Barnsley Fern in Python

Barnsley fern is a fractal shape created by mathematician Michael Barnsley. The geometric features of this fractal resemble a natural fern and hence it gets its name. Barnsley fern is created by iterating over a large number of times on four mathematical equations, introduced by Barnsley, known as Iterated Function System (IFS). The transformation Barnsley used had the formula : where, the letters had following value :

a	b	c	d	e	f	p	PART
0	0	0	0.16	0	0	0.01	Stem
0.85	0.04	-0.04	0.85	0	1.60	0.85	Small Leaflet
0.20	-0.26	0.23	0.22	0	1.60	0.07	Large Leaflet(Left)
-0.15	0.28	0.26	0.24	0	0.44	0.07	Large Leaflet(Right)

and “p” is the probability. Thus the four equations are: With the help of above equations, the fern is created. Now lets see the Python3 implementation for the same.

# importing necessary modules
import matplotlib.pyplot as plt
from random import randint

# initializing the list
x = []
y = []

# setting first element to 0
x.append(0)
y.append(0)

current = 0

for i in range(1, 50000):

    # generates a random integer between 1 and 100
    z = randint(1, 100)

    # the x and y coordinates of the equations
    # are appended in the lists respectively.

    # for the probability 0.01
    if z == 1:
        x.append(0)
        y.append(0.16*(y[current]))

    # for the probability 0.85
    if z>= 2 and z<= 86:
        x.append(0.85*(x[current]) + 0.04*(y[current]))
        y.append(-0.04*(x[current]) + 0.85*(y[current])+1.6)

    # for the probability 0.07
    if z>= 87 and z<= 93:
        x.append(0.2*(x[current]) - 0.26*(y[current]))
        y.append(0.23*(x[current]) + 0.22*(y[current])+1.6)

    # for the probability 0.07
    if z>= 94 and z<= 100:
        x.append(-0.15*(x[current]) + 0.28*(y[current]))
        y.append(0.26*(x[current]) + 0.24*(y[current])+0.44)

    current = current + 1

plt.scatter(x, y, s = 0.2, edgecolor ='green')

plt.show()

Output : Note: The whole output depends upon the coefficients of the equations. One experiment might be to change the coefficients and get a new pattern every time.