For over a century and a half, mathematicians have operated under the comforting, if slightly rigid, assumption that if you know two key things about a compact surface - its metric (how distances work on it) and its mean curvature (how it bends in space) - you can figure out its exact shape. This principle, originating with French mathematician Pierre Ossian Bonnet, has now been gently but firmly kneaded into a new form by researchers from the Technical University of Munich (TUM), the Technical University of Berlin, and North Carolina State University.

They have constructed the first explicit counterexample to this long-held rule. The team built two compact, self-contained surfaces shaped like doughnuts, known as tori. These two tori share identical values for both metric and mean curvature at every point, yet their overall structures are not the same. This type of example, a pair of surfaces that are locally identical but globally different, had been sought for decades.

Mathematicians were already aware that Bonnet's rule had its limits, with known exceptions involving non-compact surfaces that extend infinitely or have edges. Compact surfaces like spheres were thought to be safe from such ambiguity. For torus-shaped surfaces, theory had suggested a single set of metric and mean curvature values could correspond to up to two different shapes, but no one had ever baked a concrete example.

"After many years of research, we have succeeded for the first time in finding a concrete case that shows that even for closed, doughnut-like surfaces, local measurement data do not necessarily determine a single global shape," said Tim Hoffmann, Professor of Applied and Computational Topology at the TUM School of Computation, Information and Technology. The finding resolves a decades-old problem, proving that even with complete local information, a surface's full shape cannot always be uniquely pinned down.