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If the mean curvature of a surface vanishes, we call it a minimal surface. Minimal surfaces are candidates for surfaces with least area among all surfaces with a given boundary. Sometimes, they can be modelled as a soap film by dipping a wire (a.k.a. the boundary) into bubble solution.

Every image was rendered in L^{a}T_{e}X using the
packages `tikz` and `tikz-3dplot`.

Dip two close, parallel rings of wire into bubble solution and you obtain a soap film which has the shape of a Catenoid (the golden surface). Mathematically, there exists a second narrower Catenoid with the two rings as boundary (the silver surface). This one however is too instable to be realised as a single soap film.

Pull the rings apart and the two Catenoids approach each other until they touch. That is the largest distance two unit circles bounding a Catenoid can have. After that the soap film collapses into two flat discs. The neck pinch is modelled by mean curvature flow.

Next: Helicoid.