Smale's paradox

Smale's paradox

In differential topology, Smale's paradox states that it is possible to turn a sphere inside out in 3-space with possible self-intersections but without creating any crease, a process often called sphere eversion ("eversion" means "to turn inside out").More precisely, let

:fcolon S^2 o R^3

be the standard embedding; then there is a regular homotopy of immersions

:f_tcolon S^2 o R^3

such that f_0=f, and f_1=-f,.

History

This 'paradox' was discovered by Stephen Smale in 1958. It is difficult to visualize a particular example of such a turning, although some digital animations have been produced that make it somewhat easier. The first example was exhibited through the efforts of several mathematicians, including one who was blind, Bernard Morin. On the other hand, it is much easier to prove that such a "turning" exists and that is what was done by Smale.

The legend says that when Smale was trying to publish this result, the referee's report stated thatalthough the proof is quite interesting, the statement is clearly wrong 'due to invariance of degree of the Gauss map'.Fact|date=February 2007 Indeed, the degree of the Gauss map must be preserved in such "turning"—in particular it follows that there is no such "turning" of S^1, in R^2. But the degree of the Gauss map for the embeddings f, and -f, in R^3 are both equal to 1. In fact the degree of the Gauss map of all immersions of a 2-sphere in R^3 is 1; so there is in fact no obstacle.

See "h"-principle for further generalizations.

Proof

Smale's original proof was nonconstructive: 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 S^2 , in R^3 vanishes, the standard embedding and the inside-out one must be regular homotopic, but it does not produce a regular homotopy.

There are two classes of constructive proofs:
* the method of half-way models: 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. A more recent and definitive refinement (1980s) is minimax eversions, which is a variational method, and consist of special homotopies (they are shortest paths with respect to Willmore energy). The original half-way model homotopies were constructed by hand, and worked topologically but weren't minimal.

* Thurston's corrugations: this is a topological method and generic; it takes a homotopy and perturbs it so that it becomes a regular homotopy.

ee also

* Boy's surface
* Morin surface
* Whitney-Graustein theorem
* Eversion

References

*Nelson Max, "Turning a Sphere Inside Out", International Film Bureau, Chicago, 1977 (video)
*Anthony Phillips, "Turning a surface inside out", "Scientific American", May 1966, pp. 112-120.
*Smale, Stephen "A classification of immersions of the two-sphere." Trans. Amer. Math. Soc. 90 1958 281–290.

External links

* [http://video.google.com/videoplay?docid=-6626464599825291409 Outside In] , full video
* [http://new.math.uiuc.edu/optiverse/ Optiverse video] , portions available online
* [http://torus.math.uiuc.edu/jms/Papers/isama/color/opt2.htm A History of Sphere Eversions]
* [http://www.geom.uiuc.edu/docs/outreach/oi/history.html A brief history of sphere eversions]
* [http://www.cs.berkeley.edu/~sequin/SCULPTS/SnowSculpt04/eversion.html "Turning a Sphere Inside Out"]
* An MPG movie of [http://www.th.physik.uni-bonn.de/th/People/netah/cy/movies/sphere.mpg turning a sphere inside out]
* [http://www.dgp.utoronto.ca/~mjmcguff/eversion/ Software for visualizing sphere eversion]


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать реферат

Look at other dictionaries:

  • Stephen Smale — (né le 15 juillet 1930 à Flint dans le Michigan) est un mathématicien américain, lauréat de la médaille Fields en 1966, récompensé pour ses remarquables travaux en topologie différentielle. Sa réputation est due à une démonstration de la… …   Wikipédia en Français

  • List of mathematics articles (S) — NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… …   Wikipedia

  • List of paradoxes — This is a list of paradoxes, grouped thematically. Note that many of the listed paradoxes have a clear resolution see Quine s Classification of Paradoxes.Logical, non mathematical* Paradox of entailment: Inconsistent premises always make an… …   Wikipedia

  • Homotopy principle — In mathematics, the homotopy principle (or h principle) is a very general way to solve partial differential equations (PDEs), and more generally partial differential relations (PDRs). The h principle is good for underdetermined PDEs or PDRs, such …   Wikipedia

  • Eversión de la esfera — Saltar a navegación, búsqueda Superficie intermedia durante la eversión de la esfera denominada superficie de Morin. En topología se demuestra que es posible evertir una esfera sin efectuar ningún corte en ella, aunque en el proceso se intersecta …   Wikipedia Español

  • List of differential geometry topics — This is a list of differential geometry topics. See also glossary of differential and metric geometry and list of Lie group topics. Contents 1 Differential geometry of curves and surfaces 1.1 Differential geometry of curves 1.2 Differential… …   Wikipedia

  • Bernard Morin — is a French mathematician, especially a topologist, born in 1931, who is now retired. He has been blind since age 6 due to glaucoma, but his blindness did not prevent him from having a successful career in mathematics.Morin was a member of the… …   Wikipedia

  • Morin surface — seen from the top Morin surface seen from t …   Wikipedia

  • Scientific phenomena named after people — This is a list of scientific phenomena and concepts named after people (eponymous phenomena). For other lists of eponyms, see eponym. NOTOC A* Abderhalden ninhydrin reaction Emil Abderhalden * Abney effect, Abney s law of additivity William de… …   Wikipedia

  • Hypercomputation — refers to various hypothetical methods for the computation of non Turing computable functions (see also supertask). The term was first introduced in 1999 by Jack Copeland and Diane Proudfoot [Copeland and Proudfoot,… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”