Imagine you want to know the most efficient way to make a torus—a donut-shaped mathematical object—out of origami paper. But this torus, which is a surface, looks quite different from the exterior of a glazed bakery donut. Instead of looking almost perfectly smooth, the torus you imagine is jagged with many faces, each of which is a polygon. In other words, you want to build a polyhedral torus whose faces are shaped like triangles or rectangles.
It will be more difficult to create your unique looking shape than a shape with a smooth surface. The complexity of the problem only increases when you decide you want to visualize the construction in something similar but in four or more dimensions.
In a recent study, Brown University mathematician Richard Evan Schwartz tackled this problem by working backwards from an existing polyhedral torus to answer the question of what would be required to create it from scratch. He posted my findings On the preprint server in August 2025.
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Schwartz was able to find the solution to a long-standing question: What is the minimum number of vertices (corners) required to build a polyhedral tori with the property called intrinsic flatness? The answer Schwartz found is eight heads. He demonstrated for the first time that seven peaks are not enough. He then found an example of an internally flat polyhedral torus with eight vertices.
“It’s amazing that Rich Schwartz was able to solve this famous problem so completely,” says Jean-Marc Schlenker, a mathematician at the University of Luxembourg. “The problem appears to be primary but was open for several years.”
Schwartz’s discovery essentially provides the minimum number of vertices that a polyhedral torus needs so that it can be flattened. But one detail – what it means to be “intrinsically flat” rather than just “flat” – is a bit complicated to analyze. This notion is also central to connecting Schwartz’s results to the question of constructing polyhedral tori from scratch.
Mathematicians have known since the 1960s that intrinsically flat versions of mathematical objects exist. Actually finding those items is a different animal, notes Schwartz. Describing polyhedral tori as intrinsically flat is not equivalent to simply saying that they are as flat as a piece of paper. Instead it means that these surfaces have the same dimensions as tori (or, as mathematicians say, “are isometric”) that are flat surfaces. “Another way to say this is that if you calculate the angle sum around each vertex, it adds up to 2π everywhere,” says Schwartz.
According to Schlenker, Schwartz’s discovery is a great fit for his expertise. Yet for many years Schwartz was so troubled by the problem that he put it aside.
He first heard about this dilemma in 2019, when two of his mathematician friends—Alba Málaga Sabogal and Samuel Lelièvre—posed it to him. “They thought I’d be interested in it because I had solved this thing called Thompson’s problem, which was about electrons on a sphere,” says Schwartz. “He thought (Thompson’s problem was) about searching through a configuration space and trying to see which configuration was the best among infinite possibilities, and these origami tori have a similar kind of flavor.”
But Schwartz was not convinced at first. “Basically, they pushed it in my face, and over a period of time, several years passed. I really thought it was a very difficult problem,” he says. This difficulty arose from the seemingly large dimensions involved. “Even for just seven or eight (peaks), it seems like you have to look at 20-something-odd-dimensional space,” he says.
But when the three mathematicians met again in 2025, Schwartz learned that Lelièvre’s roommate, Vincent Tugue, had found an example that worked with nine vertices. “It was really a beautiful thing” that the Ph.D. Tugaye, a high school teacher with. in physics, demonstrated at mathematics outreach fairs in Paris, says Schwartz. “I thought, ‘Okay, this must be for the best,'” says Schwartz, who then set out to decide whether his intuition was correct.
To approach the question of whether cases with seven or eight vertices would work, Schwartz focused on answering “How do I reduce the dimension?” They generated a lot of ideas about how to do this in the case of the seven vertices. Yet eventually he found a mathematical gift of sorts: a little-known 1991 paper that “goes about 80 percent of the way to proving that you can’t do it with seven vertices,” he says. “Then I ended it.”
Still thinking that the case of eight vertices would not work as well, he then tried to use a similar approach to prove that claim. When he found that he could not rule out some cases, he decided to find out what properties would be required for an eight-vertex torus to be intrinsically flat. Using an approach he describes as “highly supervised machine learning”, Schwartz then found an eight-vertex example that worked.
Schlenker says, “I think the most amazing thing is that this is another example of the distinctive skill that Rich Schwartz has developed by combining traditional mathematical investigation with computational methods.” “He finds beautiful geometric ideas to prove certain results but also writes elaborate programs to try and find examples. Very few mathematicians have been able to bring those two aspects together so harmoniously.”
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