Vertex-Minimal Paper Tori

2025/07/20 by Richard Evan Schwartz, Schwartz, Richard Evan · 1 voice · 1 citation
Mathematics · #math.MG

paper · pdf · doi:10.48550/arxiv.2507.14998

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.

Citations

• Vertex-minimal hyperbolic origami 2-torus
• Realizing abstract simplicial complexes with specified edge lengths

Cited by

• Collapsibility and Near Universality for Vertex Minimal Paper Tori

Discussions

• I’m really enamored with Richard Schwartz’s result that an embedded origami flat torus must have at least 8 vertices, but… can anyone explain how to see his “pup tents” as tori? arxiv.org/abs/2507.149 [bsky, 6 points, 1 comments]

Related

• Collapsibility and Near Universality for Vertex Minimal Paper Tori
• Polyhedral tori with minimal coordinates
• Knotted Polyhedral Tori
• A Universal Triangulation for Flat Tori
• Minimal Triangulations of Manifolds