We don't expect this sphere eversion to have any direct applications. Of course, often mathematics proves to have surprising and unexpected applications. (Famous examples include Riemann's geometry of curved 3- and 4-dimensional spaces, which seemed irrelevant to the real world until Einstein's theory of relativity used it to explain gravity. Or various results in number theory, which have recently been used for secure encryption on the internet.) But we investigated this sphere eversion purely for its own interest.

The bending energy for surfaces that we used to drive the calculations, on the other hand, is quite applicable. Whereas surfaces like soap films minimize their surface area due to surface tension, other surfaces, notably the bilipid membranes around cells, seem to minimize our kind of bending energy. The characteristic shape of a red blood cell is due to minimizing bending (fixing the surface area and enclosed volume). In the laboratory, lipid vesicles have been observed attaining other characteristic shapes for minimum bending energy. (For instance, some of my earlier numerical work on minimizing this energy was used by biophysicists in France---see the article by Michalet and Bensimon in Science 269 (Aug 95) p666.)

The mathematical rules for a sphere eversion prevent the surface of the sphere from ripping or tearing, pinching or creasing. But they do allow the surface of the sphere to pass through itself. There doesn't seem to be any material in the real world that can do this---if one sheet of material crosses another, they must interact. Thus, I don't expect that we can build a physical model that really shows the eversion happening.

Thus we were driven to use computer graphics for our visualization. Our programs run on desktop workstations but also in the CAVE virtual reality environment. This is a cubical room, about 3m on a side, with computer graphics displays on three walls and the floor. Wearing special stereo goggles, you get a wonderful feeling of being inside the objects displayed. Thus, we can view a large version of the everting sphere, with convincing three-dimensional effects.

Mathematicians have built models out of wire mesh ("chicken wire") to show the several different stages of an eversion. In fact a famous set of such models used to hang in the math department at UC Berkeley, until they were stolen. But these could not move to show the eversion process.

As I mentioned above, there are physical surfaces (in particular, biological membranes) which do move to minimize the bending energy we used for our eversion. But these cannot self-intersect, so they couldn't be used to build a sphere eversion.

The problem of sphere eversions has interested mathematicians simply because it is possible. It's impossible to turn a circle inside out by the same rules, and nobody thought it could be possible to turn a sphere inside out. But then Steve Smale proved a very abstract theorem which implied that an eversion was possible. Still nobody knew how to do it. Even after the first explicit eversions were described, they have been hard to visualize.

Thus problem has remained like a "Mt. Everest" for mathematicians, a challenge out there to be conquered. We have been working on optimization problems for shapes, and figured that the sphere eversion would be a wonderful test problem. We're very happy with the results, as we find this new eversion more esthetically pleasing than any of the previous ones.

As I have described, the bending energy, when applied to surfaces that don't pass through themselves, has many real-world applications, so we hope that our increased understanding of its behavior may be helpful in these applications.

My own research recently has concentrated more on surfaces minimizing area (as in soap bubbles) instead of bending energy. But I hope to get back to the bending energy problems. Perhaps the same method can be applied to other mathematical "homotopies". For instance, if we start with a torus (the shape of a donut or inner-tube), it is possible to move it so as to interchange the small loop with the big loop. Perhaps bending energy can be used to find an optimal way to do this.

We created the eversion by a principle of energy minimization. Think of the sphere being made of an elastic material that likes to bend as little as possible. Thus it prefers to be a round sphere (perhaps inside-out, perhaps not). However, to turn it inside-out, we need to twist it up (adding energy), and then let it go again, returning to the round sphere.

It's like winding up a spring, and then releasing it. There are many physical systems like this, that have two possible states (each sometimes called a "potential well"). The system would be happy to sit in either state. But to get from one to the other, we have to add energy temporarily.

The usual analogy is to climbing over mountains, from one valley to another. We have to go up first, then back down. It's easiest to pick a route over a saddle-point, or mountain-pass, rather than going all the way up to a mountain-peak.

The computer program we used (Brakke's "Surface Evolver") does computations for energy minimization. That is, it computes a route downhill. Thus, we really only computed the second half of the eversion. We started the computation from what we call the "halfway model". That is the complicated surface with four lobes, where two of the lobes show the red inside of the sphere, and two show the blue outside. This "halfway model" is thus halfway inside-out. It is the middle of the eversion. For surfaces with its special kind of 4-fold symmetry, it has the least bending energy. (That's why it corresponds to the saddle-point or mountain-pass.) But if we perturb it slightly, it falls off downhill towards one valley or the other. If we perturb it so the red lobes get a little bigger, eventually we proceed downhill to get a sphere with the red side showing. And the same with blue.

Thus, our computational process was as follows: We started with a surface with the special 4-fold symmetry of the halfway model. We minimized energy, keeping this symmetry to get the halfway model. Then we perturbed it slightly to break the symmetry. Finally, we followed a path downhill decreasing the bending energy until we arrived at the round sphere. We saved a sequence of several hundred shapes from this process.

The movie of the eversion was then created by showing this sequence first backwards (going uphill from the round sphere to the halfway model), then forwards (back downhill to the round sphere, but on the other side of the "mountain range" so that we arrive at the inside-out sphere.