The construction of the polygonal diamond surface via bent rhombi can be varied. If we take as the bent rhombus two adjacent faces of the regular octahedron instead of the tetrahedron and follow the same rule to extend the surface by 180 degree rotations of rhombi about edges, we first get a less crooked hexagon,
four of which can be assembled to a translational fundamental piece
of a triply periodic polyhedral surface
that approximates Schwarz’ so-called primitive surface.
In this case, the ribbon representation has a much simpler appearance than the rhombic image.
After all, apparent complexity is often only a matter of the presentation.
If you want to make quick paper models of either the diamond or the primitive surface, cut out lots of equilateral triangles, divide the edges into thirds, and bend the three triangles at the corners upwards. These smaller triangles serve as flaps that you glue to the front sides of the central hexagons inside the original triangles.