ZAP // Ruben Yevgeny Hymov
The “impossible” 3D toroidal polyhedron
Mathematician Richard Evan Schwartz managed to find a solution to a question that had been going on for a long time: what is the minimum number of vertices needed to make polyhedral tori with a property called intrinsic flatness.
Imagine you want to know what is the most efficient way to make a log — a mathematical object I’m shaped like a donut — from origami paper. But this torus, which is a surface, has a drastically different appearance than the exterior of a glazed pastry donut.
Instead of appearing almost perfectly smooth, the torus you imagine it is irregularwith many faces, each of which is a polygon. In other words, you want to construct a polyhedral torus with faces that are shapes like triangles or rectangles.
Her peculiar-looking figure it will be more complicated to build than one with a smooth surface. The complexity of the problem only increases if you decide that you want to imagine building something similar, but in 4 or more dimensions.
The mathematician Richard Evan Schwartzof Brown University, tackled the problem in a recent study, working backwards from an existing polyhedral torus to answer questions about what it would take to build it from scratch.
In your , pre-published last year in arXivSchwartz managed to find a solution to an issue that has been going on for a long time: what is the minimum number of vertices (edges) needed to make polyhedral tori with a property called intrinsic flatness?
The answer, Schwartz discovered, is eight vertices. First he demonstrated that seven vertices are not enough. He then discovered an example of an intrinsically flat polyhedral torus with 8 vertices.
“It is very remarkable that Rich Schwartz was able to completely solve this well-known problem,” he says. Jean-Marc Schlenkermathematician at the University of Luxembourg, at . “The problem seems elementary, but it has been open for many years.”
Schwartz’s discovery essentially provides the minimum number of vertices a polyhedral torus needs so that it can be flattened. But one detail — what does it mean be “intrinsically flat” rather than simply “flat” — is a little complicated to analyze. This notion is also central to relating Schwartz’s results to the issue of constructing root polyhedral tori.
Since the 1960s, mathematicians have known that intrinsically flat versions of mathematical objects exist. But find effectively these objects is another story, notes Schwartz. Describe polyhedral tori as intrinsically flat not exactly equivalent to say simply that they are plans like a sheet of paper.
Instead, means that these surfaces have the same dimensions that (or, as mathematicians say, “are isometric to”) logs that are crushed until they become flat. “Another way of saying it is that if we calculate the sums of the angles around each vertex, these add up to 2π everywhere“, explica Schwartz.
According to Schlenker, Schwartz’s discovery is very much in line with his specialization. However, for many years, Schwartz was so perplexed with the problem that left him aside.
The mathematician, who even has a page where fans discuss his work, heard about the puzzle for the first time in 2019, when two of his mathematician friends, Alba Málaga Sabogal and Samuel Lelièvrethey presented it to him.
“They thought I would be interested in this because I had solved something called the Thompson problem, which was about electrons in a sphere“, says Schwartz. “They thought the problem had to do with searching a configuration space and trying to see which configuration was the best among an infinite number of possibilities, and these origami logs have a similar kind of flavor“.
Mas Schwartz was not initially convinced. “Basically, they shoved me in the face and, at one point, years passed. I actually thought it was a problem too difficult“, it says.
The difficulty came from the large dimensions that appeared to be involved. “Even for just seven or eight vertices, it seems like we would have to look at a space of twenty-odd dimensions“, he states.
But when the three mathematicians met again in 2025, Schwartz learned that Lelièvre’s housemate, Vincent Tugayéhad found a example that worked with nine vertices.
“It was a very beautiful thing” that Tugayé, a high school teacher with a doctorate in physics, displayed at mathematical fairs in Paris, says Schwartz. “I thought: well this has to be the best“, adds Schwartz, who then set out to find out if his intuition was correct.
To find out if cases with 7 or 8 vertices would workSchwartz focused on responding to “How do I reduce the size?”, and generated many ideas on how to do it for the case of 7 vertices.
However, he ended up stumbling in a kind of mathematical gift: a little-known article from 1991 that “covers about 80% of the way to prove that it can’t be done with seven vertices”, he says. “Then I simply finished the work.”
Continuing to think that the eight-vertex case wouldn’t work either, he then tried using a similar approachto prove this statement. When he discovered that could not exclude some casesdecided to discover what properties an eight-vertex torus would need to have to be intrinsically flat.
With an approach he describes as “heavily supervised machine learning,” Schwartz then found an example of eight vertices that worked.
“What’s most notable, I think, is that it’s yet another example of the specific skills that Rich Schwartz has developed, combining traditional mathematical investigation with computational methods,” says Schlenker.
“He find beautiful geometric ideas to prove some resultsbut also writes elaborate programs to search and find examples. Very few mathematicians are able to bring these two aspects together in such a harmonious way”, concludes Schlenker.
