# 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 x_{0}, 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 x_{0}A_{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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There are three contractions F_{A}, F_{B}, F_{C}, each towards one of the vertices of ΔABC. For a given point x,

### 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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