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 3-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).

**References**

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. *Making Waves: A Guide to the Ideas Behind Outside In.* Wellesley, MA: A. K. Peters, 1995.

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. *Outside-In.* 22 minute 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. *Islands of Truth: A Mathematical Mystery Cruise.* New York: W. H. Freeman, pp. 240-244, 1990.

Petersen, 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.

Smale, S. ``A Classification of Immersions of the Two-Sphere.'' *Trans. Amer. Math. Soc.* **90**, 281-290, 1958.

© 1996-9

1999-05-26