The simplest deformation of the Schwarz P surface deforms the cubical lattice into a primitive orthorhombic lattice, meaning the lattice has as a fundamental cell a rectilinear box.

The branched values of the Gauss map lie at the vertices of a (usually different) box. The coordinate planes and their lattice translates cut the surface into congruent minimal hexagons, each of which the image of the gray rectangle below that encodes the divisor of the the square of the Gauss map on the quotient torus under a 180º rotation about the z-axis.

The red crosses are the branched points of the Gauss map, they are mapped to the centers of the minimal hexagons and become centers of point reflections that leave the surface invariant (but exchanges inside and outside.

The divisor satisfies Re a + Re b =1/2. The case that Re a = Re b =1/4 corresponds to more symmetric P-surfaces, admitting an order 4 rotational symmetry and vertical straight lines. What other triply periodic minimal surfaces are there with an order 4 rotational symmetry? Below is an animation of the entire 1-parameter family with order 4 symmetry..

Limits of the general 2-parameter family include singly periodic Scherk surfaces and noded planes.

Resources

Mathematica Notebook

PoVRay Sources