The chaos game is played as follows. Pick a starting point at random. Then at each subsequent step, pick a triangle vertex at random and move half way from the current position to that vertex.
The result looks like a fractal called the Sierpinski triangle or Sierpinski gasket.
Here’s an example:
If the random number generation is biased, the resulting triangle will show it. In the image below, the lower left corner was chosen with probability 1/2, the top with probability 1/3, and the right corner with probability 1/6.
Here’s Python code to play the chaos game yourself.
from scipy import sqrt, zeros
import matplotlib.pyplot as plt
from random import random, randint
def midpoint(p, q):
return (0.5*(p[0] + q[0]), 0.5*(p[1] + q[1]))
# Three corners of an equilateral triangle
corner = [(0, 0), (0.5, sqrt(3)/2), (1, 0)]
N = 1000
x = zeros(N)
y = zeros(N)
x[0] = random()
y[0] = random()
for i in range(1, N):
k = randint(0, 2) # random triangle vertex
x[i], y[i] = midpoint( corner[k], (x[i-1], y[i-1]) )
plt.scatter(x, y)
plt.show()
Source: Chaos and Fractals
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