Schwarz Hexagonal Surface (H-1)

One of the early minimal surfaces I have neglected so far is the H-surface of Hermann Amandus Schwarz.H double

Think about it as the triangular catenoids. Two copies make a translational fundamental domain, i.e. the 10 boundary edges can be identified in pairs by Euclidean translations, thus making the surface triply periodic. As a quotient surface it has genus 3, which implies that the Gauss map has 8 branched points. They occur at the triangle vertices and midpoints of triangle edges. Thus the branched values lie at the north and south pole of the sphere, and at the vertices of two horizontal equilateral triangles in parallel planes. In particular, they are not antipodal, making these surfaces the earliest examples of triply periodic minimal surfaces that lie not in the 5-dimensional Meeks family.

H catenoid

Above is a larger portion of the H-surface with the triangle planes close to each other. In the limit we get parallel planes joined by tiny catenoidal necks. When we pull the planes apart, we get Scherk surfaces:

 

H Scherk

The spiderweb for this surface looks also pretty:

H spider

Among the crude polyhedral approximations, there is one that tiles the surface with regular hexagons. The valencies are 4 and 6, so the tiling is not platonic.

H poly

Next week we will look at one of its more surprising features.

 

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