This surface of genus 5 is one of many that has the symmetries of a box and comes in a 2-parameter family. The divisor of the squared Gauss map is shown below, we need a+b+c=d+1/2.

The special case that b=d and a+c=1/2 has additional horizontal straight lines and was first discussed by Valério Batista in his thesis.

In this case, the family limits at the Callahan-Hoffman-Meeks surface and one end, and doubly periodic Scherk surfaces stitched with catenoidal necks at the other end. This is the simplest example I know of what I have been calling the stitching constructions that uses and end-to-end gluing of Scherk surfaces while simultaneously employing catenoidal necks, so this would be a good starting point to study this sort of limit.

The general family has another interesting limit, namely the doubly periodic Callahan-Hoffman-Meeks surface. Below is an animation showing the family for fixed τ=0.8i.

Here is a new challenge for fans of Costa-surfaces: Can one make a triply periodic surface like the P-surface, but with Costa-necks towards the faces of a cube?

Resources

V. Batista: A Family of Triply Periodic Costa Surfaces, Pacific Journal of Math. 212 (2003), 347-370.

Mathematica Notebook and CDF

PoVRay Sources