This donut-shaped discovery just shattered a 150-year math rule

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April 22, 2026, 5:49 AM

• •Bonnet's rule: if you know two key properties of a compact surface at every point, its metric and its mean curvature, then you can determine its exact shape.
• •Researchers created two torus-shaped surfaces that matched in local measurements but differed globally.
• •The finding resolves a long-standing question in geometry.
• •The study highlights that a surface's full shape cannot always be uniquely determined even with complete local information.

For over 150 years, mathematicians believed that a surface's shape could be determined by its metric and mean curvature, a principle known as Bonnet's rule. Researchers at the Technical University of Munich, the Technical University of Berlin, and North Carolina State University have disproven this by constructing two torus-shaped surfaces that share the same local measurements but have different global shapes. This finding provides the first concrete example that challenges the long-accepted idea in geometry, resolving a decades-old problem. This discovery highlights that complete local information is not always enough to determine a surface's full shape.

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