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

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