Julia fractal in Python

Introduction to julia set

In the context of complex dynamics, a topic of mathematics, the Julia set and the Fatou set are two complementary sets (Julia ‘laces’ and Fatou ‘dusts’) defined from a function. Informally, the Fatou set of the function consists of values with the property that all nearby values behave similarly under repeated iteration of the function, and the Julia set consists of values such that an arbitrarily small perturbation can cause drastic changes in the sequence of iterated function values. Thus the behavior of the function on the Fatou set is ‘regular’, while on the Julia set its behavior is ‘chaotic’. The Julia set of a function f is commonly denoted J(f), and the Fatou set is denoted F(f). These sets are named after the French mathematicians Gaston Julia and Pierre Fatou whose work began the study of complex dynamics during the early 20th century. [Source Wiki] The equation to generate Julia fractal is f_c(z) = z^2 + c

where c is a complex parameter. The Julia set for this system is the subset of the complex plane given by:

J(f_{c})=\left \{ z\in \mathbb{ C}:\forall n\in\mathbb{N},|f_{c}^{n}(z)|\leq 2 \right \}

So let’s now try to create one of the fractals in the above image. To do so we need the Pillow

module of Python which makes it easy to deal with images and stuff. To install pillow through pip type the following command in the command prompt.

pip install Pillow

Now using this library to create the fractal image.

Python3

# Python code for Julia Fractal
from PIL import Image
 
# driver function
if __name__ == "__main__":
  
    # setting the width, height and zoom 
    # of the image to be created
    w, h, zoom = 1920,1080,1
 
    # creating the new image in RGB mode
    bitmap = Image.new("RGB", (w, h), "white")

    # Allocating the storage for the image and
    # loading the pixel data.
    pix = bitmap.load()
   
    # setting up the variables according to 
    # the equation to  create the fractal
    cX, cY = -0.7, 0.27015
    moveX, moveY = 0.0, 0.0
    maxIter = 255
 
    for x in range(w):
        for y in range(h):
            zx = 1.5*(x - w/2)/(0.5*zoom*w) + moveX
            zy = 1.0*(y - h/2)/(0.5*zoom*h) + moveY
            i = maxIter
            while zx*zx + zy*zy < 4 and i > 1:
                tmp = zx*zx - zy*zy + cX
                zy,zx = 2.0*zx*zy + cY, tmp
                i -= 1

            # convert byte to RGB (3 bytes), kinda 
            # magic to get nice colors
            pix[x,y] = (i << 21) + (i << 10) + i*8

    # to display the created fractal
    bitmap.show()