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In differential topology, sphere eversion is the process of turning a sphere inside out in a three-dimensional space. (The word ''eversion'' means "turning inside out".) Remarkably, it is possible to smoothly and continuously turn a sphere inside out in this way (with possible self-intersections) without cutting or tearing it or creating any 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 : be the standard embedding; then there is a regular homotopy of immersions : such that ''ƒ''0 = ''ƒ'' and ''ƒ''1 = −''ƒ''. ==History== An existence proof for crease-free sphere eversion was first created by . 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. Smale's graduate adviser Raoul Bott at first told Smale that the result was obviously wrong . His reasoning was that the degree of the Gauss map must be preserved in such "turning"—in particular it follows that there is no such ''turning'' of S1in R2. But the degree of the Gauss map for the embeddings ''f'', −''f'' in R3 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 R3 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. See ''h''-principle for further generalizations. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Sphere eversion」の詳細全文を読む スポンサード リンク
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