In his second paper about minimal surfaces from 1835, Heinrich Ferdinand Scherk summarizes his earlier findings from 1830 and gives equations for five new minimal surfaces, the first new ones since the catenoid and helicoid.
Equation 7 describes the doubly periodic Scherk surface in general form (the orthogonal case is equation 6). This is the first non-trivial deformation family of minimal surfaces.
Equation 9 is easily recognized as the associate family deformation of catenoid to helicoid, parametrized as screw motion invariant surfaces. These parametrizations are not conformal, and no complex analysis is involved. If only someone had realized that these surfaces share the same Gauß map, the discovery of the Enneper-Weierstraß representation could have happened decades earlier.
Equation 16 is a mystery to me, I couldn’t verify that it satisfies the minimal surface equation.
Equation 20, Scherk’s fourth surface, is also quite complicated, but one of the components of the implicitly given surface does satisfy the minimal surface equation.
the equation reads (slightly modernized)
To find its Enneper-Weierstraß representation and make a decent image, I looked at the level curve for x=π, which simplifies to
This turns out to be a symmetry curve of the surface, so its normal lies in the plane x=0, and the Schwarz-Björling formula can be used to find the Enneper-Weierstraß representation:
From here we can see that the surface is singly periodic with two annular and two helicoidal ends, and is also singular (at the points corresponding to 0 and infinity).
Above you can see one half of the surface, with (parts of) both helicoidal ends and one of the annular ends. The singular point is where the horizontal symmetry curve in the middle meets the intersection of the two helicoidal ends, which is a straight line on the surface. Rotating about it gives a fundamental piece; below are three copies of it.
For details, see the notebook under the resource below.
Amusingly, there is a simpler surface with the same type of ends that I accidentally discovered a while ago.
Finally, there is equation 30, giving the orthogonal case of Scherk’s singly periodic surface. Scherk does note some similarities to his doubly periodic surface.
Mathematica Notebook for Scherk IV