The H-surfaces of Hermann Amandus Schwarz form a natural 1-parameter family. There is a surprising deformation into a 2-dimensional family. To explain the construction, we turn an H-surface sideways, and cut it along vertical symmetry planes and horizontal lines into a minimal octagon that still has a normal symmetry about its center:

This octagon is the image of the shaded rectangle below that shows the divisor of the (squared) Gauss map. The points a, a’, b, b’ correspond to the vertices, where the Gauss map has its zeroes and poles.

They need to satisfy a+a’=1/2=b+b’, and there is a period condition that determines the rectangle height in terms of a and b. By changing the values of a and b, we can slide the inward and outward necks up and down. Let’s first move them so that they line up:

When a+b=1/2, they are perfectly aligned, and we have an additional horizontal symmetry plane. There is no period condition in this case. In fact, these surfaces are known as special surfaces in the oP deformation of the Schwarz P surface. Hence we found a way to deform an H surface into the P surface. This is remarkable, because the H surface does not belong to the explicit 5-dimensional Meeks family that contains most known examples of embedded triply periodic minimal surfaces of genus 3.

Above are two extreme cases of this new deformation. The left image arises when a and b are close: We obtain singly periodic Scherk surfaces over a rhombic pattern. The right image arises when b approaches .5, here we see doubly periodic KMR surfaces as limits.


Mathematica Notebook

PoVRay Sources

H. Chen and M. Weber: An orthorhombic deformation family of Schwarz’ H surfaces