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

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.

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

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

About Fractals

• Dot Patterns, Pascal Triangle and Lucas Theorem
• Sierpinski Gasket Via Chaos Game
• The Chaos Game: Address Space vs IFS
• Sierpinski Gasket By Common Trema Removal
• Sierpinski Gasket By Trema Removal
• Sierpinski's Gasket and Dihedral Symmetry
• Collage Theorem And Iterated Function Systems
• Variations on the Theme of (Triangular) Tremas
• Variations on the Theme of (Rectangular) Tremas
• Sierpinski Gasket and Tower of Hanoi
• Fractal Curves and Dimension
• Color Cycling on the Mandelbrot Set
• Iterations and the Mandelbrot Set
• Mandelbrot and Julia sets
• Emergence of Chaos
• Logistic Model