The Mandelbrot Set and Julia Sets

Scalings in the Mandelbrot Set

Hurwitz-Robucci scaling - the Robucci Constant

Finding the left-most zeros of f_n, n > 2, locates the centers, c_n, of the cardioids of the left-most n-cycle midget Mandelbrot sets along the real axis.

These accumulate at c = -2 as n -> infinity.

A natural question is this: does these accumulations follow a pattern?

Adam Robucci, D. Philip, K. Philip, and I investigated the ratios (c_n - c_n-1)/(c_n+1 - c_n) and found as n -> infinity, these ratios approach ...

4, exactly 4, 4 followed by as many 0s as you like.

Click here to see the first 50 ratios, in a new window.

Click here to see the next 50 ratios, in a new window.

If the ratio apporaches an integer limit, there must be a reason. Henry Hurwitz, David Peak, and I found the reason.

Return to Hurwitz-Robucci scaling.