# Branched Cover from (4 4 3/2) Schwarz Triangle

A Schwarz triangle is a spherical triangle that can be used to generate a tiling of a branched covering of the sphere by repeatedly reflecting this triangle across its edges. Sometimes we get an actual tiling of the sphere, but in general we get a branched covering, because the same point can lie in the interior of several triangles, and there may be branch points at the corners of the triangles.

# Pentagon-Decagon Branched Covering

Two regular pentagons and a regular decagon fit snugly at a point: their interior angles sum to 360°. Despite this, you cannot tile the plane with regular pentagons and decagons. However, there is a branched covering of the plane tiled with pentagons and decagons, which map to regular pentagons and decagons on the plane. Here Greg Egan has drawn a portion of this branched covering.

# Weierstrass Elliptic Function

The Weierstrass elliptic function is built up as a sum of terms, one for each point in a lattice in the complex plane. Each term has a pole at one lattice point. The picture here shows the very first term, namely $1/z^2$. That’s why it’s bright in the middle and the colors go twice around the color wheel as you go around. If you continue reading, you’ll see a movie made by David Chudzicki where further terms are added one at a time!

# Atomic Singular Inner Function

This picture, drawn by Elias Wegert, uses colors to show the phase $f/|f|$ of the complex function

$$f(z) = \prod_{k=1}^5 \exp\left(\frac{z+\omega^k}{z-\omega^k}\right)$$

where $\omega$ is a nontrivial fifth root of unity. This is an ‘atomic singular inner function’. To understand what that means, it helps to start with some complex analysis.