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| [[Image:MorinSurfaceFromTheTop.PNG|thumb|A [[Morin surface]] seen from "above"]]
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| In [[differential topology]], '''Smale's [[paradox]]''' states that it is possible to turn a sphere inside out in a [[three-dimensional space]] with possible self-intersections but without creating any crease, a process more commonly and historically called '''sphere eversion''' (''[[wikt:eversion#English|eversion]]'' means "to turn inside out"). This is surprising, to those who understand regular homotopy, and is hence deemed a [[veridical paradox]]. (The concept 'Smale's Paradox' generally does not appear in the traditional mathematical literature.)
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| More precisely, let
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| :<math>f\colon S^2\to \R^3</math>
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| be the standard [[embedding]]; then there is a [[regular homotopy]] of [[immersion (mathematics)|immersions]]
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| :<math>f_t\colon S^2\to \R^3</math> | |
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| such that ''ƒ''<sub>0</sub> = ''ƒ'' and ''ƒ''<sub>1</sub> = −''ƒ''.
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| ==History==
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| This 'paradox' was discovered by {{harvs|txt|authorlink=Stephen Smale|first=Stephen|last= Smale|year= 1958}}.
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| 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 [[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.
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| Smale's graduate adviser [[Raoul Bott]] at first told Smale that the result was obviously wrong {{harv|Levy|1995}}.
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| 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 degree of the Gauss map for the embeddings ''f'', −''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 a 2-sphere 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 is no documented attempt to argue that 2-sphere eversion was, or was not, possible, and accordingly, subsequent attempts at explicit eversion, or to argue that it was impossible, are in hindsight. Accordingly, there never was an historical paradox associated with Smale's actual sphere eversion, merely an appreciation of the conceptual visualization subtleties by those confronting the idea for the first time.
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| See [[h-principle|''h''-principle]] for further generalizations.
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| ==Proof==
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| Smale's original proof was indirect: he identified (regular homotopy) classes of immersions of spheres which can be unwound to produce an explicit regular homotopy, but this is not easy to do.
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| There are several ways of producing explicit examples and beautiful [[mathematical visualization]]:
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| * [[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.
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| * [[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', easily found by searching Youtube.
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| * Aitchison's 'holiverse' (2010): this uses a combination of topological and geometric methods, and is specific to the actual regular homotopy between a standardly embedded 2-sphere, and the embedding with reversed orientation. This provides conceptual understanding for the process, revealed as arising from the concrete structure of the 3-dimensional projective plane and the underlying geometry of the Hopf fibration. Understanding details of these mathematical concepts is not required to conceptually appreciate the concrete eversion that arises, which in essence only requires understanding a specific embedded circle drawn on a torus in 3-space. George Francis suggested the name "holiverse", derived from the word "holistic", since (after some thought) the complete eversion can be conceptually grasped from beginning to end, without the visual aids provided by animation. In spirit, this is closer to the ideas originally suggested by Shapiro, and in practice provides a concrete proof of eversion not requiring the abstraction underlying Smale's proof. This is partially illustrated in a Povray computer-graphics animation, again easily found by searching YouTube.
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| ==See also==
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| * [[Sphere]]
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| * [[Whitney–Graustein theorem]]
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| ==References==
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| * [http://arxiv.org/abs/1008.0916 Iain R. Aitchison (2010) The `Holiverse': holistic eversion of the 2-sphere in R^3], preprint. arXiv:1008.0916.
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| * John B. Etnyre (2004) Review of "h-principles and flexibility in geometry", {{MathSciNet|id=1982875}}.
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| * {{Citation | last1=Francis | first1=George K. | title=A topological picturebook | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-34542-0 | mr=2265679 | year=2007}}
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| * George K. Francis & [[Bernard Morin]] (1980) "Arnold Shapiro's Eversion of the Sphere", [[Mathematical Intelligencer]] 2(4):200–3.
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| * {{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 | chapterurl=http://www.geom.uiuc.edu/docs/outreach/oi/history.html}}
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| * Max Nelson (1977) "Turning a Sphere Inside Out", International Film Bureau, Chicago, (video).
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| * Anthony Phillips (May 1966) "Turning a surface inside out", ''Scientific American'', pp. 112–120.
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| * {{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 | pages=281–290}}
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| ==External links==
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| * [http://video.google.com/videoplay?docid=-6626464599825291409 Outside In], full video (short clip [http://www.th.physik.uni-bonn.de/th/People/netah/cy/movies/sphere.mpg here])
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| * [http://www.youtube.com/watch?v=wO61D9x6lNY] Above video hosted on youtube.com
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| * [http://torus.math.uiuc.edu/jms/Papers/isama/color/opt2.htm A History of Sphere Eversions]
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| * [http://www.cs.berkeley.edu/~sequin/SCULPTS/SnowSculpt04/eversion.html "Turning a Sphere Inside Out"]
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| * [http://www.dgp.utoronto.ca/~mjmcguff/eversion/ Software for visualizing sphere eversion]
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| * [http://www.youtube.com/watch?v=876a_0WAoCU/ Mathematics visualization: topology. The holiverse sphere eversion (Povray animation)]
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| {{DEFAULTSORT:Smale's Paradox}}
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| [[Category:Differential topology]]
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| [[Category:Mathematics paradoxes]]
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You never know exactly what a new video game world may contain. There are horrors and bad in every nook and cranny. This post includes advice about optimizing your gaming duration with tricks and indications you might not is aware of. Go forward reading for more important info.
Trying the higher level: the most important thing when it comes to game, but when seeking at Clash of Clans, you get a lot more subtle actions. Despite making use of clash of clans hack tools, you in addition be acquire experience points as a result of matching on top linked to other players. Increased metabolism purpose of earning Player vs player is to enable any enhancements for your indigneous group. The restoration consists of better the battle equipment, properties, troops as well tribe people.
Temperance is essential in on the whole things, and enjoying clash of clans cheats is no some other. Playing for hours on finish is not just good-for-you, bodily or on an emotional level. There are some games out truth be told there which know this and include measures to remind in order to take rests. Take the initiative yourself, although! Place an alarm and also that don't play for more than an hour right.
Time and again check several distinct locations before purchasing a gaming. Be sure - look both online combined with in genuine brick then mortar stores in a new region. In case you have any kind of issues about exactly where and also how you can utilize clash of clans hack tool (straight from the source), you'll be able to contact us with our web site. The cost of a video movie may differ widely, unusually if a game is probably not brand new. By performing a hardly any additional leg work, it is possible to arrive clash of clans.
The organization testing has apparent that this appraisement algorithm strategy consists of a alternation of beeline band clips. They are not things to consider variants of arced graphs. I will explain why later.
This particular information, we're accessible in alpha dog substituting values. Application Clash of Clans Cheats' data, let's say towards archetype you appetite 1hr (3, 600 seconds) to be able to bulk 20 gems, and then 1 day (90, 400 seconds) to help largest percentage 260 gems. It's appropriately stipulate a action for this kind about band segment.
You dont necessarily desire one of the cutting edge troops to win advantages. A mass volume of barbarians, your first-level troop, will totally destroy an assailant village, and strangely its quite enjoyable to the the virtual carnage.