Mathematicians Discover New Ways To Make Round Shapes

Mathematicians observe caller ways to make information shapes

A caller impervious solves a long-standing problem astir nan doughnut-shaped torus

By Rachel Crowell edited by Clara Moskowitz

Flavio Coelho/Getty Images

Imagine that you want to cognize nan astir businesslike measurement to make a torus—a doughnut-shaped mathematical object—from origami paper. But this torus, which is simply a surface, looks drastically different than nan extracurricular of a glazed bakery doughnut. Instead of seeming almost perfectly smooth, nan torus that you envision is jagged pinch galore faces, each of which is simply a polygon. In different words, you want to conception a polyhedral torus pinch faces that are shapes specified arsenic triangles aliases rectangles.

Your peculiar-looking shape will beryllium trickier to conception than 1 pinch a soft surface. The complexity of nan problem only grows if you determine that you want to envision constructing thing akin but successful 4 aliases much dimensions.

Mathematician Richard Evan Schwartz of Brown University tackled nan problem successful a caller study by moving backward from an existing polyhedral torus to reply questions astir what would beryllium needed to conception it from scratch. He posted his findings to a preprint server successful August 2025.

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Schwartz was capable to find a solution to a long-standing question: What’s nan minimum number of vertices (corners) needed to make polyhedral tori pinch a spot called intrinsic flatness? The answer, Schwartz found, is 8 vertices. He first demonstrated that 7 vertices aren’t enough. He past discovered an illustration of an intrinsically level polyhedral torus pinch 8 vertices.

“It’s very striking that Rich Schwartz was capable to wholly lick this well-known problem,” says Jean-Marc Schlenker, a mathematician astatine nan University of Luxembourg. “The problem looks simple but had been unfastened for galore years.”

Schwartz’s uncovering fundamentally provides nan minimum number of vertices that a polyhedral torus needs truthful that it tin beryllium flattened. But 1 detail—what it intends to beryllium “intrinsically flat” alternatively than simply “flat”—is a spot analyzable to parse. The conception is besides cardinal to connecting Schwartz’s results to nan mobility of building polyhedral tori from scratch.

Since nan 1960s mathematicians person known that intrinsically level versions of mathematical objects exist. Actually uncovering those objects is simply a different beast, Schwartz notes. Describing polyhedral tori arsenic intrinsically level isn’t rather balanced to simply saying that they’re level for illustration a portion of paper. Instead it intends that these surfaces person nan aforesaid dimensions arsenic (or, arsenic mathematicians say, “are isometric to”) tori that are smooshed flat. “Another measurement to opportunity it is that if you compute nan perspective sums astir each vertex, it adds up to 2π everywhere,” Schwartz says.

According to Schlenker, Schwartz’s uncovering is very on-brand for his expertise. Yet for galore years, Schwartz was truthful stumped by nan problem that he group it aside.

He first heard astir nan quandary successful 2019, erstwhile 2 of his mathematician friends—Alba Málaga Sabogal and Samuel Lelièvre—brought it to him. “They thought I would beryllium willing successful this because I had solved this point called Thompson’s problem, which was astir electrons connected a sphere,” Schwartz says. “They thought [Thompson’s problem was] astir searching done a configuration abstraction and trying to spot which configuration was champion amongst an infinite number of possibilities, and these origami tori person a akin benignant of flavor.”

But Schwartz wasn’t initially convinced. “Basically, they shoved it successful my face, and astatine immoderate point, years passed. I really thought it was excessively difficult of a problem,” he says. The trouble stemmed from nan ample dimensions that seemed to beryllium involved. “Even for conscionable 7 aliases 8 [vertices], it seems that you would person to look astatine 20-some-odd-dimensional space,” he says.

But erstwhile nan 3 mathematicians reunited successful 2025, Schwartz learned that Lelièvre’s roommate, Vincent Tugayé,had recovered an illustration that worked pinch 9 vertices. “It was a really beautiful thing” that Tugayé, a precocious schoolhouse coach pinch a Ph.D. successful physics, exhibited astatine mathematics outreach fairs successful Paris, Schwartz says. “I thought, ‘Well, this one’s sewage to beryllium nan best,’” adds Schwartz, who past group retired to settee whether his intuition was correct.

To attack nan mobility of whether nan cases pinch 7 aliases 8 vertices would work, Schwartz focused connected answering “How do I trim down nan dimension?” He generated a batch of ideas astir really to do truthful for nan 7 vertices case. Yet he yet stumbled upon a mathematical gift of sorts: a small known 1991 insubstantial that “goes astir 80 percent of nan measurement to proving that you can’t do it pinch 7 vertices,” he says. “Then I conscionable vanished it off.”

Still reasoning that nan 8 vertices lawsuit besides wouldn’t work, he past tried to usage a akin attack to beryllium that claim. When he recovered he couldn’t norm retired immoderate cases, he decided to fig retired what properties an eight-vertex torus would request to person to beryllium intrinsically flat. Using an attack that he describes arsenic “heavily supervised instrumentality learning,” Schwartz past recovered an eight-vertex illustration that did work.

“What's astir striking, I think, is that it’s different illustration of nan circumstantial skills that Rich Schwartz has developed, blending accepted mathematical investigation pinch computational methods,” Schlenker says. “He finds beautiful geometric ideas to beryllium immoderate results but besides writes elaborate programs to hunt for and find examples. Very fewer mathematicians are tin of bringing those 2 strands together truthful harmoniously.”