The Minimax Sphere Eversion

Abstract

We consider an eversion of a sphere driven by a gradient flow for elastic bending energy. We start with a halfway model which is an unstable Willmore sphere with 4-fold orientation-reversing rotational symmetry. The regular homotopy is automatically generated by flowing down the gradient of the energy from the halfway model to a round sphere, using the Surface Evolver. This flow is not yet fully understood; however, our numerical simulations give evidence that the resulting eversion is isotopic to one of Morin's classical sphere eversions. These simulations were presented as real-time interactive animations in the CAVE automatic virtual environment at Supercomputing'95, as part of an experiment in distributed, parallel computing and broad-band, asynchronous networking.

Computing Sphere Eversions

“…Earlier we described a minimax sphere eversion [14] computed numerically by minimizing a surface bending energy. In fact, that eversion seems to be only the first in an infinite family of similar eversions with higher-order symmetry.…”

“…There are several discretizations of the Willmore energy available in the Evolver [15]. In fact, we made use of these before to implement the minimax eversion with p = 2, which is equivalent to Morin's original eversion [14]. However, there we did not make use of the two-fold symmetry.…”

“…In an earlier paper [14], we described a minimax sphere eversion, driven automatically by minimization of the Willmore bending energy for surfaces. Here, we consider three extensions of that work, which are of interest in the areas of optimal geometry, splined surfaces and regular homotopy theory.…”

“…These fields are some of the many areas of geometry and computation which converge so nicely on the problem of finding new ways to evert the sphere. We consider this paper a continuation of [14], to which we refer the reader for background; further information on sphere eversions can be found in [20,10,24].…”

“…Events in the regular homotopy correspond to critical points for the height function t on X. For the eversion we presented in [14] (equivalent to one in [1]), counting these critical points shows that X has Euler characteristic −1 and thus is a Dyck's surface (a sphere with three cross-caps) immersed in R 4 , which we want to visualize. This involves computing the double locus at each stage of the eversion, splicing these level curves together into a surface, and then viewing the resulting immersion in R 4 by projection or slicing.…”