Mandelbrot at higher powers

Written by Paul Bourke
February 2004

The traditional Mandelbrot is created by considering the behaviour of the series z_n+1 = z_n² + z_o for each position z_o on the complex plane. A more general equation might be z_n+1 = z_n^M + z_o. The resulting shapes are less frequently explored, a graphical exploration of M space is given below.

Integer powers M = 1

z_n+1 = z_n¹ + z_o
This is of course hardly very interesting, nor fractal.

M = 2

z_n+1 = z_n² + z_o
The traditional Mandelbrot, M = 2.

M = 3

z_n+1 = z_n³ + z_o

M = 4

z_n+1 = z_n⁴ + z_o
In general there are M-1 main lobes which result in M-1 degrees of rotational symmetry.

M = 5

z_n+1 = z_n⁵ + z_o

M = 6

z_n+1 = z_n⁶ + z_o

M = 7

z_n+1 = z_n⁷ + z_o

M = 8

z_n+1 = z_n⁸ + z_o

M = 10

z_n+1 = z_n¹⁰ + z_o
The shapes get increasingly circular looking, but the detail on zooming in remains fractal.

Non-Integer powers

M = 2.1

z_n+1 = z_n^2.1 + z_o

M = 2.3

z_n+1 = z_n^2.3 + z_o

M = 2.5

z_n+1 = z_n^2.5 + z_o
Real valued M tend to appear like transitions between the lower and higher integer powers with a split along the negative real axis.

M = 2.7

z_n+1 = z_n^2.7 + z_o

Negative powers

M = -1

z_n+1 = z_n^-1 + z_o

M = -2

z_n+1 = z_n^-2 + z_o

M = -3

z_n+1 = z_n^-3 + z_o

M = -10

z_n+1 = z_n^-10 + z_o

M = -2.3

z_n+1 = z_n^-2.3 + z_o

M = -2.5

z_n+1 = z_n^-2.5 + z_o