Vertex-Minimal Paper Tori
Metric Geometry
2026-01-16 v9
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 vertices, and that there does exist a paper torus with vertices. This settles the question of the minimum number of vertices needed for a paper torus.
Keywords
Cite
@article{arxiv.2507.14998,
title = {Vertex-Minimal Paper Tori},
author = {Richard Evan Schwartz},
journal= {arXiv preprint arXiv:2507.14998},
year = {2026}
}
Comments
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