After Jean Baptiste Meusnier proved in 1776 that the Helicoid is a minimal surface, the second one after Euler’s catenoid , there was silence for a while. It took until 1830, when Heinrich Ferdinand Scherk submitted a paper where clever choices of functions allowed to separate variables and find new explicit solutions to the minimal surface equation. His first new example has the simple equation

$e^z = \frac{\cos y}{\cos x}$

and it is refreshing to see that we can read a text that is almost 200 years old without difficulty.

From Scherk’s equation it is annoyingly difficult to make a good picture of the surface, because it is, alas, not a graph over the plane, but only over a the black squares of a checker board patter.

What helps is to first switch to the Enneper-Weierstraß representation, which can be done by computing and (locally) inverting the surface normal, using the inverse of the stereographic projection. This gives

$G(z) = z \qquad\text{and}\qquad dh =\frac{z}{z^4-1}\, dz \ .$

It gets even better for drawing when one switches to a parametrization by level curves using

$G(z) = \sqrt{\coth(z)} \qquad\text{and}\qquad dh = dz \ .$

The quotient surface by its translational symmetries is a 4-punctured sphere.

Scherk also showed the existence of a deformation where the orthogonal top and bottom ends make an arbitrary non-zero angle. When the angle converges to 0, one can see helicoids with alternating spin forming.

Hippolyte Lazard-Holly and William Meeks proved in 2000 that these are the only complete, properly embedded doubly periodic minimal surfaces whose quotient have genus 0.

Resources:

H.F. Scherk: De propietatibus superficiei, quae hac continetur aequatione (1+q²)r− 2pqs + (1 + p²)t = 0 disquisitiones analyticae, Actis Soc. Jablon. nova 4 (1832), 204-80.

Note: I would love to see a copy of this.

H. F. Scherk, Bemerkungen über die kleinste Fläche innerhalb gegebener Grenzen, J. r. angew. Math. 13 (1835), 185-208.

H. Lazard-Holly and W. Meeks: Classification of doubly-periodic minimal surfaces of genus zero, Invent. math. 143 (2001), 1–27.

Mathematica Notebook for orthogonal case. PoVRay sources.

Mathematica Notebook for orthogonal case, polar coordinates. PoVRay sources.

Mathematica Notebook for general case. PoVRay sources.