The Mandelbrot Set and Julia Sets

Scalings in the Mandelbrot Set

Hurwitz-Robucci scaling - Locating Midgets

For large n,

gn(ε) = 0 implies 0 = g(ε) = 2cos(√ε)

So ε = ((2j+1)π/2)2.

Then fn(cn,j) = 0 when

cn,j = -2 + ((2j+1)2π2)/(4rn) = -2 + (6(2j+1)2π2)/(4n+1)

So we have

(cn - cn-1)/(cn+1 - cn) → 4 as n → ∞

for all j.

So not only do the left-most (j=0) Mandelbrot midgets scale this way, so do the next-to-the-leftmost (j=1), the next-to-the-next-to-the-leftmost (j=2), and so on.

Return to Hurwitz-Robucci scaling.