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
| (1) |
xk+1 = Fr(xk), k = 0, 1, 2, ...
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where Fr is a member of the IFS randomly selected for every iteration. If x0 AF, 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.
There are three contractions FA, FB, FC, each towards one of the vertices of triangle ABC. For a given point x,
| (2) |
FA(x) = (x + A)/2, FB(x) = (x + B)/2, FC(x) = (x + C)/2.
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Thus, for example, FA(x) lies halfway between x and A.
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
Copyright © 1996-2008 Alexander Bogomolny
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