Vertex-Minimal Paper Tori

TL;DR

This paper determines the minimum number of vertices required for a paper torus, proving no such torus exists with 7 vertices but one exists with 8 vertices, thus settling a key geometric question.

Contribution

It establishes the minimal vertex count for paper tori, resolving an open problem in geometric topology.

Findings

No paper torus with 7 vertices exists.

A paper torus with 8 vertices does exist.

The minimal vertex number for a paper torus is 8.

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.