The Naming of Things

Studying minimal surfaces has become a bit like hunting for rare new species of plants or animals. Having a newly discovered specimen named after a person might be considered an honor to some, but a confusion to others, because it typically says more about the name giver than the actual specimen.

Let’s, for instance, consider the minimal surface above, which is part of a triply periodic surface. It has genus 5 (after identifying opposite sides by translations), and no name yet. In my book it comes with the code name (1,0|1,1). To explain this code, look at how the symmetry planes cut the piece above into eight pieces, and look just at the fron-top-left. There are, on the piece, four points where the surface normal is vertical: Two of them are on the left side, with normal pointing up and down, and two more in the middle of the picture above with normal pointing up at both points. So the 1 stands for up, the 0 for down, and the vertical bar separates the two boundary components.

You can see this also in the frieze pattern associated to this surface via the Weierstrass representation. The uppr contour repeats left-left-right-right turns, while the bottom just alternates left-right-left-right. Replace left by 1, right by 0, to get 110011001100… and 1010101010…, respectively, and take only the first two digits of each sequence, to get the name code. Below is a larger copy of a deformed version with the same code.

That seemed a clever thing to do, because there are at least four more such codes for genus 5 surfaces (one of them has been named Schoen’s Unnamed Surface 12 before…). Unfortunately, codes can be deceiving, because there is also this surface:

It follows the same up/down pattern of the surface normal and hence gets the same code, but the top edge bends differently. Neither the frieze pattern

nor larger chunks of deformed versions

allow to codify the difference. The mystery can be resolved by looking at what are called the divisors of the Gauss map, and use Abel’s theorem to distinguish them. But not today anymore.