Sierpinski Gasket Via Chaos Game

The Chaos Game is a process of generating an approximation to a deterministic set by random means. It could be used with any Iterated Function System F to approximate its fixed point A_F. The process is iterative. Starting with a point x₀, form a sequence

x_k+1 = F_r(x_k), k = 0, 1, 2, ...

where F_r is a member of the IFS randomly selected for every iteration. If x₀A_F, the same holds for all successive points x_k. In this case it follows from Elton's Ergodic Theorem [Barnsley, p. 370] that, with probability 1, the set {x_k} is dense in A_F. Following [The Science of Fractal Images, Ch. 5.3, Chaos and Fractals, Ch. 6], another explanation is available.

The applet below demonstrates the emergence of the Sierpinski gasket as the result of the chaos game.

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Sierpinski Chaos Game

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There are three contractions F_A, F_B, F_C, each towards one of the vertices of ΔABC. For a given point x,

Sierpinski's gasket through the Chaos game

References

M. Barnsley, Fractals Everywhere, Academic Press, 1988
The Science of Fractal Images, H.-O. Peitgen and D.Daupe (eds), Springer-Verlag, 1988
H.-O. Peitgen, H. Jürgens, D. Saupe, Chaos and Fractals: New Frontiers of Science, Springer-Verlag, 1992

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