2010 Bridges Conference

# Carlo H. Séquin

## Statement

My 2010 entries focus on the theme: "Math Becomes Art." Visualization models, constructed to gain an understanding of some mathematical concept, are enhanced to emphasize their aesthetic qualities. This is demonstrated with two topics; the first one concerns "Simple Knots," the second one "Regular maps." -- In 2009, together with a few students, we explored "The Beauty of Knots." For a few simple knots at the beginning of the ubiquitous knot table, we looked for aesthetically pleasing and truly 3-dimensional realizations and then created small sculpture models on a rapid prototyping machine. -- For the last few years I have been trying to find explicit 3D models for the embedding of regular maps on surfaces of appropriate genus. "Regular Maps" are networks of high symmetry in which all vertices, edges, and faces are indistinguishable from one another. There are 76 such regular maps on surfaces of genus-2 through genus-5. So far I have found models for about half of them.

## Artworks

"Regular Maps" are networks of edges and vertices embedded in closed 2-manifolds of arbitrary genus. The most familiar examples are the five Platonic solids, which represent such maps on surfaces of genus zero. There are 20 different regular maps of genus 3. They can readily be depicted in the Poincaré disk. It is a bigger challenge to find nice symmetrical embeddings on a handle-body of suitable genus. Here I have taken the regular map R3.2_{3,8}, comprising 12 vertices and 32 triangles, which always join eight to a vertex, and mapped it onto a Tetrus surface, maintaining the full 12-fold symmetry of the oriented tetrahedron.
Another simple knot turned into a sculpture model.
A simple knot turned into a model for a monumental sculpture.