The two simplest minimal cylinders with triangular boundaries are the Schwarz rPD and H surfaces. In both cases, the triangles are equilateral. In Alan Schoen’s R2 surface from 1970, the triangles are replaced by 45º-45º-90º triangles.

The four cylinders above constitute one-half of a translational fundamental piece of a genus 9 triply periodic minimal surface. It is fascinating to see how simple construction recipes sometimes lead to complicated topology.

The distance between the two triangles can be changed. As for the catenoid between two circles, there are usually two solutions. In one limit, we obtain noded planes:

In the other limit, the catenoidal nodes have grown. We see a tiling of the plane by 8-ended and 4-ended singly periodic Scherk surfaces.

Resources

Mathematica Notebook

PoVRay Sources