Editing Sphere eversion (section)

==Proof==
Smale's original proof was indirect: he identified (regular homotopy) classes of immersions of spheres with a homotopy group of the [[Stiefel manifold]]. Since the homotopy group that corresponds to immersions of <math>S^2 </math> in <math>\R^3</math> vanishes, the standard embedding and the inside-out one must be regular homotopic. In principle the proof can be unwound to produce an explicit regular homotopy, but this is not easy to do.

There are several ways of producing  explicit examples and [[mathematical visualization]]:
[[File:Minimax Sphere Eversion.webm|thumbtime=17|thumb|Minimax sphere eversion; see the [[:commons:File:Minimax Sphere Eversion.webm|video's Wikimedia Commons page]] for a description of the video's contents]]
* [[Half-way model]]s: these consist of very special homotopies. This is the original method, first done by Shapiro and Phillips via [[Boy's surface]], later refined by many others.  The original half-way model homotopies were constructed by hand, and worked topologically but weren't minimal. The movie created by Nelson Max, over a seven-year period, and based on Charles Pugh's chicken-wire models (subsequently stolen from the Mathematics Department at Berkeley), was a computer-graphics 'tour de force' for its time, and set the bench-mark for computer animation for many years. A more recent and definitive graphics refinement (1980s) is [[minimax eversion]]s, which is a [[calculus of variations|variational]] method, and consist of special homotopies (they are shortest paths with respect to [[Willmore energy]]). In turn, understanding behavior of Willmore energy requires understanding solutions of fourth-order partial differential equations, and so the visually beautiful and evocative images belie some very deep mathematics beyond Smale's original abstract proof.
[[File:Thurston Sphere Eversion.webm|thumbtime=7|thumb|Sphere eversion using Thurston's corrugations; see the [[:commons:File:Thurston Sphere Eversion.webm|video's Wikimedia Commons page]] for a description of the video's contents]]
* [[William Thurston|Thurston]]'s corrugations: this is a [[topological]] method and generic; it takes a homotopy and perturbs it so that it becomes a regular homotopy. This is illustrated in the computer-graphics animation ''Outside In'' developed at the [[Geometry Center]] under the direction of Silvio Levy, Delle Maxwell and [[Tamara Munzner]].<ref>{{cite web|title=Outside In: Introduction|url=http://www.geom.uiuc.edu/docs/outreach/oi/|website=The Geometry Center|access-date=21 June 2017}}</ref>
* Combining the above methods, the complete sphere eversion can be described by a set of closed equations giving minimal topological complexity <ref name=sev-eq>{{cite journal|title=Analytic sphere eversion using ruled surfaces | arxiv =1711.10466 | last1 =Bednorz | first1 =Adam | last2 =Bednorz | first2 =Witold | journal =Differential Geometry and Its Applications | year =2019 | volume =64 | pages =59–79 | doi =10.1016/j.difgeo.2019.02.004 | s2cid =119687494 }} </ref>