Smale (1958) proved that it is mathematically possible to turn a sphere inside-out without introducing a sharp crease at any point. This means there is a regular homotopy from the standard embedding of the 2-sphere in Euclidean three-space to the mirror-reflection embedding such that at every stage in the homotopy, the sphere is being immersed in Euclidean space. This result is so counterintuitive and the proof so technical that the result remained controversial for a number of years.

In 1961, Arnold Shapiro devised an explicit eversion but did not publicize it. Phillips (1966) heard of the result and, in trying to reproduce it, actually devised an independent method of his own. Yet another eversion was devised by Morin, which became the basis for the movie by Max (1977). Morin's eversion also produced explicit algebraic equations describing the process. The original method of Shapiro was subsequently published by Francis and Morin (1979).

The Season 1 episode "Sniper Zero" (2005) of the television crime drama NUMB3RS mentions sphere eversion.

REFERENCES:

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." https://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." https://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. https://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. https://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. https://www.mathematicaguidebooks.org/.

Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, 1991.