivated by applications in the geometric modeling of nanotori and the determination of their symmetry groups.

Symmetry groups of single-wall nanotubes.

This work investigates the symmetry properties of single-wall carbon nanotubes and their structural analogs, which are nanotubes consisting of different kinds of atoms. The symmetry group of a nanotube is studied by looking at symmetries and color fixing symmetries associated with a coloring of the tiling by hexagons in the Euclidean plane which, when rolled, gives rise to a geometric model of the nanotube. The approach is also applied to nanotubes with non-hexagonal symmetry arising from other isogonal tilings of the plane.

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.