Barnsley’s fern: an iterated function system

Barnsley’s fern is the attractor of just four contractive affine maps, drawn with the chaos game: start anywhere, and at each step apply one of the maps chosen at random with Barnsley’s probabilities. Hutchinson’s theorem says the fern is the unique set equal to the union of its four images under the maps, which the second figure shows directly.

import matplotlib.pyplot as plt
import numpy as np

from mathematicskit.fractals_chaos import BarnsleyFern
from mathematicskit.fractals_chaos.visualizers.plots import plot_ifs_points

The fern from the chaos game

fern = BarnsleyFern()
points = fern.generate(60000, seed=0)
ax = plot_ifs_points(points, color="darkgreen")
ax.set_title("Barnsley fern (60 000 chaos-game points)")

for name, (A, _, p) in zip(["stem", "main copy", "left leaflet", "right leaflet"], fern.transforms):
    print(f"{name:14s} p = {p:.2f}, contraction |det A| = {abs(np.linalg.det(A)):.3f}")

stem           p = 0.01, contraction |det A| = 0.000
main copy      p = 0.85, contraction |det A| = 0.724
left leaflet   p = 0.07, contraction |det A| = 0.104
right leaflet  p = 0.07, contraction |det A| = 0.109

The fern is four smaller ferns

Applying each map to the whole fern gives four pieces – the stem, the fern minus its lowest leaflets, and the two lowest leaflets – whose union is the fern again.

fig, ax = plt.subplots(figsize=(5, 8))
colors = ["saddlebrown", "darkgreen", "royalblue", "darkorange"]
for (A, b, _), color in zip(fern.transforms, colors):
    image = points @ A.T + b
    ax.scatter(image[:, 0], image[:, 1], s=0.2, color=color, linewidths=0)
ax.set_aspect("equal")
ax.set_title("Images of the fern under its four maps")

Text(0.5, 1.0, 'Images of the fern under its four maps')