Mathematician Finds Minimum Number of Folds to Create an Origami Torus

A mathematician has proven that creating a doughnut-shaped surface from a flat sheet of paper requires at least 24 folds. Using computer simulations and analysis, Richard Schwartz of Brown University determined the most efficient origami technique. The finding, published in PNAS, also involved AI assistance, though a program hallucination nearly derailed the work.

The proof builds on decades of geometric folding research. Burago and Zalgaller first demonstrated in 1960 that such constructions were possible, and later researchers refined the approach using triangular arrangements. Tugayé's 2025 construction with nine vertices set the benchmark that Schwartz sought to challenge, prompting his investigation into whether fewer vertices could suffice.

Schwartz's earlier work on the shortest origami Möbius strip informed his methodology. His proof that seven vertices cannot work relied on a straightforward argument about curvature requirements. The eight-vertex question proved more difficult, and his consultation with ChatGPT initially produced misleading information about possible triangle arrangements, though the program's hallucination was ultimately overcome through further analysis.