Vertex-Minimal Paper Tori

Abstract

A paper torus is an embedded polyhedral torus that is isometric to a flat torus in the intrinsic sense. We prove that there does not exist a paper torus with $7$ vertices, and that there does exist a paper torus with $8$ vertices. This settles the question of the minimum number of vertices needed for a paper torus.

This version fixes a notational glitch. In Chapter 3 (and a bit in the introduction) I had been using indices 0,1,2 and I meant to use 1,2,3. I fixed this