Here, in reverse chronological order, are the papers that have been published as part of this research project.

Abstract

In the quest for new visually interesting polyhedra with regular faces, we define and present an infinite class of solids, constructed by placing regular polygons at the rotational axes of a polyhedral symmetry group. This new technique can be used to generate many existing polyhedra, including most of the Archimedean solids. It also yields novel families of attractive symmetric polyhedra.

On-line documents

Complete article (PDF, 431KB)
Project page

Abstract

Islamic star patterns are a beautiful and highly geometric art form whose original design techniques are lost in history. We describe one procedure for constructing them based on placing radially-symmetric motifs in a formation dictated by a tiling of the plane, and show some styles in which they can be rendered. We also show some results generated with a software implementation of the technique.

On-line documents

Complete article (PDF, 806KB)
An online version of the paper, published in the online journal Visual Mathematics
Project page

Abstract

This paper introduces and presents a solution to the "Escherization" problem: given a closed figure in the plane, find a new closed figure that is similar to the original and tiles the plane. Our solution works by using a simulated annealer to optimize over a parameterization of the "isohedral" tilings, a class of tilings that is flexible enough to encompass nearly all of Escher's own tilings, and yet simple enough to be encoded and explored by a computer. We also describe a representation for isohedral tilings that allows for highly interactive viewing and rendering. We demonstrate the use of these tools -- along with several additional techniques for adding decorations to tilings -- with a variety of original ornamental designs.

On-line documents

Complete article (PDF, 1216KB)
Project page

Abstract

A set of points in the plane induces a Voronoi diagram, a division of the plane based on proximity to points in the set. Voronoi diagrams have been used extensively in engineering and scientific disciplines, but the possibility of using them for creating abstract ornamental designs is largely unexplored. I present some techniques for creating attractive ornamental designs using Voronoi diagrams. I focus on two features of Voronoi diagrams that make them particularly useful artistic tools: their conservation of symmetry, which be used to construct interesting tilings of tne plane, and their continuity with respect to changes in the generators, which makes possible smooth, organic animations of tilings.

On-line documents

Complete article (PDF, 488KB)
Project page