# Barnsley Fern in Python

• Last Updated : 22 Oct, 2018

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 :

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 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.

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