Fractals from #Numberphile/Chaos Theory

fractals.py

import numpy as np
import matplotlib.pyplot as plt


def sierpinski_triangle():
    """Generates a Sierpinski triangle.

    Given 3 anchor points and a trace point, the
    next trace point is half-way between its current
    location and a randomly chosen anchor.
    """
    anchors = np.array([[0, 0], [1, 0], [0.5, 0.5]])
    t = np.array([0.2, 0.2])
    pts = [t]
    ids = np.random.choice(len(anchors), size=10000)
    for idx in ids:
        pts.append((pts[-1] + anchors[idx]) * 0.5)
    pts = np.stack(pts)
    plt.scatter(pts[:, 0], pts[:, 1], c="b", s=1)
    plt.scatter(anchors[:, 0], anchors[:, 1], c="r", s=1)
    plt.show()


def barnsley_fern():
    """Generates a Barnsley fern.

    Randomly chooses one of four affine transformations and
    applies it to the current location of the trace point
    to advance to the next trace point.
    """
    f1 = np.array([[0, 0, 0], [0, 0.16, 0], [0, 0, 1]])
    f2 = np.array([[0.85, 0.04, 0], [-0.04, 0.85, 1.6], [0, 0, 1]])
    f3 = np.array([[0.2, -0.26, 0], [0.23, 0.22, 1.6], [0, 0, 1]])
    f4 = np.array([[-0.15, 0.28, 0], [0.26, 0.24, 0.44], [0, 0, 1]])
    fs = [f1, f2, f3, f4]
    ps = np.array([0.01, 0.85, 0.7, 0.7])
    ids = np.random.choice(len(fs), size=1000000, p=ps / ps.sum())
    pts = [np.array([0.0, 0.0, 1.0])]
    for idx in ids:
        pts.append(fs[idx] @ pts[-1])
    pts = np.stack(pts)
    plt.gca().set_aspect("equal")
    plt.scatter(pts[:, 0], pts[:, 1], c="b", s=1)
    plt.show()


if __name__ == "__main__":
    sierpinski_triangle()
    barnsley_fern()

The episode is here
https://www.youtube.com/watch?v=kbKtFN71Lfs

Outputs:

Triangle

Fern