Sphere eversion

In differential topology, sphere eversion is the process of turning a sphere inside out in a three-dimensional space (the word eversion means "turning inside out"). Remarkably, it is possible to smoothly and continuously turn a sphere inside out in this way (allowing self-intersections of the sphere's surface) without cutting or tearing it or creating any crease. This is surprising, both to non-mathematicians and to those who understand regular homotopy, and can be regarded as a veridical paradox; that is something that, while being true, on first glance seems false.

More precisely, let

${\displaystyle f\colon S^{2}\to \mathbb {R} ^{3}}$

be the standard embedding; then there is a regular homotopy of immersions

${\displaystyle f_{t}\colon S^{2}\to \mathbb {R} ^{3}}$

such that ƒ₀ = ƒ and ƒ₁ = −ƒ.