This post consists mainly of pictures relating to elliptic functions and the conformal map from a square to disk. Useful background can be found in Chamberlain Fung’s Analytical Methods for Squaring the Disc, which inspired the work here. See also the Wikipedia article Jacobi Elliptic Functions.

Figure 1 is a rendering of theJacobi elliptic function cn(z,m), with parameter m=1/2. It uses a domain coloring method, meaning that the color at a point z depends on the value of w = cn(z,m). Note that we have alternating diamond shapes, some with a bull’s-eye pattern, some without. If you tilt your head you can see it is a checkboard pattern.

The practical application of all this is that it gives a conformal mapping of the square (±1,±1) to the unit circle.