Fibonacci and the Mandelbrot set

[claude]

the size-ratio of bulbs that are fibonacci neighbours do converge to phi^2 the higher the fibonacci period value gets?

Yep!  Great work with your conjecture, it's a pretty cool result

(Note that I ran the code for 24 bulbs, but later bulbs start to deviate and go totally wrong around 32 when the bulb size is smaller than the resolution of double precision - so it'd be worth checking with higher precision at some point...)

[lkmitch]

Mandelbrot himself came up with a model for the size of the disks.  It's something like:

d = sin(m/n * pi)/n^2,

where m/n is the fraction associated with the disk (1/2 for the largest one, 1/3 for the big one at the top, 2/3 for the big one at the bottom, etc.).  For the disks in the series 2/3, 3/5, 5/8, 8/13, etc., the fraction converges to 1/Phi ~ 0.6180339, so there's probably some way to get the size convergence from Mandelbrot's model.

Note, the model is only approximate; the deviations from the model seem the be fractal!