Mandelbrot Set for Generative Art

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Are you ready to dive into the mind-bending world of fractals? Buckle up, because we're about to explore the Mandelbrot set, a mathematical marvel that has been a source of fascination for mathematicians, artists, and programmers alike. Named after the mathematician Benoît B. Mandelbrot, this set is not only a cornerstone of complex numbers but also a fantastic tool for generating intricate and captivating art.

What is the Mandelbrot Set?

Imagine a world where you can zoom in forever, revealing more and more detail, all generated by a simple mathematical formula. Sounds like science fiction? Nope! Welcome to the Mandelbrot set. At its core, the Mandelbrot set is a collection of complex numbers that do not escape to infinity when iterated through a specific function.

In simpler terms, we start with a complex number ( c ). Then, we iterate the function ( f(z) = z^2 + c ) starting with ( z = 0 ). If the sequence of ( z ) values remains bounded (doesn't go off to infinity), then ( c ) is part of the Mandelbrot set.

Plotting the Mandelbrot Set

Alright, enough talk. Let's fire up some Python and see this bad boy in action. We'll use the matplotlib library to visualize our Mandelbrot set.

Step 1: Setting Up

First, let's install matplotlib if you haven't already:

pip install matplotlib

Now, let's import the necessary libraries and set up our canvas.

import matplotlib.pyplot as plt
import numpy as np

# Set up dimensions
width, height = 800, 800

# Generate a 2D grid of complex numbers
real = np.linspace(-2.0, 1.0, width)
imag = np.linspace(-1.5, 1.5, height)
X, Y = np.meshgrid(real, imag)
C = X + 1j * Y

Step 2: Iteration Function

Next, let's define our iteration function to determine which points belong to the Mandelbrot set.

def mandelbrot(c, max_iter):
    z = 0
    n = 0
    while abs(z) <= 2 and n < max_iter:
        z = z*z + c
        n += 1
    return n

Step 3: Creating the Fractal

We will now apply our iteration function to each point in our grid and store the results.

max_iter = 100
mandelbrot_set = np.zeros((width, height))

for i in range(width):
    for j in range(height):
        mandelbrot_set[i, j] = mandelbrot(C[i, j], max_iter)

Step 4: Visualizing

Finally, let's visualize our Mandelbrot set using matplotlib.

plt.imshow(mandelbrot_set.T, extent=[-2, 1, -1.5, 1.5], cmap="inferno")
plt.colorbar()
plt.title("Mandelbrot Set")
plt.show()

Creating Generative Art

Now that we've got the basics down, let's take our fractal game to the next level. By tweaking parameters, adding colors, and layering images, we can create truly unique and stunning pieces of generative art.

Adding Color

One way to spice things up is by coloring the points based on the number of iterations it took to determine their membership in the Mandelbrot set.

plt.imshow(mandelbrot_set.T, extent=[-2, 1, -1.5, 1.5], cmap="twilight_shifted")
plt.colorbar()
plt.title("Colorful Mandelbrot Set")
plt.show()

Zooming In

Want to explore deeper? Let's zoom into a specific region of the Mandelbrot set.

# Define zoom region
zoom_real = np.linspace(-0.75, -0.7, width)
zoom_imag = np.linspace(0.1, 0.15, height)
X_zoom, Y_zoom = np.meshgrid(zoom_real, zoom_imag)
C_zoom = X_zoom + 1j * Y_zoom

# Apply the iteration function
mandelbrot_zoom = np.zeros((width, height))

for i in range(width):
    for j in range(height):
        mandelbrot_zoom[i, j] = mandelbrot(C_zoom[i, j], max_iter)

# Visualize the zoomed region
plt.imshow(mandelbrot_zoom.T, extent=[-0.75, -0.7, 0.1, 0.15], cmap="magma")
plt.colorbar()
plt.title("Zoomed Mandelbrot Set")
plt.show()

Experimenting with Different Functions

Want more variety? Try iterating different functions or using different escape conditions. The possibilities are endless!

def mandelbrot_variant(c, max_iter):
    z = 0
    n = 0
    while abs(z) <= 2 and n < max_iter:
        z = z**3 + c  # Cubic function instead of quadratic
        n += 1
    return n

mandelbrot_var_set = np.zeros((width, height))

for i in range(width):
    for j in range(height):
        mandelbrot_var_set[i, j] = mandelbrot_variant(C[i, j], max_iter)

# Visualize the variant Mandelbrot set
plt.imshow(mandelbrot_var_set.T, extent=[-2, 1, -1.5, 1.5], cmap="plasma")
plt.colorbar()
plt.title("Mandelbrot Set Variant")
plt.show()

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