The most efficient origami torus

Abstract

A paper torus, also called an origami torus, is a torus in three-dimensional space that is made from finitely many triangles which fit together in such a way that the sum of the angles around each vertex is 2 π . This article announces the result that there does not exist an origami torus with 7 vertices and there does exist an origami torus with 8 vertices. These results settle the question about the most efficient way, in terms of the number of vertices, to construct an origami torus.