Scherk's doubly periodic surface and its deformations (Heinrich Ferdinand Scherk)

This classical example is the oldest complete, embedded, doubly periodic minimal surface. It was found by Scherk in 1835. The most symmetric example can be considered as two families of equally spaced, parallel, vertical halfplanes meeting orthogonally. In the region between the both halfplane families the surface looks like saddles over black arrays of a checkerboard. The following picture shows half a fundamental domain of this surface.



The other surfaces of Scherk's family can be considered as two families of halfplanes meeting at an angle between zero and pi/2. In this case the saddles lying over alternating regions of a plane tiled by rhombi. See the following pictures. If the angle between the halfplane families goes to zero, the Scherk family goes to the helicoid.



How do I get the complete fundamental domain of translation when using GRAPE?

The input file name of these surfaces is k_lsst.am.

Fist of all you have to choose the TASK: CONJUGATE, since this surface is the conjugate of the Scherk tower. Then you should load the status-file k_lsss_g.st to see the computed piece of the surface. This piece is bounded by four straight lines. Now carry out two 180-degree rotations around the both horizontal straight lines. Then you should read in the patch-status-file. After an additional 180-degree rotation around a vertical straight line you see the fundamental domain of translation of this surface. If you want to see a larger part of the surface, you can continue to rotate around vertical straight lines.


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