The Mandelbrot Set

Question

The Mandlelbrot set is the set of all points in the complex plane for which, starting with $z_0 = 0$, the iteration $z_{n+1}=z_n^2 + c$ remains bounded. It is the case that if $|z_n|$ ever becomes larger than 2, then the sequence diverges (i.e. is not bounded).

The Mandelbrot set may be visualised as an image in which the $(x, y)$ coordinates of each pixel are the real and imaginary parts of $c = x + iy$, and the pixel is coloured black or white according to whether $c$ is in the set or not.

Write a program to display the Mandelbrot set in the region of the complex plane bounded by $-3 \le x \le 1, -2 \le y \le 2$, by displaying a space if a point is in the set or an asterisk if it isn't.

Hints:

• Map the region of interest to an array 40 rows of 80-character strings, and inspect each point in this array:

cx, cy = -1., 0.
xsize, ysize = 80, 40
for j in range(ysize):
    for i in range(xsize):
        x = cx + 4. * (i - xsize / 2.) / xsize
        y = cy + 4. * (j - ysize / 2.) / ysize
        ...

• Either use Python's builtin support for complex numbers, _e.g._:

z = 3. + 8.5j

or handle the real and imaginary parts as separate floats.

• When iterating, in addition to checking if $|z_n| \ge 2$, keep track of the number of iterations performed and bail (assume $c$ is in the set) if this number exceeds some maximum (say, 1000).

(Extra credit). Extend your answer to the previous exercise by generating a 500 x 500 pixel greyscale image of the Mandelbrot set. The shade of grey a pixel is coloured should reflect the number of iterations required to determine that the corresponding point is not in the set. Use PIL, the Python Imaging Library as follows:

from PIL import Image

im = Image.new('RGB', (xsize, ysize))
...
im.putpixel((i, j), (ired, igreen, iblue))
...
im.save('mandelbrot.png', 'PNG')

where a pixel is placed at $(i, j)$ and coloured with red, green and blue components ired, igreen, iblue (each an integer from 0 to 255, such that (0,0,255) is pure blue, (255, 255, 255) is white, (0, 0, 0) is black, etc).

Solution

# mandelbrot1.py

max_it = 1000
cx, cy = -1., 0.
xsize, ysize = 80, 40

for j in range(ysize):
    row = []
    for i in range(xsize):
        x, y = (cx + 4. * (i - xsize / 2.) / xsize,
                cy + 4. * (j - ysize / 2.) / ysize)

        a, b = (0., 0.)
        it = 0
        while (a**2 + b**2<= 4.0 and it < max_it):
            a, b = a**2 - b**2 + x, 2*a*b + y
            it += 1
        if it == max_it:
            char = ' '
        else:
            char = '*'
        row.append(char)
    print(''.join(row))

# mandelbrot2.py

from PIL import Image

max_it = 1000
cx, cy = -1., 0.
xsize, ysize = 500, 500

im = Image.new('RGB', (xsize, ysize))
for i in range(xsize):
    for j in range(ysize):
        c =  complex(cx + 4. * (i - xsize / 2.) / xsize,
                     cy + 4. * (j - ysize / 2.) / ysize)

        z = 0. + 0.j
        it = 0
        while (abs(z) <= 2.0 and it < max_it):
            z = z**2 + c
            it += 1
        if it == max_it:
            col = 0
        else:
            col = it * 20 % 255
        im.putpixel((i, j), (col, col, col))

im.save('mandelbrot.png', 'PNG')