#mathematica

cn

I went to see if I've talked about elliptic curves before, and I actually talked about this exact issue here. But now I've leveled up in Mathematica so I can do the visuals better.

Oversimplified, an elliptic curve is an equation that looks like y^2 = x^3+ax+b for some integers a and b. And they have lots of interesting number theoretic properties if you look for solutions that are rational numbers, none of which I want to talk about right now.

If you graph these equations over the real numbers you get pictures like one of these two (left: y^2=x^3+7x+1; right: y^2=x^3-7x+1):

And we like to imagine points at infinity in each direction, so the picture on the left looks like one giant circle (going through infinity at the top/bottom), and the picture on the right looks like two disconnected circles. And the difference is basically whether we get one or three solutions to the cubic when we set y equal to zero.

But over the complex numbers they should all look the same. The problem is, graphs of complex functions are four-dimensional and that's hard to display. But in that four-dimensional space, we should get a torus: a circle moved through a circular path (which still includes the point at infinity).

But we can make three-dimensional graphs, and then we can let them vary over time. So here is an animation of y^2=x^3+7x+1, with time controlling the imaginary coordinate of the output:

And here is y^2=x^3-7x+1:

You can see the moment in the middle where they "snap" into the real-plane-only version.

Vertex normals in a Mathematica Manipulate[]