For decades, mathematicians have wondered about the simplest way to construct a doughnut-shaped object from flat pieces, like origami paper. The catch? This “doughnut” isn’t the smooth, glazed kind we eat. Instead, it’s a polyhedral torus—a jagged, many-sided surface resembling a child’s geometric toy. Now, researchers have finally determined the minimum number of corners (vertices) required to build such a shape while ensuring it remains mathematically “flat.”
The Problem: Flatness in Higher Dimensions
The challenge lies in understanding what “flat” even means in this context. It’s not about physical flatness like a sheet of paper, but intrinsic flatness. This means the surface has the same geometric properties as a smooth torus that has been squashed flat, like a piece of clay. The problem becomes significantly harder when considering shapes in four or more dimensions, where intuition fails and calculations become complex.
The Breakthrough: Eight Vertices is the Key
Mathematician Richard Evan Schwartz of Brown University recently solved this problem, publishing his findings in August 2025. He proved that a polyhedral torus cannot be intrinsically flat with fewer than eight vertices, and then demonstrated an example of an eight-vertex torus that is flat.
“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 method involved working backward from an existing polyhedral torus to uncover the necessary conditions for its construction. To ensure flatness, the sum of angles around each vertex must equal 2π—a core requirement for intrinsic flatness.
From Electrons on a Sphere to Origami Toruses
The inspiration for this research came from a seemingly unrelated problem: finding the optimal arrangement of electrons on a sphere. Schwartz’s colleagues suggested the connection, noting that both problems involve searching through complex configurations to find the most efficient solution. For years, Schwartz dismissed the origami torus problem as too difficult, even after his friends pushed it on him.
The breakthrough came when he learned that another researcher had found a nine-vertex solution. This discovery spurred Schwartz to revisit the challenge, and a little-known 1991 paper provided an important clue. He then completed the proof that seven vertices are insufficient, and then used “heavily supervised machine learning” to find the working eight-vertex example.
A Blend of Tradition and Computation
Schwartz’s success highlights his unique approach: combining rigorous mathematical investigation with computational methods. He writes programs to search for examples, complementing his traditional geometric insights. This skill set is rare among mathematicians, according to Schlenker.
This finding provides a definitive answer to a long-standing question in geometry, proving that eight vertices are the bare minimum needed to construct an intrinsically flat polyhedral torus. The discovery underscores the power of combining theoretical rigor with computational tools in modern mathematics.
