# The Barnsley fern

The Barnsley Fern is a fractal which resembles the Black Spleenwort species of fern. It is constructed by plotting a sequence of points in the $(x,y)$ plane, starting at $(0,0)$, generated by the following affine transformations $f_1$, $f_2$, $f_3$, and $f_4$ where each transformation is applied to the previous point and chosen at random with probabilities $p_1 = 0.01$, $p_2 = 0.85$, $p_3 = 0.07$ and $p_4=0.07$. \begin{align*} f_1(x,y) &= \left( \begin{array}{ll} 0 & 0\\0 & 0.16 \end{array}\right) \left( \begin{array}{l}x\\y\end{array} \right)\\ f_2(x,y) &= \left( \begin{array}{ll} 0.85 & 0.04\\-0.04 & 0.85 \end{array}\right) \left( \begin{array}{l}x\\y\end{array} \right) + \left( \begin{array}{l}0\\1.6\end{array} \right)\\ f_3(x,y) &= \left( \begin{array}{ll} 0.2 & -0.26\\0.23 & 0.22 \end{array}\right) \left( \begin{array}{l}x\\y\end{array} \right) + \left( \begin{array}{l}0\\1.6\end{array} \right)\\ f_4(x,y) &= \left( \begin{array}{ll} -0.15 & 0.28\\0.26 & 0.24 \end{array}\right) \left( \begin{array}{l}x\\y\end{array} \right) + \left( \begin{array}{l}0\\0.44\end{array} \right) \end{align*}

This algorithm is implemented in the program below.

import numpy as np
import matplotlib.pyplot as plt
import matplotlib.cm as cm

f1 = lambda x,y: (0., 0.16*y)
f2 = lambda x,y: (0.85*x + 0.04*y, -0.04*x + 0.85*y + 1.6)
f3 = lambda x,y: (0.2*x - 0.26*y, 0.23*x + 0.22*y + 1.6)
f4 = lambda x,y: (-0.15*x + 0.28*y, 0.26*x + 0.24*y + 0.44)
fs = [f1, f2, f3, f4]

npts = 50000
# Canvas size (pixels)
width, height = 300, 300
aimg = np.zeros((width, height))

x, y = 0, 0
for i in range(npts):
# Pick a random transformation and apply it
f = np.random.choice(fs, p=[0.01, 0.85, 0.07, 0.07])
x, y = f(x,y)
# Map (x,y) to pixel coordinates.
# NB we "know" that -2.2 < x < 2.7 and 0 <= y < 10

ix, iy = int(width / 2 + x * width / 10), int(y * height / 12)
# Set this point of the array to 1 to mark a point in the fern
aimg[iy, ix] = 1

plt.imshow(aimg[::-1,:], cmap=cm.Greens)
plt.show()