Mathematicians discover new ways to make round shapes

image via New Scientist

February 9, 2026, 12:30 PM

• •Schwartz found the minimum number of vertices for intrinsically flat polyhedral tori is eight.
• •He proved that seven vertices are insufficient and provided an example using eight.
• •The research blends traditional mathematical methods with computational approaches.
• •The problem has been open for many years and is considered elementary yet complex.

Mathematician Richard Evan Schwartz of Brown University solved a long-standing problem concerning the minimum number of vertices required to construct intrinsically flat polyhedral tori. Schwartz determined that the answer is eight vertices, disproving the possibility of seven vertices and providing an example of an eight-vertex torus. This finding addresses a complex issue in geometry, building on previous work from the 1960s and utilizing both traditional mathematical investigation and computational methods. This discovery has significant implications for understanding the construction and properties of these unique mathematical shapes.

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