Mathematicians have solved a tricky geometry question about constructing polyhedral approximations of a torus, the doughnut-shaped surface familiar from both topology and everyday life, says Scientific American. In a paper posted on a preprint server in August 2025, Richard Evan Schwartz, a mathematician at Brown University, identified the exact minimum number of vertices (corners) needed to create a “flat” polyhedral torus, one that locally behaves like a smooth torus even though its surface is built from flat faces. His work settles a problem that had remained open for years, showing that seven vertices are not enough but that eight suffice to produce an intrinsically flat polyhedral torus.
The question dates back decades and stems from an attempt to understand how complex surfaces can be represented in a piecewise linear form. A torus built from flat polygons differs from a smooth surface because each vertex brings curvature; the challenge was to find the sparsest such network of polygons that still preserves intrinsic flatness, meaning that the sum of angles around each vertex matches that of a smooth torus. Schwartz first proved that no configuration of seven vertices could satisfy the necessary conditions, drawing on earlier research and computational exploration. Then, by constructing and analyzing an example with eight vertices, he demonstrated that the long-sought minimum exists.
Schwartz’s result matters because it illuminates deep connections between combinatorial geometry, topology, and discrete mathematics. Although the shapes he studied are highly abstract, and even harder to visualize than typical origami projects, they reflect fundamental questions about how curved spaces can be approximated by simpler building blocks. As Jean-Marc Schlenker, a mathematician at the University of Luxembourg, points out, the problem appears elementary on the surface, yet resisted solution for years before Schwartz’s breakthrough.