A recipe for creating random fractals

Last week I gave an example of a randomly generated fractal and mentioned that it was “a particularly special case of a more general algorithm for generating similar fractals found in [1].”

Here’s the general pattern. First, create a non-singular matrix M with integer entries and let k be the determinant of M.

Let P be the parallelogram spanned by the columns of M. Choose a set of column vectors r_i for i = 1, 2, 3, …, k from the two columns of M and from the interior of P.

Pick a random starting vector v then iterate

v = M⁻¹ v + r_i

where i is chosen at random on each iterations.

Here’s an example suggested as an exercise in [2]. We start with

and look at the parallelogram spanned by the columns of M.

Then the algorithm described above is implemented in the following Python code.

import numpy as np
import matplotlib.pyplot as plt

A = np.linalg.inv(np.array([[2,  -1], [1, 2]]))
R = np.array([[2, -1, 1,  1,  0],
              [1,  2, 0, -1, -1]])

v = np.random.random(size=2)
for _ in range(100000):
    i = np.random.choice([0, 1, 2, 3, 4])
    v = np.dot(A, v) + R[:, i]
    plt.plot(v[0], v[1], 'bo', markersize=1)
plt.gca().set_aspect("equal")
plt.show()

This produces the following fractal.

[1] Darst, Palagallo, and Price. Fractal Tilings in the Plane. Mathematics Magazine [71]:1, 1998.

[2] Lorelei Koss, Fractal Rep-tiles and Geometric Series. Math Horizons. Vol 30, Issue 1, September 2022.