Gordon I. Williams

Representing the sporadic Archimedean polyhedra as abstract polytopes

We present the results of an investigation into the representations of Archimedean polyhedra (those polyhedra containing only one type of vertex figure) as quotients of regular abstract polytopes. Two methods of generating these presentations are discussed, one of which may be applied in a general setting, and another which makes use of a regular polytope with the same automorphism group as the desired quotient. Representations of the 14 sporadic Archimedean polyhedra (including the pseudorhombicuboctahedron) as quotients of regular abstract polyhedra are obtained, and summarised in a table. The information is used to characterize which of these polyhedra have acoptic Petrie schemes (that is, have well-defined Petrie duals).

Minimal covers of the prisms and antiprisms

Abstract This paper contains a classification of the regular minimal abstract polytopes that act as covers for the convex polyhedral prisms and antiprisms. It includes a detailed discussion of their topological structure, and completes the enumeration of such covers for convex uniform polyhedra. Additionally, this paper addresses related structural questions in the theory of string C-groups.

Polytopes with Preassigned Automorphism Groups

We prove that every finite group is the automorphism group of a finite abstract polytope isomorphic to a face-to-face tessellation of a sphere by topological copies of convex polytopes. We also show that this abstract polytope may be realized as a convex polytope.

The monodromy group of the n-pyramid

The monodromy group is an important tool for encoding and understanding the combinatorial structure of polytopes, both convex and abstract. In the current work we investigate the structure of the monodromy groups of the infinite family of pyramids over n-gons, as well as the structure of their minimal regular covers.

Pyramids Over Regular 3-Tori

In this paper we exhibit the first infinite family of abstract 4-polytopes whose connection groups are not string C-groups. In addition we present some new results on methods of representing the connection group of a polytope in terms of its automorphism group. We also analyze the connection groups of all of the pyramids over the finite regular 3-tori.

Fully truncated simplices and their monodromy groups

Abstract We describe a simple way to manufacture faithful representations of the monodromy group of an n-polytope. This is used to determine the monodromy group for 𝓣n, the fully truncated n-simplex. As by-products, we get the minimal regular cover for 𝓣n, along with the analogous objects for a prism over a simplex.

Operations on Oriented Maps

A map on a closed surface is a two-cell embedding of a finite connected graph. Maps on surfaces are conveniently described by certain trivalent graphs, known as flag graphs. Flag graphs themselves may be considered as maps embedded in the same surface as the original graph. The flag graph is the underlying graph of the dual of the barycentric subdivision of the original map. Certain operations on maps can be defined by appropriate operations on flag graphs. Orientable surfaces may be given consistent orientations, and oriented maps can be described by a generating pair consisting of a permutation and an involution on the set of arcs (or darts) defining a partially directed arc graph. In this paper we describe how certain operations on maps can be described directly on oriented maps via arc graphs.

Using the Fractal Paintbrush

Since their popularization in the 1980s, fractals have been used by artists and computer scientists as a tool to create compellingly intricate images and to imitate organic forms. This article explores some new methods for generating artwork with iterated functions systems, the computational tool central to the creation of many fractal images. Details of the method and examples are included, with special attention given to the mathematical and computer science techniques used to produce the images.