Mathematician Richard Evan Schwartz has solved a puzzle that stumped the mathematical world for years: what is the absolute minimum number of folds needed to turn a flat sheet of paper into a three-dimensional donut shape? The answer, he has now rigorously proven, is eight.
The question may sound obscure to those outside mathematics, but it sits at the heart of how we think about efficiency. In origami, a torus—the formal name for a donut's geometry—is constructed by folding paper into triangles that fit together so precisely that the angles around each vertex sum to exactly 360 degrees, like pizza slices arranged into a perfect pie. The number of vertices, or fold points, becomes a measure of how efficiently the shape is made. Fewer vertices mean fewer folds, less material wasted, and a more elegant solution.
For decades, early examples of paper tori required thousands of vertices. More recent work proved that nine or ten vertices could work. But a nagging mathematical question remained: could you go lower? Theoretically, mathematicians knew the absolute floor had to be at least seven vertices—below that, the geometry simply doesn't allow a torus to exist. Yet whether the true minimum was seven, eight, or nine remained an open question until Schwartz's breakthrough.
Using a combination of rigorous mathematical analysis and computer experiments, Schwartz determined that seven vertices is impossible—the geometry fails. But eight vertices works. In his paper, published in the Proceedings of the National Academy of Sciences, he provides both a formal proof and a computer-aided method for constructing the eight-vertex solution. He even describes the shape playfully as a "pup tent"—a specific family of folded forms that any eight-vertex origami torus must satisfy.
What makes the achievement particularly charming is Schwartz's candid admission: he cannot actually fold his own template. "I have to admit that I cannot successfully fold my own template," he writes, "but my origami-skilled friends can do it easily." The paper includes a link to a template that anyone with patience and a steady hand can attempt to fold, though Schwartz's origami-skilled collaborators clearly have an edge.
For mathematicians, this represents a satisfying closure to a long-standing optimization question. But the implications extend far beyond pure mathematics. Work like this illuminates principles of efficient design that matter in architecture, materials science, and industrial design—anywhere the goal is to achieve a desired form using the minimum amount of folding or construction. Educational value emerges too: the problem bridges abstract geometry with tangible art, showing how mathematical elegance manifests in physical form.
Schwartz, who has also solved the puzzle of the shortest possible Möbius strip, has demonstrated that even seemingly impractical questions can yield surprising depth. What appears to be merely about origami folds turns out to be fundamentally about optimization, geometry, and the hidden efficiency rules that govern how three-dimensional forms emerge from flat materials. With his proof that eight is the magic number, he has drawn a definitive line under one of mathematics' quieter mysteries.