Willmore
Steve Trettel
| Medium | Mathematica Notebook |
|---|---|
| The Art | An approximation to a Willmore torus, embedded in the 3-sphere. The surface is viewed perspectively correct from a position in 4-dimensional space off of the surface of the sphere, as the sphere rotates below it. | The Math | The Willmore energy of an embedded surface is the integral of the square of its mean curvature. A Willmore torus is an embedding of the torus (here into the 3-sphere) which is a critical point of this functional. Infinitely many of these exist, and one method of production (due to Ulrich Pinkall) involves considering the preimage of certain curves on the 2-sphere under the Hopf map. |
| Categories | Mathematical Illustration, Surfaces, Low Dimensional Topology, Geometric Topology |