Turning a sphere inside-out without tearing — Smale's impossible theorem made visible
Stephen Smale proved in 1958 that a sphere can be turned inside-out through immersions (self-intersections allowed, no tears or pinches). This uses the "halfway model" approach: the sphere passes through itself, with colors tracking which side is which. At t=0.5, we reach the halfway model — a beautiful, symmetric self-intersecting surface.