Here’s a short blog post for the holiday season, inspired by this article from Wolfram MathWorld. The topic is Kummer Surfaces, which are a particular family of algebraic varieties in 3-dimensional space. They make beautiful mathematical pictures, like these from their wikipedia page:

A Kummer surface is the points in space where a particular equation is satisfied. One way to describe them is as the zero-sets of equations like:

.

The variables are coordinates in 3-dimensional space, and and are two parameters, related by the equation . As we change the value of the parameter, the equation changes, and its zero set changes too.

What does the Kummer Surface look like as the parameter changes?

When the parameter , the non-linearity of the Kummer surface disappears, the surface degenerates to a union of four planes.

When the parameter is close to 3, we’re between planes and Kummer surfaces:

And for , we see the 16 singular points surrounding five almost-tetrahedra, in the center. A zoomed in version is in my other blog post that featured Kummer Surfaces.

Ok, I can see “planes” and “Kummer surface”, but what about “trains”? Well, I guess you say that when a parameter is changing, often something is being trained. Though, er, not here.

This equation is not for a Kummer surface, but it’s not so dissimilar either. It came up recently in one of my research projects:

P.S. The code (language=Mathematica) that I used to make the video is here: