CC Spheres Explained, i.e. Math

I thought I would take a post to explain what the shapes I made for my friend are, or rather how you can form them. To remind you, below are the shapes of radius 1, 2, and 3. (Imagine the shape is reflected–the imagines below are just the top half, but the bottom half looks exactly the same.)

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You start with the discrete Heisenberg group. This is simply the matrix group \left(\begin{array}{ccc}1&x&y\\ 0&1&z\\ 0&0&1\end{array}\right), where x, y, and z range over all integers. So for each matrix you can kind of think of a point in space.

The next thing you do is pick a set of generator matrices for the group. A set of generator matrices is a set of matrices that you can multiply together in a bunch of different ways to generate every matrix in the group. It’s weird, but you don’t need many matrices to generate every matrix. So let’s pick the following generator matrices, because why not:

\left(\begin{array}{ccc}1&1&0\\ 0&1&0\\ 0&0&1\end{array}\right) \left(\begin{array}{ccc}1&0&1\\ 0&1&0\\ 0&0&1\end{array}\right) \left(\begin{array}{ccc}1&0&0\\ 0&1&1\\ 0&0&1\end{array}\right)

Okay, now we’ll make a graph. Imagine a point in space for every matrix in the Heisenberg group. We’re going to draw lines between specific points, and the lines are determined by how you “get” to those points using the generator matrices. So if you have matrix a and multiply it by a generator matrix to get matrix b then those two matrices (points in the graph) have a line between them. Do this infinitely – all the lines to get to all the points.

Now we have this huge graph! Last thing to do is this little exercise: Starting at the origin, take one billion edge-steps (i.e. move along the lines in the graph) in all possible directions. Then zoom out (in your mind) and look at the shapes that the end points create. There is it! There’s the shape!

Okay, so what does “radius” mean? Radius is the number of billion edge-steps you take. So to make a shape of radius 1, you take 1 billion edge-steps. For a shape of radius 2, take 2 billion edge-steps, etc.

That’s it. That’s the shape. You can imagine that by picking different generator matrices you’ll get different shapes. However, Moon tells me that no matter which generator matrices you pick you’ll always get similar shapes: you’ll always get a dimple in the middle, and a curvy top and bottom, and flat walls.


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