Draw Barnsley’s Fern with Chaos Game. The equation for construction is provided in this Wikipedia article.

Solution:

import turtle
import random

screen = turtle.Screen()
screen.title('Barnsley\'s Fern Chaos Game with Python Turtle')
screen.setup(1000,1000)
screen.setworldcoordinates(-6,-1,6,11)
screen.tracer(0,0)
turtle.hideturtle()
turtle.speed(0)
turtle.up()

n = 100000 # number of points to draw
p = (0,0) 
t = turtle.Turtle()
t.up()
t.hideturtle()
for i in range(n):
    t.goto(p)
    t.dot(2,'green') 
    r = random.uniform(0,1)
    if r < 0.01:
        p = (0,0.16*p[1])
    elif r < 0.86:
        p = (0.85*p[0] + 0.04*p[1], -0.04*p[0] + 0.85*p[1] + 1.6)
    elif r < 0.93:
        p = (0.2*p[0] - 0.26*p[1], 0.23*p[0] + 0.22*p[1] + 1.6)
    else:
        p = (-0.15*p[0] + 0.28*p[1], 0.26*p[0] + 0.24*p[1] + 0.44)
        
    if i % 1000 == 0: # update for every 1000 moves, this part is for performance reason only
        t = turtle.Turtle() # use new turutle
        t.up()
        t.hideturtle()
        screen.update()