Julia set fractal generator created a Poincaré disk?

Question

This happened to me when I was playing around on this site

did I stumble upon a link between the Julia set and the geometry of a Poincaré disk? Does anyone know if there are documented occurrences of this already out there? I have left the webpage open in a tab so as not to lose any valuable data that could be pulled if needed.

Here's a screenshot:

Answer 1

That is very cool!

It looks to me like the software takes an input image then repeatedly computes inverse images of that input under the complex function $z^n+c$. The result should look like the Julia set of $z^n+c$. You can choose the point $c$ by clicking on the image. If you click near the center of the image, you generate something like the Julia set of $z^n$, which is a circle. The curves that you see approaching the circle are the images of the line segments bounding the input image under the $n^{\text{th}}$ root. Alas, I don't think that the image of a line under an $n^{\text{th}}$ root function is a circle; it's not even bounded. It still looks pretty cool!

I've got a similar page set up that computes the square root of an image obtained from a camera. Iteration can be achieved via video feedback by pointing the camera at the screen. You can add a $+c$ by shifting the camera. You can get some pretty nice look Julia sets.

Here's the web page and here's a screenshot:

Answer 2

Your picture seems to have a conformal tiling of the disc. It has a tree-like structure where each ring is has 5x the number of copies, each 5x smaller, with the nearest point to the center of each ring being 5x nearer the edge. But this tiling is not uniform or regular, because the center point is special with 5 neighbours, while the equivalent points in the other rings have 2 neighbours (one inwards, one outwards).

The Poincaré disc is a model of hyperbolic geometry in which geodesics (the equivalent of straight lines in Euclidean flat geometry) are circular arcs (or diameters) that intersect the containing circle at right angles. As Mark McClure states in his answer, the curved lines in your picture are not circular arcs, which means the tiles have some edges that are curved in hyperbolic space.

A similar tiling can be constructed of the Poincaré half-plane, this one has straight edges (the curves are circular arcs that intersect the bottom axis at right angles). However, the pentagons are not regular (having internal angles 120/120/60/120/60 degrees), and there is no special center point - I'm not sure what it would look like transformed to the Poincaré disc but it would probably not be symmetrical.