A basic implementation of Julia Set. For more: https://en.wikipedia.org/wiki/Julia_set

julia-set.py

import time
from PIL import Image

# Area of complex space to investigate
x1, x2, y1, y2 = -1.8, 1.8, -1.8, 1.8
c_real, c_imag = -0.62772, -0.42193


def calculate_z_serial_purepython(maxiter, zs, cs):
    """Calculate output list using Julia update rule"""
    output = [0] * len(zs)
    for i in range(len(zs)):
        n = 0
        z = zs[i]
        c = cs[i]
        while abs(z) < 2 and n < maxiter:
            z = z * z + c
            n += 1
        output[i] = n
    return output


def calc_pure_python(desired_width, max_iterations):
    """Create a list of complex coordinates (zs) and complex parameters (cs),
    build Julia set, and generate image with PIL"""
    x_step = (x2 - x1) / desired_width
    y_step = (y1 - y2) / desired_width
    x = []
    y = []
    ycoord = y2
    while ycoord > y1:
        y.append(ycoord)
        ycoord += y_step
    xcoord = x1
    while xcoord < x2:
        x.append(xcoord)
        xcoord += x_step

    zs = []
    cs = []
    for ycoord in y:
        for xcoord in x:
            zs.append(complex(xcoord, ycoord))
            cs.append(complex(c_real, c_imag))

    print("Length of x:", len(x))
    print("Total elements:", len(zs))
    start_time = time.time()
    output = calculate_z_serial_purepython(max_iterations, zs, cs)
    end_time = time.time()
    secs = end_time - start_time
    print(calculate_z_serial_purepython.__name__ + " took", secs, "seconds")
    assert sum(output) == 33219980

    # Generate image using PIL
    image = Image.new("RGB", (desired_width, desired_width))
    pixels = image.load()
    for i in range(desired_width):
        for j in range(desired_width):
            value = output[i * desired_width + j]
            color = int(value * 255 / max_iterations)
            pixels[j, i] = (color, color, color)  # grayscale
    image.save("julia_set.png")
    print("Image saved as julia_set.png")


if __name__ == "__main__":
    calc_pure_python(desired_width=1000, max_iterations=300)

Output

Length of x: 1000
Total elements: 1000000
calculate_z_serial_purepython took 2.905118227005005 seconds
Image saved as julia_set.png

Output julia_set.png