Mathematicians discover new ways to make round shapes

Mathematicians discover new ways to make round shapes

A new proof solves a long-standing problem about the doughnut-shaped torus

By Rachel Crowell edited by Clara Moskowitz

Imagine that you want to know the most efficient way to make a torus—a doughnut-shaped mathematical object—from origami paper. But this torus, which is a surface, looks drastically different than the outside of a glazed bakery doughnut. Instead of seeming almost perfectly smooth, the torus that you envision is jagged with many faces, each of which is a polygon. In other words, you want to construct a polyhedral torus with faces that are shapes such as triangles or rectangles.

Your peculiar-looking shape will be trickier to construct than one with a smooth surface. The complexity of the problem only grows if you decide that you want to envision constructing something similar but in four or more dimensions.

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

On supporting science journalism

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

Schwartz was able to find a solution to a long-standing question: What’s the minimum number of vertices (corners) needed to make polyhedral tori with a property called intrinsic flatness? The answer, Schwartz found, is eight vertices. He first demonstrated that seven vertices aren’t enough. He then discovered an example of an intrinsically flat polyhedral torus with eight vertices.

“It’s very striking that Rich Schwartz was able to entirely solve this well-known problem,” says Jean-Marc Schlenker, a mathematician at the University of Luxembourg. “The problem looks elementary but had been open for many years.”

Schwartz’s finding 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 simply “flat”—is a bit complicated to parse. The notion is also central to connecting Schwartz’s results to the question of building polyhedral tori from scratch.

Since the 1960s mathematicians have known that intrinsically flat versions of mathematical objects exist. Actually finding those objects is a different beast, Schwartz notes. Describing polyhedral tori as intrinsically flat isn’t quite equivalent to simply saying that they’re flat like a piece of paper. Instead it means that these surfaces have the same dimensions as (or, as mathematicians say, “are isometric to”) tori that are smooshed flat. “Another way to say it is that if you compute the angle sums around each vertex, it adds up to 2π everywhere,” Schwartz says.

According to Schlenker, Schwartz’s finding is very on-brand for his expertise. Yet for many years, Schwartz was so stumped by the problem that he set it aside.

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

But Schwartz wasn’t initially convinced. “Basically, they shoved it in my face, and at some point, years passed. I actually thought it was too hard of a problem,” he says. The difficulty stemmed from the large dimensions that seemed to be involved. “Even for just seven or eight [vertices], it seems that you would have to look at 20-some-odd-dimensional space,” he says.

Bu