The fourth type of triply periodic surfaces in Hermann Amandus Schwarz Preisschrift solves a Plateau problem for a hexagonal contour that can be obtained by following six edges of a box that belong to just two faces. Alan Schoen named them CLP-surfaces, because the skeletal graphs can be described as crossed layers of parallels.

Rotating about the edges creates a triply periodic surface that resembles an array of singly periodic Scherk surfaces. The shape of the Plateau contour depends on 2 parameters (disregarding scaling), and Schwarz provides us with the general solution.

The conjugate surfaces are also triply periodic and of genus 3. They usually lack  the horizontal straight lines but have vertical symmetry planes instead.

The branched values of the Gauß map lie on the equator of the sphere, at the vertices of two rectangles whose edges are parallel to the x- and y-axis. There are more symmetric cases where the original Plateau contour uses a box over a square. In this case, the two rectangles of the branched values of the Gauß map are congruent but rotated by 90 degrees.  Then, a conjugate surface will be congruent to another CLP surface with these symmetries. This is the reason why the conjugates of CLP are also called CLP, causing for confusion. Among these cases one is particularly interesting, namely when both rectangles are squares, i.e. the branched values lie at the vertices of a regular octagon.

In this case, the surface (shown above) is congruent to its conjugate. To convince oneself that this might be true, it helps to locate the horizontal straight lines on the right surface that are so evident on the left.

Alan Schoen also remarks in his NASA report that “CLP does not appear to have been analyzed”. All other examples of Schwarz have been the cause of surprises in the past. I suspect that CLP will also have something in store for us.

##### Resources

H.A. Schwarz:  Bestimmung einer speciellen Minimalfläche, Eine von der Königlichen Akademie der Wissenschaften zu Berlin am 4. Juli 1867 gokrönte Preisschrift. Nebst einem Nachtrage und einem Anhange.

Mathematica Notebook

PoVRay Sources