How to animate the convergence of the Mandelbrot set in one interesting lattice of the complex plane

animate_mandelbrot_set.py

import numpy as np
import matplotlib.pyplot as plt
import matplotlib.animation as animation

x_start, y_start = -2, -1.5  # an interesting region starts here
width, height = 3, 3  # for 3 units up and right
density_per_unit = 250  # how many pixles per unit

# real and imaginary axis
re = np.linspace(x_start, x_start + width, width * density_per_unit )
im = np.linspace(y_start, y_start + height, height * density_per_unit)

fig = plt.figure(figsize=(10, 10))  # instantiate a figure to draw
ax = plt.axes()  # create an axes object

def animate(i):
    ax.clear()  # clear axes object
    ax.set_xticks([], [])  # clear x-axis ticks
    ax.set_yticks([], [])  # clear y-axis ticks
    
    X = np.empty((len(re), len(im)))  # re-initialize the array-like image
    threshold = round(1.15**(i + 1))  # calculate the current threshold
    
    # iterations for the current threshold
    for i in range(len(re)):
        for j in range(len(im)):
            X[i, j] = mandelbrot(re[i], im[j], threshold)
    
    # associate colors to the iterations with an iterpolation
    img = ax.imshow(X.T, interpolation="bicubic", cmap='magma')
    return [img]
 
anim = animation.FuncAnimation(fig, animate, frames=45, interval=120, blit=True)
anim.save('mandelbrot.gif',writer='imagemagick')