The Hexagonal or H-surface is one of the four triply periodic minimal surfaces discussed by Hermann Amandus Schwarz in his Preisschrift from 1867. It comes in a natural 1-parameter family with no easily distinguishable representative.

 

Like his other examples, this surface has genus 3 when divided by its translational symmetries. It is in may ways analogous to his P-surface. For instance, it also solves a Plateau problem for two equilateral triangles in parallel planes.H-fund

Here the triangles are translates of each other, while for the P-surface they are rotated against each other by 180 degrees. The 1-parameter family that maintains all symmetries has two extreme cases: In one limit, one obtains parallel planes joined by catenoidal neck, in the other translation invariant Scherk surfaces with 6 ends.

There is, however, a dramatic difference: While the P-surface belongs to an explicit, 5-dimensional family of embedded triply periodic minimal surfaces where the 8 branched values of the Gauss map come in pairs of antipodal points (the Meeks family), this is not the case for the H-family, where these branched values are at the vertices of a triangular prism and its triangle centers. It has only recently been shown that this surface can be further deformed into surfaces that also belong to the Meeks family.

H-spider

Particularly attractive is the translation structure associated to G dh above. If you identify all parallel edges, you get a genus 3 surface with a single cone point of cone angle 10π.

Resources

H.A. Schwarz:  Bestimmung einer speciellen Minimalfläche, Eine von der Königlichen Akademie der Wissenschaften zu Berlin am 4. Juli 1867 gokrönte Preisschrift. Nebst einem Nachtrage und einem Anhange.

Mathematica notebooks 1, 2, 3

PoVRay sources

Related Surfaces
  • H. Chen’s tD family
  • The Lidinoid