Exploring The Julia Set Fractals With Python

Introduction

In this blog post, we'll be diving into the fascinating world of fractals, specifically the Julia set. The Julia set is a mathematical fractal that can be visually represented using complex number mathematics and recursive programming. We'll be using Python and the popular library, matplotlib, to visualize the Julia set fractals.

The Julia Set

The Julia set is described as the set of complex numbers, c, for which the function:

f(z) = z^2 + c

does not diverge when iterated from z = 0. This means that it stays confined within a certain range and doesn't tend toward infinity.

Visualization with Python and Matplotlib

To get started, let's import the necessary libraries and define some parameters for our visualization.

import numpy as np
import matplotlib.pyplot as plt

width, height = 800, 800
pixels = np.zeros((width, height))

Next, we'll write a function that will perform the necessary calculations on the complex number to determine whether or not it belongs to the Julia set.

def julia(c, max_iterations=1000):
    for y in range(height):
        for x in range(width):
            zx, zy = x * (3.0 / width) - 1.5, y * (3.0 / height) - 1.5
            z = zx + zy * 1j
            iterations = 0

            while (abs(z) < 2) and (iterations < max_iterations):
                z = z * z + c
                iterations += 1

            # Assign the number of iterations to the corresponding pixel
            pixels[y, x] = iterations

With our julia function defined, let's call it with a sample complex number, c, and visualize the resulting Julia set.

c = -0.7 + 0.27015j
julia(c)

plt.imshow(pixels, cmap='viridis')
plt.colorbar()
plt.show()

This code will generate an image of the Julia set fractal for the given complex constant c. You can experiment with different values of c to explore the various beautiful patterns and shapes that the Julia set creates.

Conclusion

In this blog post, we have learned about the Julia set fractal and how to visualize it using Python and the matplotlib library. By diving into the world of fractals, we have uncovered just one of the countless wonders that mathematics and programming have to offer. Try experimenting with different parameters and values for c to discover new patterns and intriguing shapes in the Julia set.