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. The Jacobi elliptic function cn(z,m), using the domain coloring method.
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.
Figure 2. Annotations show a quarter period and a full period, along with some poles and zeroes. A green diamond at the upper right shows a diamond with a bull’s-eye.Figure 3. Domain coloring of the complex plane. The outer green circle is the unit circle. Inside it there are another nine concentric blue circles. The black dot at the center is the origin.Figure 4. The function cn(z,1/2) rotated (by 45 degrees) and scaled so that the image of the unit circle is aligned with the square with corners (±1,±1).Figure 5: Closeup of two sub-tiles. Together, these map rectangle to the entire complex plane. Left sub-tile maps to the interior of the unit disk, right sub-tile maps to the exterior of the unit disk.Figure 6: a single period of this function. Note that colors and patterns match on the left and right boundaries. Same for the top and bottom boundaries. These can be identified (glued together) to make a torus. Although this is an ideal flat torus with no curvature or distortion, a 3D version (with curvature and distortion) can be viewed at https://codepen.io/brainjam/full/daoWNo. We saw in Figure 5 that two of the sub-tiles mapped to the entire complex plane. So the four sub-tiles that comprise a period tile will cover the complex plane twice. “So, our function is a branched double cover of the sphere by the torus, which has four branch points.” (see http://math.ucr.edu/home/baez/week229.html)
The practical application of all this is that it gives a conformal mapping of the square (±1,±1) to the unit circle.
Figure 7: The 2×2 square, centered at the origin.Figure 8: the square, conformally mapped to the unit disk.Figure 9: a closup of the positive quadrant of the disk.Figure 10: An animation showing how half a torus can form a single cover for the Riemann sphere. This can be extended (just add the mirror image) to the full torus forming a double cover of the sphere, with the four branch points located at the ends of the two slits. Animation by Jim Belk, from https://math.stackexchange.com/a/1352275/1257.
Complex Dynamics 