Alan Schoen’s FR-D surface extends handles towards the faces of a rhombic dodecahedron (or, the edges of a cube) and hence has genus 6:

Schoen calls the two skeletal graphs the F-graph (the graph dual to the tiling of space by rhombic dodecahedra, or the graph that, in Schoen’s description, connects nearest vertices in face-centered cubic lattice) and the RD-graph, consisting of the edges of the rhombic dodecahedra tiling. Below is a view emphasizing the RD-perspective.

The conjugate surface was known, like the conjugate of the I-WP surface, to Berthold Steßmann. It solves the Plateau problem for a quadrilateral in a box of dimensions 1\times1\times\sqrt2:

The dimensions of the box are chosen so that the quadrilateral has angles of 90º,90º,60º,45º.

I neither know an algebraic equation for this surface, nor a simple polyhedral approximation. To find either will be an amusing exercise.


Mathematica Notebook and CDF

PoVRay Sources