This is a 1-parameter family of triply periodic surfaces of genus 5 which has equilateral triangles in parallel planes, centered above each other, and rotated by 180º, as the genus 3 surfaces of the rPD family. Therefore I am cautiously naming these complementary rPD surfaces, consistent with Alan Schoen’s convention. In contrast to the rPD surfaces, these here do not have a horizontal normal symmetry line. It might well be that they appear somewhere in the literature.
The divisor of the cubed Gauss map has the following divisor on the quotient torus by the order 3 rotational symmetry:
There are two periods to close by adjusting the López-Ros factor and the parameter a. To establish existence and embeddedness, one can use Traizet regeneration with one of the limits that consists of a simple pattern of noded planes:
The other limit is more exciting, one obtains Costa-Hoffman-Meeks surfaces of order 3. Therefore one could also call this a triply periodic Costa-Hoffman-Meeks surface. This is left as an exercise to the reader…
Below is an animation showing the 1-parameter deformation.