Editing Sphere eversion

{{distinguish|Sphere inversion}}
{{short description|Topological operation of turning a sphere inside-out without creasing}}
[[Image:MorinSurfaceFromTheTop.PNG|thumb|A [[Morin surface]] seen from "above"]]
[[File:Evshort2.webm|thumb| Sphere eversion process as described in <ref name=sev-eq/>]]
[[File:Eversion flat.jpg|thumb|Paper sphere eversion and Morin surface]]
[[File:Eversion six flat.jpg|thumb|Paper Morin surface (sphere eversion halfway) with hexagonal symmetry]]

In [[differential topology]], '''sphere eversion''' is a theoretical process of turning a [[sphere]] inside out in a [[three-dimensional space]] (the word ''[[wikt:eversion#English|eversion]]'' means "turning inside out"). It is possible to smoothly and continuously turn a sphere inside out in this way (allowing [[self-intersection]]s of the sphere's surface) without cutting or tearing it or creating any [[Line (geometry)|crease]]. This is surprising, both to non-mathematicians and to those who understand [[regular homotopy]], and can be regarded as a [[veridical paradox]]; that is something that, while being true, on first glance seems false.

More precisely, let

:<math>f\colon S^2\to \R^3</math>

be the standard [[embedding]]; then there is a [[regular homotopy]] of [[immersion (mathematics)|immersions]]

:<math>f_t\colon S^2\to \R^3</math>

such that ''ƒ''<sub>0</sub>&nbsp;=&nbsp;''ƒ'' and ''ƒ''<sub>1</sub>&nbsp;=&nbsp;&minus;''ƒ''.

==History==
An [[existence proof]] for crease-free sphere eversion was first created by {{harvs|txt|authorlink=Stephen Smale|first=Stephen|last= Smale|year= 1958}}.
It is difficult to visualize a particular example of such a turning, although some [[computer animation|digital animations]] have been produced that make it somewhat easier. The first example was exhibited through the efforts of several mathematicians, including [[Arnold S. Shapiro]] and [[Bernard Morin]], who was blind. On the other hand, it is much easier to prove that such a "turning" exists, and that is what Smale did.

Smale's graduate adviser [[Raoul Bott]] at first told Smale that the result was obviously wrong {{harv|Levy|1995}}. 
His reasoning was that  the [[Degree of a continuous mapping|degree]] of the [[Gauss map]] must be preserved in such "turning"—in particular it follows that there is no such ''turning'' of '''S'''<sup>1</sup> in '''R'''<sup>2</sup>. But the degrees of the Gauss map for the embeddings ''f'' and &minus;''f'' in '''R'''<sup>3</sup> are both equal to 1, and do not have opposite sign as one might incorrectly guess. The degree of the Gauss map of all immersions of '''S'''<sup>2</sup> in '''R'''<sup>3</sup> is 1, so there is no obstacle. The term "veridical paradox" applies perhaps more appropriately at this level: until Smale's work, there was no documented attempt to argue for or against the eversion of '''S'''<sup>2</sup>, and later efforts are in hindsight, so there never was a historical paradox associated with sphere eversion, only an appreciation of the subtleties in visualizing it by those confronting the idea for the first time.

See [[h-principle|''h''-principle]] for further generalizations.

==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>

==Variations==
* A six-dimensional sphere <math>S^6</math> in seven-dimensional euclidean space <math>\mathbb{R}^7</math> admits eversion.<ref>{{cite book
 | last = Goryunov | first = Victor V.
 | contribution = Local invariants of mappings of surfaces into three-space
 | isbn = 0-8176-3883-0
 | pages = 223–255
 | publisher = Birkhäuser | location = Boston, Massachusetts
 | title = The Arnold–Gelfand mathematical seminars
 | year = 1997}}</ref> With an evident case of an 0-dimensional sphere <math>S^0</math> (two distinct points) in a real line <math>\mathbb{R}</math> and described above case of a two-dimensional sphere in <math>\mathbb{R}^3 </math> there are only three cases when  sphere <math>S^n</math> embedded in euclidean space <math>\mathbb{R}^{n+1}</math> admits eversion.

==Gallery of eversion steps==
{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
|+Surface plots
|{{multiple image |align= center | header = Ruled model of halfway with quadruple point | width =100 | image1= hw-a.png| caption1=top view|
image2 = hw-b.png| caption2= diagonal view|
image3 = hw-c.png| caption3= side view|direction=|total_width=|alt1=}}
|{{multiple image |align= center | header = Closed halfway | width =100 | image1= morin-a.png| caption1=top view|
image2 = morin-b.png| caption2= diagonal view|
image3 = morin-c.png| caption3= side view|direction=|total_width=|alt1=}}
|{{multiple image |align= center | header = Ruled model of death of triple points | width =100 | image1= Ttw-b.png| caption1=top view|
image2 = Ttw-b1.png| caption2= diagonal view|
image3 = Ttw-b2.png| caption3= side view|direction=|total_width=|alt1=}}
|-
|{{multiple image |align= center | header = Ruled model of end of central intersection loop | width =100 | image1= Ddw-b.png| caption1=top view|
image2 = Ddw-b1.png| caption2= diagonal view|
image3 = Ddw-b2.png| caption3= side view|direction=|total_width=|alt1=}}
| colspan="2" |{{multiple image |align= center | header = Ruled model of last stage | width =100 | image1= Ww-b.png| caption1=top view|
image2 = Ww-b1.png| caption2= diagonal view|
image3 = Ww-b2.png| caption3= side view|direction=|total_width=|alt1=}}
|}



