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.
