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How can you create a doughnut-shaped surface from a flat sheet of paper? The most elegant solution is probably to use the art of paper folding called origami. Mathematician Richard Schwartz of Brown University wanted to find out how to achieve this feat in as few steps as possible.
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In the journal Proceedings of the National Academy of Sciences USA, Schwartz presented his result: the paper must be folded at least 24 times. This creates 16 triangles that intersect at eight vertices. For his proof, he used computer simulations and methods of analysis. He also consulted artificial intelligence—and a hallucination by the program almost undid his efforts.
Creating a Curve from Something Flat
That a doughnut shape—called a torus by mathematicians—can be formed from a single folded sheet of paper is by no means a given. In fact, you have to curve a sheet of paper to form a doughnut-shaped surface: first, it is rolled into a cylinder, and then the two open ends are brought together. This last step cannot be done without creating creases, however. The same is true when trying to form a sphere from a sheet of paper. But the principles of origami come to the rescue. Through folding, it becomes possible to create certain curves from a flat sheet of paper. In this way, you can form a torus, though not a sphere.
The first examples of such a construction were determined by mathematicians Yuri Burago and Viktor Zalgaller in 1960. In the decades that followed, experts constructed various types of origami tori, all composed of triangular shapes. The vertices of the triangles, which meet at the corners (also called vertices), must have an angle sum of 360 degrees. Most recently, in 2025, researcher Vincent Tugayé constructed an origami torus with only nine vertices. Schwartz then wondered whether this value could be further reduced.
The mathematician, who had previously proven the shortest origami Möbius strip dimensions, was able to prove that a torus with only seven vertices is impossible. “The proof was quite simple,” Schwartz says, “so I was surprised that no one had presented it before.” As he demonstrated, a curvature is necessary to form a torus from triangles with only seven vertices—something that cannot be achieved using pure origami folding techniques. Then arose the question of whether it was possible to construct an origami torus with eight vertices or whether Tugayé had already found the optimal solution.
Schwartz initially tried to demonstrate that an origami torus of eight vertices could not exist, as he had for a torus with seven vertices. But he failed. So he looked for examples of such structures.
His first goal was to determine how many different ways triangles could be arranged on the sheet in the eight vertex case. In other words, how many folds are theoretically possible? To simplify things, he asked ChatGPT this question. The AI chatbot told him there are thousands of possible solutions and linked to a website where the answer should be available. “I found that very discouraging because I didn’t want to work my way through thousands of patterns,” Schwartz says. So he briefly abandoned the project.
But the topic wouldn’t let him go. He started going through the various arrangements of triangles with a pencil and paper. “It seemed highly improbable that there could really be thousands of different solutions,” Schwartz says. “I finally looked at the website linked by ChatGPT, and it turned out there were only seven possibilities!” From these arrangements, the mathematician picked out the ones that looked most promising. One of them reminded him of the solution to the origami Möbius strip puzzle.
Using a computer program, he then tried to fold the possible arrangements of triangles to create a torus—and succeeded. The result resembles a doghouse more than a doughnut. But from a mathematical perspective, it was exactly what he was looking for: a ring-shaped structure with a hole.
“The most difficult part of the study was probably debugging the computer programs,” Schwartz says. “Whenever you’re working with computer-aided mathematics, you have to be absolutely certain that the program is telling you the truth.” He estimates that the final proof, by which he demonstrated the existence of an origami torus with eight vertices, accounted for only about 10 percent of his time.
But with that, the smallest origami doughnut has now been found. Schwartz hasn’t folded it from real paper yet, however, citing his skill in origami: “I’m terribly bad at it.”
This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission. It was translated from the original German version with the assistance of artificial intelligence and reviewed by our editors.
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