The Mandelbrot Set and Julia Sets

Scalings in the Mandelbrot Set

Hurwitz-Robucci scaling - Finding the Universal Function

Using this we can find g(ε) with a lot of combinatorics of the Taylor series, or we can recall the trigonometric identity cos(2x) = 2cos2(x) - 1 and note

g(ε) = 2cos(√ε)

satisfies g(4ε) = g(ε)2 - 2:

g(4ε) = 2cos(√(4ε)) = 2cos(2√ε) = 2(2cos2(√ε) - 1) = g(ε)2 - 2

Here is an anmiation illustrating the convergence of gn(ε) to g(ε).

Click the picture to animate.

Return to Hurwitz-Robucci scaling.