It took a while, but Hoffmann and Sageman-Furnas were finally able to convince themselves that rhinos were worth taking seriously. And if such a likely example of a discrete pair of Bonnets could be found, perhaps the smooth case was not so hopeless after all. Hoffmann and Sageman-Furnas spent that steamy summer searching for tracks of the rhinoceros, sometimes sitting in video chats for eight to 12 hours at a time, looking for unusual features and geometric constraints that might help them narrow down where to look for smooth Bonnet bulls.
As September rolled around, they finally found a new lead that seemed so promising that Bobenka was dragged back into the problem he had left decades ago.
Closed loops
The clue had to do with the specific lines that curved around the rhinoceros along its edges.
These lines were already known to provide important information about the curvature of the rhinoceros – they trace the directions in which it bent and flexed the most and the least. Since the rhinoceros is a two-dimensional surface that lives in three-dimensional space, mathematicians expected these lines to carve out paths in 3D space as well. But instead they always lay either flat or on a sphere. It was extremely unlikely that these alignments occurred by chance.
“That indicated to us that something really special was going on,” Sageman-Furnas said. It was “awesome”.
Unlike discrete surfaces, smooth surfaces have no edges. However, you can still draw “curvature lines” that mark the paths of maximum and minimum bending. Sageman-Furnas, Bobenko and Hoffmann decided to look for a smooth counterpart to the rhinoceros, whose curves were similarly restricted to life in planes or on spheres. Perhaps a launch surface with these characteristics could give rise to a smooth Bonnet tori.
But it was not clear whether such a surface existed at all.
Then Bobenko realized that more than a century ago, the French mathematician Jean Gaston Darboux had laid out almost exactly what mathematicians need now.
Darboux came up with formulas for generating surfaces that had the right kinds of curvature. The problem was that his formulas didn’t create curvature curves that would loop in on themselves. Instead, “they look like spirals and go to infinity,” Bobenko said. “There is no way to close them. Which meant that while curvature curves could live on planes and spheres, the overall surface would not be a torus.
After years of toil, mathematicians—using a combination of pen and paper techniques and computational experiments—figured out how to modify Darboux’s formulas to close the curvature curves. They finally found their smooth rhino counterpart (although the two didn’t look much alike).
Additionally, they hoped, this smooth rhino could create a pair of new tori that had the same mean curvature and metric data, but different overall structures. The team finally got an answer to Bonnet’s original problem: Some tori, after all, cannot be uniquely defined by their local characteristics.
But when they figured out what this pair of Bonnets really looked like, they discovered that the two bulls were mirror images of each other. “Technically it wasn’t a problem,” Sageman-Furnas said. “It formally solved the problem. But he added that it was still unsatisfactory.
And so, over the next year, they tried to modify their smooth rhinoceros in different ways. Eventually they realized that if they dropped the requirement that one set of curved lines sit on the balls, they could build a new smooth rhinoceros that would do whatever they wanted. They then used this surface to generate a new pair of Bonnets – this time two very twisted tori that were much more obviously different surfaces but still had the same metric and mean curvature.
Leave a Reply