From H to P and beyond (H-3)

In March we had a look at the Schwarz H surface, and it is time to revisit it. We begin by turning it on its side, for comfort:

Hside

Then horizontal lines and vertical symmetry planes cut the surface still into simply connected pieces like this one

Octagon

The H-surfaces form a natural 1-parameter family with hexagonal symmetry. It turns out that in this representation one gains another parameter at the cost of  losing the hexagonal symmetry. This allows to deform the H-surface into new minimal surfaces, and the question arises what these look like. To get used to this view, below is yet another version of the classical H-surfaces near one of its two limits. 

HLimit

The new deformation allows to shift the catenoidal necks up and down, until they line up like so:

OPnodes

This surface is a member of the so-called orthorhombic deformation of the P-surface of Schwarz so that we can deform any H-surface into the P-surface, and from there into any other member of the 5-dimensional Meeks family.

This is remarkable because the H-surface does not belong to the Meeks family, but to another 5-dimensional family of triply periodic minimal surfaces that is much less understood. The final image is another extreme case of the newly deformed H-surfaces:

ExtremeH

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.