Recently I saw a description of spherical fractals in a blog post by Samuel Monnier.

These Julia-sets are constructed like ordinary Mandelbrots and Julias: first the argument is squared, but instead of adding a constant afterwards, a Möbius transformation is applied:

\(z = \frac{a z^2 + b}{c z^2 + d}\)

For the right choices of (complex) constants, plane-filling patterns appear.

There is an intimate connection between Möbius transformations and spherical geometry: if the plane is stereographically projected onto a sphere, a Möbius transformations corresponds to rotating and moving the sphere, and then project stereographically back to the plane (this is nicely visualized in this video).

This connection can be visualized graphically: if the plane-filling patterns are stereographically projected onto a sphere, they fit naturally on it. There are no discontinuities or voids, and no singularities near the poles.

Here I’ve used Fragmentarium to create some images of these plane-filling patterns, together with their stereographical projection onto a sphere. It was done by distance estimated ray marching, but in this case we could have used ordinary ray tracing, and calculated the exact intersections.

The Fragmentarium script can be found here.

Wow! These are really beautiful. In particular the second one. It should be possible to convert the colors into a height field on the sphere and explore those planets ;o).

And thank you for the shader.

Thanks Knighty! Actually the second one is a heightmap (although the effect is subtle). I experimented with it, and found it somewhat difficult to get results (and it is much slower!). But it can most likely be improved!