In his 1992 paper about doubly periodic minimal surfaces, Fusheng Wei also proves the existence of a 2-parameter family of triply periodic surfaces of genus 4 of which his doubly periodic surfaces are limits.

a=.1, b=.3

Below are two views of the same surface, only assembled differently to show the inside resp. the outside.

Besides the genus 2 limit, there are other limits as well, for instance vertical planes over a rhombic tiling, with the intersections resolved by singly periodic Scherk surfaces of different periodicity:

a=.1, b=.399

Below is an animation showing the deformation family for surfaces in a box with a square base:

Resources

Mathematica Notebook

PoVRay Sources

F.Wei: Some existence and uniqueness theorems for doubly
periodic minimal surfaces, Invent. Math 109 (1992), 113–136.

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