Planes, trains and Kummer Surfaces

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:

anim = Animate[
 ContourPlot3D[{(x^2 + y^2 + z^2 - 
 musq)^2 - ((3*musq - 1)/(3 - musq))*(1 - z - 
 sq2*x)*(1 - z + sq2*x)*(1 + z + sq2*y)*(1 + z - sq2*y) == 
 0}, {x, -5, 5}, {y, -5, 5}, {z, -5, 5}, 
 PerformanceGoal -> "Quality", BoxRatios -> 1, 
 PlotRange -> 1], {musq, 3.001, 1, 0.0002}];

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