A smooth deformation with no tearing or pinching — topologically possible, intuitively shocking
Can you turn a sphere inside out through smooth deformations, allowing self-intersections but no creasing or tearing?
In 1957, Stephen Smale proved this is possible using differential topology — to the disbelief of his advisor. No explicit construction existed until later.
This visualization uses the halfway model: the sphere passes through a critical shape (like Boy's surface) where inside and outside are exchanged.
Key stages: push poles toward equator → corrugate the surface → pass through the halfway model → uncorrugate → original sphere, now inside-out.
The animation uses a corrugation function that creates controlled self-intersections, demonstrated by Thurston and animated by Levy, Max & Munzner (1994).