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Mandelbrot Fractal Set visualization in Python

  • Last Updated : 03 Oct, 2018

Fractal:
A fractal is a curve or geometrical figure, each part of which has the same statistical character as the whole. They are useful in modeling structures (such as snowflakes) in which similar patterns recur at progressively smaller scales, and in describing partly random or chaotic phenomena such as crystal growth and galaxy formation.

In simpler words, a fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Driven by recursion, fractals are images of dynamic systems – the pictures of Chaos.
Geometrically, they exist in between our familiar dimensions. Fractal patterns are extremely familiar since nature is full of fractals. For instance: trees, rivers, coastlines, mountains, clouds, seashells, hurricanes, etc. Abstract fractals – such as the Mandelbrot Set – can be generated by a computer calculating a simple equation over and over.

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Mandelbrot set:
The Mandelbrot set is the set of complex numbers c for which the function  f_c(z) = z^2 + c does not diverge when iterated from z=0, i.e., for which the sequence  f_c(0), f_c(f_c(0)), etc., remains bounded in absolute value. In simple words, Mandelbrot set is a particular set of complex numbers which has a highly convoluted fractal boundary when plotted.



Installation of needed Python modules:

pip install pillow
pip install numpy

Code #1:




# Python code for Mandelbrot Fractal
  
# Import necessary libraries
from PIL import Image
from numpy import complex, array
import colorsys
  
# setting the width of the output image as 1024
WIDTH = 1024
  
# a function to return a tuple of colors
# as integer value of rgb
def rgb_conv(i):
    color = 255 * array(colorsys.hsv_to_rgb(i / 255.0, 1.0, 0.5))
    return tuple(color.astype(int))
  
# function defining a mandelbrot
def mandelbrot(x, y):
    c0 = complex(x, y)
    c = 0
    for i in range(1, 1000):
        if abs(c) > 2:
            return rgb_conv(i)
        c = c * c + c0
    return (0, 0, 0)
  
# creating the new image in RGB mode
img = Image.new('RGB', (WIDTH, int(WIDTH / 2)))
pixels = img.load()
  
for x in range(img.size[0]):
  
    # displaying the progress as percentage
    print("%.2f %%" % (x / WIDTH * 100.0)) 
    for y in range(img.size[1]):
        pixels[x, y] = mandelbrot((x - (0.75 * WIDTH)) / (WIDTH / 4),
                                      (y - (WIDTH / 4)) / (WIDTH / 4))
  
# to display the created fractal after 
# completing the given number of iterations
img.show()

Output:

Code #2:




# Mandelbrot fractal
# FB - 201003254
from PIL import Image
  
# drawing area
xa = -2.0
xb = 1.0
ya = -1.5
yb = 1.5
  
# max iterations allowed
maxIt = 255 
  
# image size
imgx = 512
imgy = 512
image = Image.new("RGB", (imgx, imgy))
  
for y in range(imgy):
    zy = y * (yb - ya) / (imgy - 1+ ya
    for x in range(imgx):
        zx = x * (xb - xa) / (imgx - 1+ xa
        z = zx + zy * 1j
        c = z
        for i in range(maxIt):
            if abs(z) > 2.0: break
            z = z * z + c
        image.putpixel((x, y), (i % 4 * 64, i % 8 * 32, i % 16 * 16))
  
image.show()

Output:




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