In 1867, Pierre Ossian Bonnet asked whether knowing the metric and mean curvature at every point of a surface determines the surface's shape. The metric tells you distances between nearby points. The mean curvature tells you how the surface bends. Together, they constitute the complete local geometry — every measurement you could make in a small neighborhood of any point. The question: is this enough to fix the global shape?
For non-compact surfaces — those with edges or extending to infinity — the answer was known to be no. Surfaces that stretch forever or have boundaries can bend in globally different ways while preserving their local geometry. But for compact, closed surfaces — the finite, bounded ones like spheres and tori that have no edges — no counterexample had been found in 150 years of searching. The conjecture hardened into expectation: complete local data determines global structure, at least for closed surfaces.
Alexander Bobenko, Tim Hoffmann, and Andrew Sageman-Furnas (published October 2025) found the counterexample. Two tori — doughnut-shaped surfaces — with identical metric and identical mean curvature at every point, that are nonetheless different shapes. Two surfaces that agree on every local measurement but disagree globally. The shapes are mirror images that pass through themselves, twisting in ways that create the same local geometry through different global configurations.
The proof technique is notable. Sageman-Furnas started not with smooth surfaces but with discrete ones — pixelated approximations. He found a spiky starter surface and discovered that certain curvature lines were constrained to lie in planes or on spheres. This led the team to Darboux's century-old formulas, adapted to construct smooth surfaces whose curvature lines close up properly. The path to the continuous result went through the discrete approximation — a coarser tool revealing structure the finer tool couldn't access directly.
The structural insight is about completeness. The local data is not merely partial or approximate — it is mathematically complete. Every infinitesimal measurement at every point is specified. There is no hidden local variable, no missing local observable, no further local refinement that would help. And it still does not determine the global object. The underdetermination is not from lack of information but from the nature of the relationship between local and global. Local completeness and global determination are independent properties.