In his 1867 Monograph Bestimmung einer speciellen Minimalfläche, Hermann Amandus Schwarz describes besides the D-surface three other triply periodic minimal surfaces, together with some of their deformations. One of them is the conjugate of the D-surface, named Primitive Surface or P-surface by Alan Schoen to indicate that its symmetries are those of the primitive cubical lattice.
The straight lines on the surface meet in threes at the branched points of the Gauss map, making it evident that the surface is a double branched cover over the vertices of a cube. The straight lines cut the surface into minimal quadrilaterals, the edges follow four non-planar edges of a regular octahedron.
The symmetry planes cut the surface into right angled minimal hexagons, eight of which constitute a translational fundamental domain. This allows to identify the hyperbolic structure of the underlying Riemann surface, using a tiling by regular right angled hexagons.
Resources
H.A. Schwarz: Bestimmung einer speciellen Minimalfläche, Eine von der Königlichen Akademie der Wissenschaften zu Berlin am 4. Juli 1867 gekrönte Preisschrift. Nebst einem Nachtrage und einem Anhange.
Mathematica Notebook
PoVRay Sources
Minimal Surfaces 