The Mandelbrot set

Colours every parameter \(c\) by how quickly the orbit of \(z_0 = 0\) under \(z \mapsto z^2 + c\) escapes. The black region, the Mandelbrot set, is the set of \(c\) whose orbit stays bounded – equivalently, whose Julia set is connected. Zooming into its boundary reveals ever finer detail, including small copies of the whole set.

import matplotlib.pyplot as plt

from mathematicskit.fractals_chaos import mandelbrot_set
from mathematicskit.fractals_chaos.visualizers.plots import plot_escape_time

The whole set

mandelbrot = mandelbrot_set(resolution=400, max_iter=200)
ax = plot_escape_time(mandelbrot)
ax.set_title("Mandelbrot set")

Text(0.5, 1.0, 'Mandelbrot set')

Zooming into the boundary

Near \(c \approx -1.77\) on the real axis, a small copy of the Mandelbrot set sits inside the boundary’s filaments.

fig, axes = plt.subplots(1, 2, figsize=(10, 4.5))
seahorse = mandelbrot_set(extent=(-0.80, -0.70, 0.05, 0.15), resolution=300, max_iter=400)
plot_escape_time(seahorse, ax=axes[0], cmap="twilight")
axes[0].set_title("Seahorse valley")
minibrot = mandelbrot_set(extent=(-1.80, -1.74, -0.03, 0.03), resolution=300, max_iter=400)
plot_escape_time(minibrot, ax=axes[1], cmap="twilight")
axes[1].set_title("A small copy near c = -1.77")
fig.tight_layout()

Membership check

\(c=-1\) and \(c=0.25\) lie in the set (bounded orbits); \(c = 0.3\) and \(c = 1\) do not.

for c in (-1.0, 0.25, 0.3, 1.0):
    z, steps = 0.0, 0
    while abs(z) <= 2.0 and steps < 1000:
        z, steps = z * z + c, steps + 1
    print(f"c = {c:5.2f}: {'bounded (in the set)' if abs(z) <= 2.0 else f'escapes after {steps} steps'}")

c = -1.00: bounded (in the set)
c =  0.25: bounded (in the set)
c =  0.30: escapes after 12 steps
c =  1.00: escapes after 3 steps