Computing Sphere Eversions

by George Francis, John M. Sullivan & Chris Hartman

Published in Mathematical Visualization, (H.-C. Hege, K. Polthier, eds.), Springer, 1998, pp 237-255. Plain-English Abstract

It was a surprising consequence of an abstract theorem of Smale that a spherical surface can be turned inside out without tearing or creasing, if we do allow the surface to pass through itself. Over the intervening forty years, mathematicians have designed different ways to see explicitly how such a ``sphere eversion'' can be achieved.

Here, we consider several tools for computing and visualizing sphere eversions. First, we discuss a family of optimal sphere eversions with rotational symmetry. These are computed by minimizing an elastic bending energy for surfaces in space called the Willmore energy. We start with a complicated self-intersecting sphere, which has the desired rotational symmetry, and is also halfway inside-out in the sense of having its inside and outside equally exposed. This halfway model is a saddle critical point for the Willmore energy. When we push off the saddle in the two directions and then flow downhill in energy to the ordinary round sphere, it is inside-out one way, but not the other way.

In any sphere eversion, the "double locus" of self-intersections is a key item to follow through time as the sphere turns inside out. We describe real-time interactive computer animation programs to compute and display the double locus of any immersed surface, and to track this along a homotopy like a sphere eversion. Finally, we consider ways to implement computationally the various eversions originally drawn by hand; this requires interpolation of splined curves in time and space.