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Vertex-Minimal Paper Tori

Richard Evan Schwartz

TL;DR

This work determines the vertex-minimality for paper tori by proving there is no $7$-vertex example and constructing an $8$-vertex realization. The approach combines combinatorial hull analysis (Hull Lemma), Crofton's formula from integral geometry, and a computer-assisted robustness argument around a specific 8-vertex construction (the pup tent) to show a nearby flat embedding exists. The results settle a long-standing question about the minimum vertex count for polyhedral tori realizations of flat tori, and connect to broader themes in origami torus constructions and universal triangulations. The work also documents extensive computational methods and provides open avenues for further refinement and exploration of vertex-minimal embeddings.

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.

Vertex-Minimal Paper Tori

TL;DR

This work determines the vertex-minimality for paper tori by proving there is no -vertex example and constructing an -vertex realization. The approach combines combinatorial hull analysis (Hull Lemma), Crofton's formula from integral geometry, and a computer-assisted robustness argument around a specific 8-vertex construction (the pup tent) to show a nearby flat embedding exists. The results settle a long-standing question about the minimum vertex count for polyhedral tori realizations of flat tori, and connect to broader themes in origami torus constructions and universal triangulations. The work also documents extensive computational methods and provides open avenues for further refinement and exploration of vertex-minimal embeddings.

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.

Paper Structure

This paper contains 19 sections, 16 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Context and Main Results
  3. Seven Vertices
  4. Eight Vertices
  5. Further Developments
  6. Computer Assistance
  7. Acknowledgements
  8. Proof of Theorem \ref{['one']}
  9. Crofton's Formula
  10. Proof of the Main Theorem
  11. Proof of the Hull Lemma
  12. Proof of Theorem \ref{['two']}
  13. Robust Embedding
  14. The Proof Modulo a Crude Bound
  15. Proof of the Crude Bound Lemma
  16. ...and 4 more sections

Key Result

Theorem 1.1

There does not exist a $7$-vertex paper torus.

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 1.3: Hull
  • Corollary 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5: Cycle Rule
  • Lemma 3.1: Robust Embedding
  • ...and 6 more