In 2018, Hao Chen discovered a new 2-parameter family of embedded triply periodic minimal surfaces of genus 3 that superficially shares many features with the Schwarz oD family.

The images above show translational fundamental pieces of a tD (green) and a tΔ (yellow) surface in the same box over a square. They share the same horizontal straight lines at the top and bottom and have likewise behaved planar symmetry curves in the vertical faces of the box. The divisors of the squared Gauss map on the quotient tori are similar, too:

For oD, we have a=b and therefore also vertical straight lines. I found it very surprising that there exists an additional less symmetric surface without the straight lines. The new parameter b is not arbitrary but is determined by a period condition. In case a+b=1/2, the surfaces live over a square and form a 1-parameter tΔ subfamily.

The animation above shows the oΔ family for a=.25. Half way through, when a=b=.25, it actually meets tD. However, this deformation of particular oD surfaces does not belong to the known and explicit 5-dimensional Meeks family but to the rather elusive family of non-Meeks surfaces to which also belong the Schwarz H-surfaces, the Gyroid and the Lidinoid.

When a is close to 0 or 1/2, the surfaces approach nodal limits. However, the corresponding Traizet balance configurations are degenerate, they also arise as limits of oD surfaces.

##### Resources

H. Chen, M. Weber: A new deformation family of Schwarz’ D surface

Mathematica Notebook

PoVRay Sources