Origami Tori

par Richard Schwartz (Brown University & IHES)

lundi 5 oct. 2026, 14:00 → 15:15 Europe/Paris

Amphithéâtre Léon Motchane (IHES)

An origami torus is a piecewise affine isometric embedding of a flat torus into R³. It is surprising that these things exist, but thanks to the work of Burago and Zalgaller in 1960 they do. 30 years later, B&Z proved that every flat torus is realized as an origami torus. In 2023, Lazarus and Tallerie proved that you can realize all flat tori using a single underlying triangulation with about 2500 vertices.

The question remained: how many vertices do you need to make an origami torus? In this talk I will prove that the minimum number of vertices needed is 8. I will also discuss joint work, with Peter Doyle, proving that almost every flat torus is represented by an 8-vertex origami torus. Finally, I will explain work-in-progress with Doyle, Fabian Lander, and Steve Trettel, which shows that in fact every flat torus is represented by an 8-vertex origami torus. If I have time, I will also discuss hyperbolic versions of this business.