### The Minimax Sphere Eversion

Published in **Visualization and Mathematics**,
(H.-C. Hege, K. Polthier, eds.), Springer, 1997, pp 3-20
**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 a new sphere eversion
computed by minimizing an elastic bending energy
for surfaces in space called the Willmore energy. We start
with a complicated self-intersecting sphere (a Morin surface)
which is 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.
Here, we can explicitly describe the double locus, and prove
that our eversion is optimal not only geometrically, but also
topologically in the sense of having the fewest catastrophes.