How to Turn a Sphere Inside Out

Turning a sphere inside out without tearing, creasing, or pinching seems physically impossible, yet differential topology proves this transformation, known as sphere eversion, is achievable. This video explores the mathematical proof pioneered by Stephen Smale in 1957, which requires the surface to pass smoothly through itself, maintaining local differentiability throughout the process. We define the constraints of regular homotopy, analyze the necessary self-intersections, and visualize explicit solutions like the highly symmetric Morin surface. Discover how the Gauss map tracks the orientation reversal and why this abstract concept is a foundational example of the h-principle in geometry. 00:00: The Impossible Topological Feat 00:46: Defining Regular Homotopy Constraints 01:26: Smale's Existential Proof 02:03: The Necessary Self-Intersection 02:45: Visualizing the Morin Surface 03:20: The Half-Turn Method 03:52: Gauss Map Tracks Orientation 04:31: Minimal Critical Points 05:06: Eversion in Higher Dimensions 05:43: Impact on Topology and Visualization #SphereEversion #Topology #DifferentialTopology #SmaleTheorem #RegularHomotopy #MorinSurface #Geometry