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 AF. The process is iterative. Starting with a point x0, form a sequence

xk+1 = Fr(xk), k = 0, 1, 2, ...

where Fr is a member of the IFS randomly selected for every iteration. If x0AF, the same holds for all successive points xk. In this case it follows from Elton's Ergodic Theorem [Barnsley, p. 370] that, with probability 1, the set {xk} is dense in AF. 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 FA, FB, FC, each towards one of the vertices of ΔABC. For a given point x,

Sierpinski's gasket through the Chaos game

References

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

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