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arXiv:2507.14998math.MG

Vertex-Minimal Paper Tori

Richard Evan Schwartz

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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 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
Journal reference
DOI
Subjects
math.MGMetric Geometry

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  1. Mathematician solves origami donut efficiency challenge with fewest folds 1 comment

    • Which is probably why Schwartz himself doesn't call them origami tori. He calls them paper tori. Here's the paper: https://arxiv.org/abs/2507.14998.

      aureateView on HNParent context

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