
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 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.
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