Figure 5: These sketches by George Francis show a French tobacco pouch, and a cutaway view of the halfway-model for the order-five tobacco-pouch eversion.

Francis and Morin realized that the Morin surface and Boy's surface are just the first in an infinite sequence of possible halfway-models for what they call the tobacco-pouch eversions (see Fig. 5, taken from [Fr]). In the 1980s, Robert Bryant [Bry] classified all critical points for the Willmore energy among spheres; they all have integer energy values which (except for the round sphere) are at least 4. Rob Kusner, being familiar with both of these results, realized that he could find, among Bryant's critical spheres, ones with the tobacco-pouch symmetries. Among surfaces with those symmetries, Kusner's have the least possible Willmore energy.

In particular, Kusner's Morin surface with four-fold orientation-reversing symmetry has energy exactly 4. If we don't enforce this symmetry, then presumably this surface is an unstable critical point: a saddle point for the energy. Pushing off from this saddle in one downhill direction, and flowing down by gradient descent, we should arrive at the round sphere, since it is the only critical point with lower energy. (Of course, there's not enough theory for fourth-order partial differential equations for us to know in advance that the surface will remain smooth and not pinch off somewhere.)

As we saw before, such a homotopy, when repeated with a twist, will give us a sphere eversion. This eversion starts at the round sphere (which minimizes energy) and goes up over the lowest energy saddle point; then it comes back down on the other side arriving at the inside-out round sphere.

Around 1995, in collaboration with Kusner and Francis, I computed this minimax eversion [FSK] using Ken Brakke's Evolver [Bra], which is a software package designed for solving variational problems, like finding the shape of soap films or (see [HKS]) minimizing Willmore energy. We were pleased that the computed eversion was not only geometrically optimal in the sense of requiring the least bending, but also topologically optimal, in that it was one of the Morin eversions with the fewest topological events.

Computations of the higher-order minimax sphere eversions [FSH] (like the one with a Boy's surface halfway-model) had to wait until Brakke and I implemented some new symmetry features in the Evolver (see [BS]). A minimax tobacco-pouch eversion, whose halfway-model has 2k-fold symmetry, will break some of this symmetry. But it will maintain k-fold symmetry throughout, and the Evolver now works with only a single fundamental domain for this symmetry. We can find the initial halfway-models by minimizing bending energy while enforcing the full symmetries.

Alternatively, we can compute them directly. Bryant's classification says that all critical spheres are obtained as conformal transformations of minimal surfaces, and Kusner gave explicit Weierstrass parameterizations for the minimal surfaces he needed. In Fig. 6 we see the minimal surface with four flat ends which transforms into our Morin halfway-model.