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.
