The helicoid was discovered by Jean Baptiste Meusnier in 1776. Minimality is easy to see: If a straight line on a surface a symmetry line, the mean curvature along this line is automatically zero, because rotating about the line changes the direction of the normal, thus  the sign of the shape operator, thus the sign of the mean curvature. For the helicoid, all its straight lines are symmetry lines, so its mean curvature vanishes everywhere.

The usual picture of a helicoid (left) is deceiving. From far away it looks more like a collection of horizontal planes with some sort of miracle happening in the middle.

That the helicoid is the only ruled minimal surface (besides the plane) is a bit more difficult. There are proofs that use only elementary differential geometry. More advanced approaches use the Björling formula or that the conjugate surface must be a surface of revolution (and the fact that the catenoid is the only minimal surface of revolution, which is easier to see).

Below is an animation showing the associate family from catenoid to helicoid, an isometric deformation. This deformation was first described by Heinrich Ferdinand Scherk around 1832, but not in the context of the associate family.


Here is the original reference. I haven’t seen it, if anybody has a scanned copy, please send.

Meusnier, J.B.:  Mémoire sur la courbure des surfaces. Mémoire des savants étrangers 10 (l1776), 477– 510.

PoVRay source for animation. Use clock range 0 to 1 and square size images.



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