Turning sphere inside out

This animation shows how to turn a sphere outside in. For a circle, it is impossible because the turning number, a topological invariant, must be conserved but it distinguishes the two oppositely oriented circles. However, all analogous topological invariants - such as those based on Morse theory - have the same sign for both orientations of a sphere.




So a smooth algorithm, ultimately obtained from a regularized wiggly version of a singular algorithm, may be found to turn the sphere inside out. String theorist Matt Headrick was one of the animators who used Mathematica. See also videos of Möbius transformations and animated fractals.

A different video starring a bubble that turns inside out.

Hat tip: Tom Weidig

Another recommended video: Richard Feynman playing bongos. The song is about "Orange Juice"; see also a related song about Darinka, a mandarine whose husband managed not to become orange juice. ;-) Well, geniuses are usually mad at the same moment. ;-)