The minimum vertex problem for origami tori

An origami torus is an embedded topological torus that is made from finitely many triangles in such a way that around each vertex the total angle sum is 2 pi. In other words, an origami torus is a torus-shaped polyhedron that is intrinsically isometric to a flat torus. Origami tori have been known to exist since the work of Burago and Zalgaller in 1960. A natural and long-standing question arises: What is the fewest number of vertices needed for an origami torus. In this talk I will explain my recent answer to this question: 8. If time permits I will discuss even more recent work, in which Peter Doyle and I prove that almost every flat torus is isometric to an 8-vertex origami torus.