When you take two non-parallel planes, they will intersect in a line. The singly periodic Scherk surfaces are the only minimal way to “desingularize” this, in the sense that they are the only known minimal surfaces asymptotic to these two planes. To show this is one of the many famous open problems about minimal surfaces.
The situation gets vastly more complicated with three planes. Nobody has yet succeeded in constructing a minimal surface that is asymptotic to the three coordinate planes. That is another open problem. A case where we do know something is that of three (or more) vertical planes. Martin Traizet has shown in 1994 that in case the planes are reasonably general one can wiggle them a little bit and desingularize them by gluing in singly periodic Scherk surfaces. The concrete and very symmetric example above was known before that.
The only requirement on the Scherk surfaces is that they have the same translational period and share a horizontal reflectional symmetry plane to ground them. But nothing prevents us from shifting one of the Scherk surfaces by a half-period, like up above. To make the image, I assumed another reflectional symmetry at a vertical plane (roughly parallel to the screen). This still left me with a 1-parameter family, whose existence is truly only guaranteed near the limit that looks like three Scherk surfaces (with one of them shifted). But nothing keeps us from looking at the other surfaces in this family.
Above I have turned it around so that one can appreciate the handles better. What emerges becomes clear when one pushed the parameter further:
A singly periodic Costa surface! There is a similar one constructed by Bastista and Martín where the Costa-necks are rotated by 45 degrees. It then loses its reflectional symmetries but gains straight lines.
The two psychedelic designs up above arise from their simplistic ancestors we looked at last time by cutting off corners. These are still two conformal annuli that also satisfy a slightly complicated condition on the lengths of their edges, which makes them responsible for a variation of the Diamond surface:
If you cut either of the psychedelic shapes into quarters, using a vertical and a horizontal cut, you get four right angled octagons, with some right angles being exterior angles. Similarly, the marked symmetry lines on the surface up above cut the surface into eight right angled curved octagons, that correspond to the psychedelic octagons via a conformal and harmonic map.
There is a 1-parameter family of such critters. Above and below are larger portions of extreme cases that also show how the surface repeats.
You can see in the image above pieces of the doubly periodic Karcher-Scherk surface reappearing. No surprise, its psychedelic polygons also arise by cutting corners in the polygons corresponding to the original Scherk surface.
Everything simple reappears.
Last week I explained a really complicated way to get from Scherk’s doubly periodic minimal surface to the helicoid, through a family of Schwarz Diamond surfaces. As was known already to Scherk, this can be done much easier, namely by “shearing” the standard Scherk surface above. I put apostrophes because a simple Euclidean shearing isn’t enough to keep the surface minimal.
Bill Meeks and Hippolyte Lazard-Holly have shown that these are the only embedded doubly periodic minimal surfaces of genus 0 (after taking the quotient by their translational periods). Things get tricky for larger genus.
First of all one needs to distinguish whether the parallel half planes “on top” are parallel to the ones “at the bottom” or not. Today we stick with the case that they are not parallel, and are in fact orthogonal. Then there is just one such surface of genus 1 (I am pretty sure, but I think nobody has written a proof). This was first constructed by Hermann Karcher. It’s pretty clear (and provable) that one can continue like this, creating doubly periodic surfaces with more handles, like the genus 2 example below.
It would be a nice theorem if they all would be unique. But I don’t think so. Below is a picture of a genus 3 surface where the handles are arranged differently.
Proving that this really exists won’t be easy, but interesting, because it would allow one to speculate what will happen if one can shear this surface like the original Scherk surface.
This innocent minimal surface, which can be obtained from Heinrich Scherk’s traditional surface by adding two wings and bending them towards each other, poses interesting challenges when printed (vertically, i.e. rotated by 90 degrees) in clay. First of all, there are three horizontal cross sections which look like branches of hyperbolas (but aren’t, not even for the original Scherk surface, in contrast what Wikipedia currently claims).
When printing this layer by layer, the nozzle has to move from branch to branch, and as the printer can’t stop printing while it skips across, it leaves hairy artifacts.
They clearly have their own charme.
Another problem arises from the saddle points that are printed without support. This leads to other imperfections and sometimes structural complications that might take away from the elegance of the original surface but contribute to wild interior landscapes.
Watching the printer work for two hours is dramatic, because failure in the form of collapsing walls can happen any minute.