Mathematician solves origami donut efficiency challenge with fewest folds

June 1, 2026 report Mathematician solves origami donut efficiency challenge with fewest folds Krystal Kasal Author Sadie Harley Scientific Editor Robert Egan Associate Editor Most people wouldn't think that it would take rigorous mathematical proof to show how many folds it takes to make a donut shape out of paper. Yet, no one could quite figure it out until recently. In a new paper, published in Proceedings of the National Academy of Sciences, mathematician Richard Evan Schwartz provides detailed proof of where the line is drawn when it comes to the fewest folds required to construct a torus—the proper name for the shape of a donut—from a piece of paper. A quest for the minimum number of vertices In practice, an origami torus is constructed by folding a finite number of triangles that fit together in such a way that the total angle of the triangles around each vertex equals 2π (or 360 degrees) when added together. A better way to visualize this is to think of adding up the angles formed by the tips of individual pizza slices to form a whole pizza. Schwartz explains in his paper that the number of vertices acts as a kind of metric for efficiency. He writes, "As with many mathematical problems, one can view paper tori through the lens of optimization. Given that they exist, how efficiently can they be made? I don't know when this question was first asked, but it seems like one of the first things you would want to know about paper tori beyond the fact of their existence. "A nice measure of efficiency is given by the number of vertices. (This is the same as asking about the minimum number of triangles in the triangulation, or the minimum number of edges you fold along.)" Early examples of paper tori consisted of thousands of vertices, while more recent examples proved that origami tori could be made with ten or nine vertices. Schwartz said that it was also apparent that a paper torus would require at least seven vertices because triangulations of a torus having fewer than seven vertices don't exist. Still, this left the question of whether the minimum was actually seven, eight, or nine. Proving the most efficient possible construction Using a combination of mathematical analysis and computer experiments, Schwartz found that the construction of a paper torus with only seven vertices was impossible. He also found that an origami torus with eight vertices does exist, making it the most efficient possible construction. His paper provides both a mathematical proof and a computer-aided approach to finding the eight-vertex solution. Although a skilled mathematician (who also determined the shortest mobius strip possible), Schwartz admits he can't actually fold the donut shape himself. He calls the shape a "pup tent," which refers to a family of examples with specific properties that an 8-vertex paper torus must satisfy in order to qualify. He writes, "Some readers might like to actually make a pup tent. My paper has a link to a template you can copy and, with effort, fold up into a pup tent. I have to admit that I cannot successfully fold my own template, but my origami-skilled friends can do it easily." Although it may seem irrelevant to non-mathematicians, work like this could provide insight into efficient design in architecture, materials science, and art where minimal folding or construction is desired. It may also be helpful as an educational tool for teaching about geometry and its link to art. Written for you by our author Krystal Kasal, edited by Sadie Harley, and fact-checked and reviewed by Robert Egan—this article is the result of careful human work. We rely on readers like you to keep independent science journalism alive. If this reporting matters to you, please consider a donation (especially monthly). You'll get an ad-free account as a thank-you. Publication details Richard Evan Schwartz, The most efficient origami torus, Proceedings of the National Academy of Sciences (2026). DOI: 10.1073/pnas.2523301123 Journal information: Proceedings of the National Academy of Sciences © 2026 Science X Network