In differential topology, Smale’s paradox states that it is possible to turn a sphere inside out in a three-dimensional space with possible self-intersections but without creating any crease, a process more commonly and historically called sphere eversion (eversion means “to turn inside out”). This is surprising, to those who understand regular homotopy, and is hence deemed a veridical paradox.
Images: The award winning computer animation Outside IN explains the amazing discover, made by Steve Smale in 1957, that a sphere can be turned inside out by means of smooth motions and self intersections, without cutting and pasting. It convincingly demonstrates how valuable vizualization can be in the communication of mathematics - Source & full size at Outsite In - geom.uiuc.edu.
Video: Turning A Sphere Insite Out on Youtube.