https://arxiv.org/pdf/1702.04862.pdf

# Willmore

Steve Trettel

Medium | Mathematica Notebook |
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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 |