Bridges 2010 Short Paper

We present models of polyhedra built from 12 heptagons meeting three per vertex. Unlike the analogous case with 12 pentagons, where there is a single unique combinatorial structure, there are six combinatorially distinct ways to combine 12 heptagons, meeting three per vertex, into a (possibly self-intersecting) polyhedron. We identified realizable (non-self-intersecting) examples for five of the six possible structures, and fabricated physical models of them. They all necessarily have genus 2 (topologically equivalent to a 2-holed donut), and they appear in a variety of aesthetically pleasing symmetries. These models demonstrate a form of art emerging from mathematics.