Mandelbrot at higher powersWritten by Paul BourkeFebruary 2004
The traditional Mandelbrot is created by considering the behaviour of the series z_{n+1} = z_{n}^{2} + 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 = 1z_{n+1} = z_{n}^{1} + z_{o} This is of course hardly very interesting, nor fractal. M = 2 z_{n+1} = z_{n}^{2} + z_{o} The traditional Mandelbrot, M = 2. M = 3 z_{n+1} = z_{n}^{3} + z_{o} M = 4 z_{n+1} = z_{n}^{4} + 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_{n}^{5} + z_{o} M = 6 z_{n+1} = z_{n}^{6} + z_{o} M = 7 z_{n+1} = z_{n}^{7} + z_{o} M = 8 z_{n+1} = z_{n}^{8} + z_{o} M = 10 z_{n+1} = z_{n}^{10} + z_{o}
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} 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} |