Collapsibility and Near Universality for Vertex Minimal Paper Tori

Abstract

A paper torus is a piecewise linear isometric embedding of a flat torus into ${\mathbb{R}}^3$. 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 $\epsilon >0$, there is an $8$-vertex paper torus within $\epsilon$ of a unit equilateral triangle in the Hausdorff metric.

Same as previous version, except that the triangulation labels are canonical now and there are more pics. As previously, every calculation in the paper is checked by two downloadable Mathematica files. The reader can also download an extensive Java program which helps visualize the objects. Download instructions are at the end of the intro