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:
  f_1 :\\     x_{n+1} = 0\\     y_{n+1} = 0.16 y_n\\ \\ f_2 :\\     x_{n+1} = 0.85 x_n + 0.04 y_n\\     y_{n+1} = -0.04 x_n + 0.85 y_n + 1.6\\ \\ f_3 :\\     x_{n+1} = 0.2 x_n - 0.26 y_n\\     y_{n+1} = 0.23 x_n + 0.22 y_n + 1.6\\ \\ f_4 :\\     x_{n+1} = -0.15 x_n + 0.28 y_n\\     y_{n+1} = 0.26 x_n + 0.24 y_n + 0.44\\
With the help of above equations, the fern is created. Now lets see the Python3 implementation for the same.

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# 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()        

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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.



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