Collapsibility and Near Universality for Vertex Minimal Paper Tori

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 $\epsilon>0$, there is an $8$-vertex paper torus within $\epsilon$ of a unit equilateral triangle in the Hausdorff metric.