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.
