Here is the algorithm of the generation of the Sierpinski triangle using the chaos game.

  If we note (xi,yi) for i=1,2,3 the coordinates of the different vertices of the triangle and (x,y) the coordinates of the moving point, we can write the algorithm of the geometrical construction as follows :
FOR the number of iterations DO
- roll the die, it means choose randomly the direction to a vertex noted a
- construct the new point (x,y) using :
x(n+1) = x(n) + 0.5 * [xa - x(n)]

y(n+1) = y(n) + 0.5 * [ya - y(n)]

- plot this point (x,y)
  • I have written the program in Matlab language.
    You can choose the original triangle vertices, the seed coordinates and the number of iterations.
    Here is the program listing :
      The table presents the generation of the Sierpinski triangle for different number of iterations. I don't have rejected the beginning points on the graphs in order to see the "attraction" of the orbit to the Sierpinski triangle even if the seed is chosen very far from the triangle (it is particularly easy to observe it for the 5,000 iterations graph).

    The parameters for all these graphs are the same :
    (x1 , y1) = (1 , 1)
    (x2 , y2) = (0 , 0)
    (x3 , y3) = (2 , 0)
    (x0 , y0) = (0.5 , 1.5)
    and the number of iterations moves from 50 to 50,000 iterations.
    50 iterations
    100 iterations
    500 iterations
    1,000 iterations
    2,000 iterations
    4,000 iterations
    5,000 iterations
    10,000 iterations
    50,000 iterations