Playing with Infinity

After Squaring the Circle and using Blaschke products to squeeze finitely many disks conformally into a single disk, the restless mathematician wants to put even more disks into a disk. Infinitely many, that is. What we need for this is a conformal map that maps the disk infinitely often onto itself. We will cheat a little and employ the exponential function. It maps the left half-plane to the punctured unit disk, and because it is 2π periodic. Each point except the center of the unit disk has infinitely many preimages.

So, my first examples map the disk via a Möbius transformation to the left half plane (that is a bijection), and then via the exponential map back to the (punctured) disk. Missing one point won’t be a big deal – what is a single pixel? Pulling back the default spider web on the left replicates it infinitely often, but in a somewhat disappointing way: The circles of the original web become horocycles that are mapped periodically onto the circles. We just have an infinite hyperbolic wall paper, periodic with respect to a parabolic isometry. We can do slightly better by taking products of several such maps.

Below are products of three such Möbius-exponential maps,

and here the two images that use four factors. I have placed the parabolic fixed points symmetrically but played with the Möbius transformations a little.

Of course one can also use infinite Black products, as long as the zeroes of the factors converge rapidly enough to the boundary circle, and as long one is patient enough to evaluate the infinite products to sufficient accuracy.

In the left example, the zeroes follow a spiral, while in the second example, the zeroes alternate between two spirals that turn the opposite way. The zeroes correspond to the circular holes.