The Mandelbrot Set and Julia Sets

Scalings in the Mandelbrot Set

Hurwitz-Robucci scaling - the Robucci Constant

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

These accumulate at c = -2 as n → ∞.

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

Adam Robucci, D. Philip, K. Philip, and I investigated the ratios (cn - cn-1)/(cn+1 - cn) and found as n → ∞, 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.