This is a 1-parameter family of embedded triply periodic minimal surfaces of genus 4 with horizontal straight lines and vertical symmetry planes.

The horizontal straight lines form a square pattern with squares half the size of the squares formed by the vertical symmetry planes. Below is the divisor of the squared Gauss map on the quotient torus.

The image of this torus under \int^z G\, dh and \int^z 1/G\, dh are shown below; parallel edges are identified to give the tori:

For fixed parameter a, the lattice parameter τ is determined to solve the 1-dimensional period problem. The surfaces limit at the Costa and the doubly periodic Karcher-Scherk surface:

The Costa limit is the simplest noded limit I know where the nodes are not catenoids but Costa surfaces. It would be interesting to have a regeneration construction using Costa nodes à la Traizet.

Resources

D. Freese, M. Weber, A.T. Yerger, R. Yol: Two New Embedded Triply Periodic Minimal Surfaces of Genus 4

Mathematica Notebook

PoVRay Sources