Turning a Sphere Inside Out

Unfolding the impossible: turning a sphere inside out, without a single crease.

This video is about the problem of turning a sphere inside out, by passing the surface through itself, without making any holes or creases. Mathematicians believed the problem to be unsolvable until 1958, when Stephen Smale proved otherwise. The motion of turning a sphere inside out, called a regular homotopy, is extremely difficult to visualize. The homotopy in this film was developed by Bernard Morin, a blind mathematician. The motion is illustrated with a sequence of chicken-wire models, built by Charles Pugh, showing the crucial stages in the motion. Commentary is provided by mathematicians Nelson L. Max, Stephen Smale, and Charles Pugh, and by physicist Judith Bregmann.