Smale (1958) 證明了在數學上可以將一個球體由內向外翻轉，而不會在任何點引入尖銳的褶皺。這意味著存在一個從二維球面在歐幾里得三維空間中的標準嵌入到映象反射嵌入的正則同倫，使得在同倫的每個階段，球體都被浸入在歐幾里得空間中。這個結果非常違反直覺，並且證明非常技術性，以至於這個結果多年來都存在爭議。
1961年，Arnold Shapiro 設計了一個顯式的外翻方法，但沒有公開。Phillips (1966) 聽說了這個結果，並在嘗試重現它的過程中，實際上設計了一種獨立的自身方法。Morin 又設計了另一種外翻方法，這成為了 Max (1977) 電影的基礎。Morin 的外翻方法還產生了描述該過程的顯式代數方程。Shapiro 最初的方法隨後由 Francis 和 Morin (1979) 發表。
電視劇犯罪劇集數字追兇第一季劇集 "狙擊手零" (2005) 提到了球面外翻。
參見
外翻,
球體
透過 探索
參考文獻
Apéry, F. "An Algebraic Halfway Model for the Eversion of the Sphere." Tôhoku Math. J. 44, 103-150, 1992.Apéry, F.; and Franzoni, G. "The Eversion of the Sphere: a Material Model of the Central Phase." Rendiconti Sem. Fac. Sc. Univ. Cagliari 69, 1-18, 1999.Bulatov, V. "Sphere Eversion--Visualization of the Famous Topological Procedure." http://www.physics.orst.edu/~bulatov/vrml/evert.wrl.Francis, G. K. Ch. 6 in A Topological Picturebook. New York: Springer-Verlag, 1987.Francis, G. K. and Morin, B. "Arnold Shapiro's Eversion of the Sphere." Math. Intell. 2, 200-203, 1979.Levy, S. "A Brief History of Sphere Eversions." http://www.geom.umn.edu/docs/outreach/oi/history.html.Levy, S.; Maxwell, D.; and Munzner, T. Making Waves: A Guide to the Ideas Behind Outside In. Wellesley, MA: A K Peters, 1995. Book and 22 minute Outside-In. videotape. http://www.geom.umn.edu/docs/outreach/oi/.Max, N. "Turning a Sphere Inside Out." Videotape. Chicago, IL: International Film Bureau, 1977.Peterson, I. "Inside Moves." Sci. News 135, 299, May 13, 1989.Peterson, I. Islands of Truth: A Mathematical Mystery Cruise. New York: W. H. Freeman, pp. 240-244, 1990.Peterson, I. "Forging Links Between Mathematics and Art." Science News 141, 404-405, June 20, 1992.Phillips, A. "Turning a Surface Inside Out." Sci. Amer. 214, 112-120, Jan. 1966.Schimmrigk, R. http://www.th.physik.uni-bonn.de/th/People/netah/cy/movies/sphere.mpg.Smale, S. "A Classification of Immersions of the Two-Sphere." Trans. Amer. Math. Soc. 90, 281-290, 1958.Toth, G. Finite Möbius Groups, Minimal Immersion of Spheres, and Moduli. Berlin: Springer-Verlag, 2002.Trott, M. The Mathematica GuideBook for Symbolics. New York: Springer-Verlag, pp. 38-39, 2006. http://www.mathematicaguidebooks.org/.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, 1991.在 上被引用
球面外翻
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Weisstein, Eric W. "球面外翻。" 出自 Web 資源。 https://mathworld.tw/SphereEversion.html
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