There are two triply periodic minimal surfaces of genus 3 which have centered equilateral triangles in parallel planes as boundary contours: For the H-surface, they form the top and bottom faces of a prism, for the rPD surfaces they are rotated into an antiprism.
There are higher genus versions of this: There is a genus 5 complementary rPD surface, and here we are looking at a genus 7 version that has in its nodal limit at the top a configuration of catenoidal necks that corresponds in the finite total curvature case to a Horgan surface of dihedral symmetry 3 (which does not, however, exist).
One of the surprising aspects of this 1-parameter family is how much you can pull them up, so to speak. Below are two rotated views of the surface at its maximal height.
After that, we approach a second, simpler nodal limit.
Resources
PoVRay Sources
Minimal Surfaces 