Thursday, 14 May 2020

Mandelbrot set

When I reviewed Matt Pearson's Generative Art, and plotted a Mandelbrot set made from random splodges, I went and looked at my own webpage about this structure.  I discovered a case of bit rot: the little Java applet I'd written many years ago had stopped working, as is the way with much old software.  Rather than rewrite it, I pottered around in Python, reproducing some of the plots.

The Mandelborot set is a highly complex fractal generated from a very simple equation. Consider the sequence \(z_0=0; z_{n+1} = z_n^2 + c\), where \(z\) and \(c\) are complex numbers.  For some values of \(c\), the sequence diverges to infinity; for other values it stays bounded.  The Mandelbrot set is all the values of \(c\) for which the sequence does not diverge.  Plotting these points in the complex plane (and plotting the points that do diverge in colours that represent how fast they diverge) gives the now well-known picture:

If \(z\) is raised to a different power, different sets are seen, still with complex shapes.

iterating \(z^4 + c\)
The power doesn't even need to be an integer.

iterating \(z^{2.5} + c\)
The discontinuities visible are due to the fractional power resulting in a logarithm in the solutoin, which is multi-values in the complex plane. The plot shows only the principal value.

We can make an animation of the shape of the set as this power \(k\) changes:

\(k = 1 .. 6\), step \(0.05\)

One thing I love about the Mandelbrot set is the sheer simplicity of the code needed to plot it:

import numpy  as np
import matplotlib.pyplot as plt

IMSIZE = 512 # image width/height 
ITER = 256

def mandelbrot(c, k=2): 
    # c = position, complex; # k = power, real
    z = c
    for i in range(1, ITER): 
        if abs(z) > 2: 
            return 4 + i % 16 #16 colours
        z = z ** k + c 
    return 0  

mandy = np.zeros([IMSIZE,IMSIZE]) 
for ix in range(IMSIZE):
    x = 4 * ix / IMSIZE - 2
    for iy in range(IMSIZE):
        y = 2 - 4 * iy / IMSIZE
        mandy[iy,ix] = mandelbrot(np.complex(x,y), k)
    plt.imshow(mandy, cmap='cubehelix')

And there's more.  But that's for another post.

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