Mathematicians discover new ways to create round shapes

Imagine you want to know the most efficient way to make a torus, donut shape. mathematical object-from origami paper. But this toruswhich is a surface, is radically different from the exterior of a glazed bakery donut. Instead of appearing almost perfectly smooth, the torus you imagine is jagged and has many faces, each of which is a polygon. In other words, you want to construct a polyhedral torus whose faces are shapes such as triangles or rectangles.

Your strange shape will be more difficult to build than a model with a smooth surface. The complexity of the problem only increases if you decide to consider building something similar but in four or more dimensions.

Mathematician Richard Evan Schwartz of Brown University tackled the problem in a recent study by working backwards from an existing polyhedral torus to answer questions about what would be needed to build it from scratch. He published its findings to a preprint server in August 2025.

On supporting science journalism

If you enjoy this article, please consider supporting our award-winning journalism by subscribe. By purchasing a subscription, you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today.

Schwartz managed to find a solution to a long-standing question: what is the minimum number of vertices (corners) needed to create polyhedral tori with a property called intrinsic flatness? According to Schwartz, the answer is eight vertices. He first demonstrated that seven summits are not enough. He then discovered an example of an intrinsically flat polyhedral torus with eight vertices.

“It is very striking that Rich Schwartz was able to completely solve this well-known problem,” says Jean-Marc Schlenker, a mathematician at the University of Luxembourg. “The problem seems elementary, but it has remained open for many years.”

Schwartz’s discovery essentially provides the minimum number of vertices a polyhedral torus needs in order to be flattened. But one detail – what it means to be “inherently flat” rather than just “flat” – is a bit complicated to parse. This notion is also central to linking Schwartz’s results to the question of constructing polyhedral tori from scratch.

Since the 1960s, mathematicians have known that there are intrinsically flat versions of mathematical objects. Actually finding these items is another matter, Schwartz notes. Describing polyhedral tori as inherently flat is not simply the same as saying that they are flat as a piece of paper. Instead, it means that these surfaces have the same dimensions as (or, as mathematicians say, “are isometric to”) the tori that are flattened. “Another way of saying it is that if you calculate the sum of the angles around each vertex, it adds up to 2π everywhere,” says Schwartz.

According to Schlenker, Schwartz’s discovery is very much in line with his expertise. Yet for many years, Schwartz was so perplexed by the problem that he put it aside.

He first heard about the dilemma in 2019, when two of his mathematician friends, Alba Málaga Sabogal and Samuel Lelièvre, introduced it to him. “They thought I would be interested because I had solved what’s called the Thompson problem, which was about electrons on a sphere,” Schwartz says. “They thought [Thompson’s problem was] it’s about searching through a configuration space and trying to see which configuration is best among an infinite number of possibilities, and these origami tori have a similar flavor.

But Schwartz wasn’t convinced at first. “They basically shoved it in my face, and at some point, years went by. I actually thought it was too difficult a problem,” he says. The difficulty came from the large dimensions which seemed involved. “Even for only seven or eight [vertices]it seems that we would have to examine a space of around twenty dimensions,” he says.

But when the three mathematicians got together in 2025, Schwartz learned that Lelièvre’s roommate, Vincent Tugayé, had found an example working with nine vertices. “It was a really lovely thing,” says Tugayé, a high school teacher with a doctorate. in physics, exhibited at mathematical popularization fairs in Paris, explains Schwartz. “I thought, ‘Well, this one has to be the best,'” adds Schwartz, who then decided to determine if his hunch was correct.

To address the question of whether seven or eight vertex cases would work, Schwartz focused on the answer “How can I reduce the dimension?” » It generated many ideas on how to proceed for the case of the seven summits. Yet he eventually stumbled upon a mathematical gift of sorts: a little-known 1991 paper that “proves 80 percent that you can’t do it with seven vertices,” he says. “Then I just finished it.”

Still thinking that the eight peaks case wouldn’t work either, he then tried to use a similar approach to prove this claim. When he realized that he could not rule out certain cases, he decided to determine what properties a torus with eight vertices would have to have to be intrinsically flat. Using an approach he describes as “strongly supervised machine learning,” Schwartz then found an eight-vertex example that worked.

“What is most striking, I think, is that this is another example of the specific skills developed by Rich Schwartz, blending traditional mathematical investigation and computational methods,” says Schlenker. “He finds beautiful geometric ideas to prove certain results, but also writes elaborate programs to search and find examples. Very few mathematicians are able to bring these two strands together so harmoniously.”