The Mandelbrot Set and Julia Sets

The Mandelbrot Set - Higher-Order Versions

For any integer n > 2 we can define the Mandelbrot set for f(z) = zⁿ + c.

This function has just one critical point, z = 0, and so the algorithm to generate the Mandelbrot set is the same as that for z² + c. That is,

start with z₀=0 and generate z_n by z_n = f(z_n-1).

If the z_n run away to infinity, c does not belong to the Mandelbrot set; if the z_n do not run away to infinity, c does belong to the Mandelbrot set .

For functions with more than one critical point, the problem is more subtle.

Here are some examples.

z3 + c	z4 + c	z5 + c
z6 + c	z7 + c	z8 + c
z10 + c	z15 + c	z20 + c

The pattern should be clear: the central component of the Mandelbrot set for zⁿ + c has n - 1 cusps.

In addition, the general extent of the Mandelbrot set decorations appears to be getting smaller.

This is correct: as n -> infinity, the Mandelbrot set for zⁿ + c approaches the unit disc.

Return to The Mandelbrot Set.