It took some time, but Hoffmann and Sageman-Furnas were eventually able to convince themselves that the rhino was worth taking seriously. And if it was possible to find such a likely example of a discrete Bonnet pair, maybe the smooth case wasn’t so hopeless after all. Hoffmann and Sageman-Furnas spent that sweltering summer scouring the rhino for clues, sometimes sitting in video chats for eight to 12 hours at a time, searching for unusual properties and geometric constraints that might help them narrow down where to look for smooth Bonnet tori.
As September rolled around, they finally found a new lead that felt so promising that it drew Bobenko back into the problem he’d abandoned decades earlier.
The clue had to do with particular lines that loop around the rhino along its edges.
These lines were already known to provide important information about the rhino’s curvature — tracing out the directions in which it bent and folded the most and least. Since the rhino is a two-dimensional surface that lives in three-dimensional space, the mathematicians had expected these lines to carve out paths throughout 3D space as well. But instead, they always lay either in a plane or on a sphere. It was exceedingly unlikely that these alignments had happened by chance.
“That suggested to us that there was really something special happening,” Sageman-Furnas said. It was “spectacular.”
Unlike discrete surfaces, smooth surfaces don’t have edges. But you can still draw “curvature lines” that trace out the paths of maximum and minimum bending. Sageman-Furnas, Bobenko, and Hoffmann decided to look for a smooth analogue of the rhino whose curvature lines were similarly restricted to living in planes or on spheres. Perhaps a starter surface with those properties could give rise to smooth Bonnet tori.
But it wasn’t clear if such a surface even existed.
Then Bobenko realized that more than a century ago, the French mathematician Jean Gaston Darboux had laid out almost exactly what the mathematicians now needed.
Darboux had come up with formulas for generating surfaces that had the right kinds of curvature lines. The problem was that his formulas wouldn’t produce curvature lines that looped back on themselves. Instead, they “look like spirals and go to infinity,” Bobenko said. “No chance to get them closed.” Which meant that while the curvature lines might live on planes and spheres, the overall surface wouldn’t be a torus.
After years of toil, the mathematicians — using a combination of pen-and-paper techniques and computational experiments — figured out how to adjust Darboux’s formulas so that the curvature lines would close up. They’d finally found their smooth analogue of the rhino (although the two didn’t look much alike).
Moreover, as they’d hoped, this smooth rhino could generate a pair of new tori that had the same mean curvature and metric data but different overall structures. The team finally had their answer to the original Bonnet problem: Some tori can’t be uniquely defined by their local features after all.
But when they worked out what this Bonnet pair actually looked like, they found that the two tori were mirror images of each other. “Technically, this wasn’t an issue,” Sageman-Furnas said. “Formally, it solved the problem.” But, he added, it was still unsatisfying.
And so over the next year, they tried to tweak their smooth rhino in various ways. Ultimately, they realized that if they dropped the requirement that one set of curvature lines had to sit on spheres, they could construct a new smooth rhino that did what they wanted. They then used this surface to generate a new Bonnet pair — this time, two very twisty tori that were much more obviously different surfaces but still had the same metric and mean curvature.