The Mandelbrot Set and Julia Sets

Scalings in the Mandelbrot Set

Hurwitz-Robucci scaling - Finding the Universal Function

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

g(e) = 2cos(sqrt(e))

satisfies g(4e) = g(e)² - 2.

g(4e) = 2cos(sqrt(4e)) = 2cos(2sqrt(e)) = 2(2cos²(sqrt(e)) - 1) = g(e)² - 2

Here is an anmiation illustrating the convergence of g_n(e) to g(e).

Click the picture to animate.

Return to Hurwitz-Robucci scaling.