click and drag to draw.

What is this?

Riemann’s mapping theorem states the following. Suppose $U$ is a non-empty, proper open subset of $ℂ$ and that $U$ is simply-connected, and let $z₀,z₁∈U$ . Then there exists a unique biholomorphism $ϕ\colon U→𝔻$ (where $𝔻$ is the unit disk), with the property that $ϕ(z₀)=0$ , and $ϕ(z₁)$ lands on the positive-real axis.

Thurston’s conjecture (also sometimes known as the discrete/finite Riemann mapping theorem) provides an insightful (and pretty!) construction/visualization of the Riemann mapping theorem, proceeding roughly as follows:

Given a (bounded) domain $U$ , approximately triangulate $U$ via a hexagonal circle-packing of radius $ϵ$ .
Carefully resize each circle to obtain a new packing with the same tangency structure that fits “maximally” within the unit disk; i.e., boundary circles end up tangent to the boundary of the disk. This results in a mapping from circles in the original domain $U$ to circles in $𝔻$ .
Map each triangle in $U$ (as defined by the centers of three mutually-tangent circles) to the corresponding triangle in $𝔻$ by a (uniquely-determined) affine map; gluing each piece together gives a map defined on the full triangulation.
Apply a (Möbius) disk automorphism to send the image of $z₀$ to $0$ , and rotate the image of $z₁$ onto the real axis.
As $ϵ$ tends to zero, the map constructed in this manner gradually tends to the Riemann map from $U$ to $𝔻$ .

What is this app?

This app is an interactive demonstration of Thurston’s conjecture. Here’s how it works:

There are two panes: the pane on the left depicts the original domain, $U$ , and the pane on the right depicts the target domain, $𝔻$ .
Start by specifying $U$ using the “set domain” tools, and use the “change zero/positive-real anchor” tools to specify $z₀$ and $z₁$ .
Once this is done, the circle-packings and triangulations (of both the original domain and the unit disk) will be automatically computed and displayed. Use the “resolution” slider to adjust $ϵ$ , the size of circles in the original packing. Hover over circles on either side to highlight corresponding circles on the other side.
Use the “add drawings” tool to mark points and curves within $U$ (in the left pane); their images in $𝔻$ will be automatically computed and rendered (in the right pane).