Vertex-Minimal Paper Tori

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.

Publication Details

Journal
ArXiv.org
Published
2025-07-20
Primary Topic
Geometric and Algebraic Topology
Type
article
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article

Vertex-Minimal Paper Tori

ArXiv.org
Geometric and Algebraic Topology
article

Vertex-Minimal Paper Tori

article en

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.

ArXiv.org
Openalex Percentile: Top 41%
Geometric and Algebraic Topology
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Vertex-Minimal Paper Tori · ArXiv.org (2025) | TGRS Research Map | TGRS