A couple of weeks ago I attended the SIAM/ACM Joint Conference on Geometric and Physical Modeling and heard a lovely talk by Richard Riesenfeld. Riesenfeld and his wife Elaine Cohen were this year’s Bézier award winners for their work in computer aided geometric design (CAGD). He spoke about his correspondence with Bézier and showed us many of the letters they sent back and forth in the early days of CAD/CAM, with their many hand drawn diagrams and the typed text with the math symbols added in by hand. I spent the time marveling at how they managed to have an effective collaboration over such an impoverished communication channel. But even with all of the wonderful 3D rendering capabilities we have today, it is still hard to communicate about 3D objects and spaces over a distance. Having a visual rendering is not sufficient. Spatial reasoning requires more. Riesenfeld mentioned Bézier’s view that “touch is more discriminative than eyes.”

This theme reminded me that I’ve been meaning to describe and send to the math factory folks a suggestion for an exhibit in the math museum. Instead, I’ll first write about it here.

Many years ago I was caught by the fascinating story surrounding the problem of “everting” the sphere. While I loved the story, and had seen a movie of a sphere eversion, I never felt I fully understood how a sphere eversion is done. I needed to be able to play with it the way you play with clay with your hands. Michael J. McGuffin has gone part of the way there, with a lovely application that supports choosing various vantage points and renderings as you step through a sphere eversion. The next step would be to a virtual reality application that supports playing with sphere deformations with your hands. The trickiest part is designing a good interface and effective haptic feedback through virtual reality gloves.

This sort of application shows virtual reality at its best: its strength is being able to do something based on reality, but going beyond it in some way. “Everting” the sphere means turning it inside out by a “regular homotopy,” which roughly means that the surface is allowed to pass through itself, but creasing and tearing are not allowed. (The movie, or McDuff’s application, will give you a better sense for what this means.) A VR interface for sphere eversion would need to support fine control of the sphere surface, allow a surface to pass through itself (which is easy in VR), and disallow moves that cause creases.

I now describe the problem and its history. Silvio Levy gives a excellent brief history. Here I give a briefer one. In 1957, Smale proved some general results having to do with sphere immersions, mappings of a sphere into three dimension spaces that are smooth but can self-intersect. The renowned mathematician Bott told him he was wrong largely because the results implied that the sphere could be everted, something that seemed impossible. The proof was checked, and held up, which meant that the sphere could be inverted. But no one knew how to do it. Smale’s proof, strictly speaking, was constructive, but as Levy puts it “It is akin to describing what happens to the ingredients of a soufflé in minute detail, down to the molecular chemistry, and expecting someone who has never seen a soufflé to follow this “recipe” in preparing the dish.”

One of the first people to understand how to evert the sphere was Bernard Morin. It is tempting to say “one of the first people to see how” except that would be inaccurate; Bernard Morin was blind. Part of the reason the story captured my imagination was that I had long felt that my spatial sense was not visual so much as physical. As Levy says, the fact that Morin was blind is “convincing proof that “visualization” goes far beyond the physical sense of sight.” Spatial sense has more in common with touch, and is closely tied to proprioception, the sense that tells us where our body parts are. It is through proprioception that we know where our hand is even when we cannot see it. My sense is that if I could just “get my hands on it,” I could understand the process of everting the sphere.

The technology to do create such an exhibit is either already here, or very close. Any takers?

Sheldonsays:This is fascinating stuff. I’ve learned a new word (“eversion”) from this post, so thanks for that.

I recently found and acquired a copy of the old Time-Life book on Mathematics that I used to enjoy as a child. The book has a step-by-step diagram of how to turn an inner tube (an inflatable torus for those of you under 30), but it needed a hole in the surface. The book also introduced me to the Klein Bottle, which is a “bottle with no inside.” It reminds me of the sphere eversion. The book notes that the Klein Bottle is not quite without an inside because it cannot physically pass through itself without the existence of a hole. If I read your post correctly, it sounds like that limit is removed or mitigated for sphere eversion.

Your insistence on a physical, “hands-on” model for grasping (literally and figuratively) these exotic shapes and operations on them makes complete sense. Without the pictures or physical approximations shown in the Time-Life book, there is no way I could have understood what that stuff was about.

For an interesting idea of what your proposed virtual system might look like, there is a scene in the movie “Iron Man” where the protagonist is designing his new armored suit using holographic projections of components that he can manipulate directly with his hands, turning them this way and that to see how they move. Very cool. it seems to me that such a system ought to be very doable.

Eleanor Rieffelsays:Sheldon,

Having some related examples, such as how to turn an inner tube insight out, would fill out such an exhibit. And yes, any immersion of a Klein bottle in three-dimensional space has self-intersections. The Klein bottle can be immersed in four dimensional space, however, without any self-intersections. My favorite source for klein bottles, of the immersed in three-dimensional space type, is Acme Klein Bottle . You might also enjoy the accompanying documentation.