## 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:

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

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.

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

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:

## The Translation Invariant Costa Surface

Out of the flurry of minimal surfaces that was inspired by the Costa surface, a particularly fundamental new surface is the Translation Invariant Costa Surface, discovered by Michael Callahan, David Hoffman, and Bill Meeks around 1989.

Like Riemann’s minimal surface, its ends are asymptotic to horizontal planes, but it is invariant under a purely vertical translation, and the connections between consecutive planes are  borrowed from the Costa surface. Surprisingly, in a few ways this surface is even simpler than Costa’s surface. To see this, let’s look at a quarter of a translational fundamental piece from the top:

It is bounded by curves that lie in reflectional symmetry planes, and cut off with an almost perfect quarter circular arc. Hence the conjugate minimal surface will have an infinite polygonal contour, like so:

It is not too hard to solve the Plateau problem for such contours, and adjust the edge length parameter so that the conjugate piece is the one used for the Translation Invariant Costa Surface. It is also possible to argue that the Plateau solution is embedded, and conclude the same for the Translation Invariant Costa Surface. All this is not so easy for the Costa surface itself.

Above is a variation with one handle added at each layer. Surprisingly, the corresponding finite surface does not exist. One can add deliberately more handles. Below is a rather complicated version that I called CHM(2,3), with a wood texture rendered in PoVRay in 1999, when I had figured out how to export Mathematica generated surface data to PoVRay.