The Bonnet Problem

Mathematicians have long studied differential geometry and coordinate systems on surfaces. One such mathematician was Pierre Ossian Bonnet (1819 – 1892) in France, who asked questions about whether relatively small amounts of information can uniquely identify whole surfaces. Apparently not and a pair of images are shown from an answer that was published on-line on 14 October, 2025 after several years of of study. (Bobenko, A.I., Hoffmann, T. & Sageman-Furnas, A.O. Compact Bonnet pairs: Isometric tori with the same curvatures. Publ.math.IHES 142, 241–293 (2025)). This article is licensed under a Creative Commons Attribution 4.0 International License.

Thus for the first time, mathematicians have found an example of a compact doughnut-like surface (as seen below) that shares its local geometric information with another surface, despite having a completely different global angular structure, nicknamed Rhino. To see an image of this  different surface, click this link:  Rhino pair This mathematical problem discussed in the above paper, was solved by computational investigations presented in almost 50 pages of mind-boggling mathematical equations. They are very complicated, but an excellent review and explanation may be found in an article in the following link:
https://www.quantamagazine.org/two-twisty-shapes-resolve-a-centuries-old-topology-puzzle-20260120/