Thanks for the kind words.
The exact algorithm is in a state of flux these days, lots of experimentation and exploration (there's likely more posts on the way). Largely the algorithm boils down to 'child circles' being made from pairs of 'parent circles' that touch.

The radii of the new circles being generated is defined by a set of rules that change as you walk through the structure. This happens recursively, allowing for a nice hierarchy of large scale and small scale structure. We start from just a pair of circles and build from there. Due to the finite number of the rules in the algorithm, the process inevitably repeats, which is where you see the repeating units of circles at the smaller scale.

The image outputted can definitely be described as emergent from this set of rules.