The Barnsley Fern

[ref: https://scipython.com/book/chapter-7-matplotlib/examples/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 f1, f2, f3, and f4 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*}

First we import necessary libraries for arrays and plotting:

In [1]:

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

Next we set up the affine transformations:

In [2]:

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]

Then we set the number of points and width and height:

In [8]:

npts = 500 
# Canvas size (pixels)
width, height = 300, 300
aimg = np.zeros((width, height))

Then we use a for loop to repeatedly apply the affine transormations:

In [4]:

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

Finally, plot the resulting image:

In [5]:

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

Out[5]:

In [6]:

ix

Out[6]:

153

In [10]:

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

Out[10]:

Thank you for generating the Barnsley Fern!

#test

#Thank you for generating the Barnsley Fern!

hello

In [0]:

The Barnsley Fern

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