PROVIDENCE — Mathematician Richard Evan Schwartz resolved a longstanding question in geometry by proving that the minimum number of folds required to construct a paper torus is 24. His solution, published in the May 26 issue of the Proceedings of the National Academy of Sciences, demonstrates that such a torus consists of 16 triangles meeting at eight vertices.

The problem centered on whether a flat sheet of paper could be folded into a torus—a doughnut-shaped surface—while satisfying a key geometric condition: at each vertex where triangles meet, the angles must sum to exactly 360 degrees. Previous work had produced a nine-vertex torus that met this criterion, and mathematicians had described a theoretical torus with seven vertices, though it remained uncertain whether all seven could comply with the angle requirement.

Schwartz proved that in any seven-vertex configuration, at least one vertex would necessarily violate the 360-degree condition, ruling out the possibility of a valid seven-vertex paper torus. To explore whether an eight-vertex version was feasible, he employed machine learning, training a computational program to search for viable configurations.

The program successfully identified a folding pattern that yields an eight-vertex torus satisfying the geometric constraints. The resulting shape resembles a pup tent with an extra flap inside. This configuration requires precisely 24 folds and forms 16 triangular facets.

Schwartz, a mathematician at Brown University in Providence, R.I., has a history of tackling geometric optimization problems; he previously determined the shortest possible Möbius strip. In addition to his theoretical contribution, he has made a foldable paper template available to the public, allowing others to construct their own minimal paper torus. His use of machine learning in this geometric context illustrates a novel intersection of computational methods and classical mathematical inquiry.