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 matrixM with integer entries and let k be the determinant ofM.

LetP be the parallelogram spanned by the columns ofM. Choose a set of column vectors r_i for i = 1, 2, 3, ..., k from the two columns ofM and from the interior ofP.

Pick a random starting vectorv then iterate

v =M^-1v +r_i

wherei 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 npimport matplotlib.pyplot as pltA = 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.

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