Collapsibility and Near Universality for Vertex Minimal Paper Tori
Peter Doyle, Richard Evan Schwartz
TL;DR
The paper establishes that every flat torus without reflection symmetry can be realized as an 8-vertex paper torus (pup tent) by proving a Near Universality theorem that connects intrinsic pup-tent deformations to the modular surface $\mathcal{M}=H^2/SL_2(\mathbb{Z})$ via a map $\Phi:\mathcal{X}\to\mathcal{M}$ with $\Phi(\mathcal{U})=\mathcal{I}$. It also proves a Collapsibility result: for any good polygon $Q$, there exists a path in the pup-tent moduli space that collapses to $Q$ in the Hausdorff metric, showing that 8-vertex pup tents asymptotically realize degenerate limiting shapes. The construction centers on the Golden Valley, a boundary/accumulation subset of $\mathcal{X}$ consisting of gracefully immersed pup tents, built from an explicit 8-vertex triangulation and parametrized by the modular domain $\mathcal{F}$; a Good Path Lemma provides the essential deformation theory ensuring robustness of embeddings under perturbations. The results yield both intrinsic universality (within $\mathcal{U}$) and extrinsic approximations to canonical shapes (square, hexagonal, and equilateral configurations), with computational verification and visualization supporting the geometric constructions. Together, these findings advance the understanding of how polyhedral tori can densely approximate all flat torus moduli within the 8-vertex polygonal framework and suggest practical realizations via explicit algorithms and visual tools.
Abstract
A paper torus is a piecewise linear isometric embedding of a flat torus into $\R^3$. Following up on the $8$-vertex paper tori discovered by the second author, we prove universality and collapsibility results about these objects. One corollary is that any flat torus without reflection symmetry is realized as an $8$-vertex paper torus. Another corollary is that, for any $ε>0$, there is an $8$-vertex paper torus within $ε$ of a unit equilateral triangle in the Hausdorff metric.