{{multiple image
| align = center
| header = Nylon string open model
| width = 100
| image1 = Q-point.jpg
| caption1 = halfway top
| image2 = Q-point2.jpg
| caption2 = halfway side
| image3 = T-point.jpg
| caption3 = triple death top
| image4 = T-point2.jpg
| caption4 = triple death side
| image5 = D-point.jpg
| caption5 = intersection end top
| image6 = D-point2.jpg
| caption6 = intersection end side
| direction = 
| total_width = 
| alt1 = 
}}



==See also==
* [[Whitney–Graustein theorem]]

==References==
{{Reflist}}

===Bibliography===
* [https://arxiv.org/abs/1008.0916 Iain R. Aitchison (2010) The `Holiverse': holistic eversion of the 2-sphere in R^3], preprint.   arXiv:1008.0916.
* John B. Etnyre (2004) Review of "h-principles and flexibility in geometry", {{MathSciNet|id=1982875}}.
* {{Citation | last1=Francis | first1=George K. | title=A topological picturebook |title-link=A Topological Picturebook| publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-34542-0 | mr=2265679  | year=2007}}
* George K. Francis & [[Bernard Morin]] (1980) "Arnold Shapiro's Eversion of the Sphere", [[Mathematical Intelligencer]] 2(4):200&ndash;3.
* {{Citation | last1=Levy | first1=Silvio | title=Making waves | publisher=A K Peters Ltd. | location=Wellesley, MA | isbn=978-1-56881-049-2 | mr=1357900  | year=1995 | chapter=A brief history of sphere eversions | chapter-url=http://www.geom.uiuc.edu/docs/outreach/oi/history.html}}
* Max, Nelson (1977) "Turning a Sphere Inside Out", https://www.crcpress.com/Turning-a-Sphere-Inside-Out-DVD/Max/9781466553941 {{Webarchive|url=https://web.archive.org/web/20160304074158/https://www.crcpress.com/Turning-a-Sphere-Inside-Out-DVD/Max/9781466553941 |date=2016-03-04 }}
* Anthony Phillips (May 1966) "Turning a surface inside out", ''Scientific American'', pp.&nbsp;112–120.
* {{Citation | last1=Smale | first1=Stephen | author1-link=Stephen Smale | title=A classification of immersions of the two-sphere | jstor=1993205 | mr=0104227  | year=1958 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=90 | issue=2 | pages=281–290 | doi=10.2307/1993205| doi-access=free }}

==External links==
* [http://torus.math.uiuc.edu/jms/Papers/isama/color/opt2.htm A History of Sphere Eversions] {{Webarchive|url=https://web.archive.org/web/20200711130735/http://torus.math.uiuc.edu/jms/Papers/isama/color/opt2.htm |date=2020-07-11 }}
* [http://www.cs.berkeley.edu/~sequin/SCULPTS/SnowSculpt04/eversion.html "Turning a Sphere Inside Out"]
* [http://profs.etsmtl.ca/mmcguffin/eversion/ Software for visualizing sphere eversion]
* [https://www.youtube.com/watch?v=876a_0WAoCU/ Mathematics visualization: topology. The holiverse sphere eversion (Povray animation)]
* The deNeve/Hills sphere eversion: [https://www.youtube.com/watch?v=FL4JoWlVj98/ video] and [http://www.chrishills.org.uk/ChrisHills/sphereeversion/interactive/index.html?a=y/ interactive model]
* [https://leanprover-community.github.io/sphere-eversion/ Patrick Massot's project] to formalise the proof in the [[Lean (proof assistant)|Lean Theorem Prover]]
* An [https://rreusser.github.io/explorations/sphere-eversion/ interactive exploration] of Adam Bednorz and Witold Bednorz method of sphere eversion 
* [https://www.youtube.com/watch?v=wO61D9x6lNY Outside In]: A video exploration of sphere eversion, created by [[Geometry Center|The Geometry Center]] of [[University of Minnesota|The University of Minnesota]]. 
{{Portal bar|Mathematics}}

{{DEFAULTSORT:Smale's Paradox}}
[[Category:Differential topology]]
[[Category:Mathematical paradoxes]]