I made the 3D visualizations for a movie about how the "Bonnet problem" was solved.
1. The Bonnet problem
Can two surfaces with the same mean curvature and metric differ by more than a rigid transformation? Two surch surfaces are called a Bonnet pair. The first question is: do Bonnet pairs even exist? If they exist, what do they look like?
2. The paper
Compact Bonnet Pairs: isometric tori with the same curvatures Alexander Bobenko, Tim Hoffmann and Andrew Sageman-Furnas proved the existence of a Bonnet pair by constructing one.
3. The movie
Ekaterina Eremenko collected a lot of video material during the research process and made a movie out of it. I was in charge along with Sara Samy to do the CGI for the movie. The movie is still being showed in festivals and movie theaters, so it is not accessible yet. But the rumors say it might land on the internet soon. Stay tuned.
4. Some pretty pictures
4.1. Curvature spheres
The two spheres in second order contact with a surface at a point. They touch the surface along the two principal curvature directions. The radius of these sphere is the inverse of the principal curvatures.
4.2. Numerical experiment
This very coarse discrete surface called the "Rhino" revealed that Bonnet pairs might exist and what special properties they probably have.
4.3. The first Bonnet pair
A render I made of the first ever found Bonnet pair.
This is only the first and there are certainly more, go find your own Bonnet pair!