where the ratios in the Mandelbrot set come from

Robert Devaney has dome a lot of work with the Mandelbrot set, and here
he explains all the cool stuff that relates the set to ratios, Fibonacci
series, etc.
http://math.bu.edu/DYSYS/FRACGEOM2/FRACGEOM2.html
See especially "How to Count"

--
John Starrett
"We have nothing to fear but the scary stuff."
http://www-math.cudenver.edu/~jstarret/microtone.html