The Mandelbrot Set and Julia Sets

Scalings in the Mandelbrot Set

Hurwitz-Robucci scaling - Locating Midgets

For large n,

g_n(e) = 0 implies 0 = g(e) = 2cos(sqrt(e))

So e = ((2j+1)pi/2)².

Then f_n(c_n,j) = 0 when

c_n,j = -2 + ((2j+1)²pi²)/(4r_n) = -2 + (6(2j+1)²pi²)/(4ⁿ⁺¹)

So we have

(c_n - c_n-1)/(c_n+1 - c_n) -> 4 as n -> infinity

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.