Originally the chaos game referred to a means of creating a fractal, using a polygon and a random point inside it. The fractal is created by finding the point a given fraction of the distance between the previous point and one of the vertices, chosen at random, a large number of times. Using a regular triangle and the factor 1/2 will result in the Sierpinski triangle. In this way, a great many shapes can be generated, all the more realistically if the original shape has a hidden fractal order. The chaos game is an example of a random process leading to a predetermined result.
Today the meaning of chaos game has been generalized and now refers to a way of generating the attractor, or the fixed point, of an iterated function system (IFS). Starting with any point x0, successive iterations are formed as xk+1 = fr(xk), where fr is a member of the given IFS randomly selected for each iteration. The iterations converge to the fixed point of the IFS. Whenever x0 belongs to the attractor of the IFS, all iterations xk stay inside the attractor and, with probability 1, form a dense set in the latter.
Fractal fern created using chaos game
The process for n functions can be described in pseudo-code
(x,y) = a random point in the bi-unit square
iterate {
i = a random integer from 0 to n-1 inclusive
(x, y) = Fi(x, y)
plot(x, y)
}
The points that are plotted show up in random order all over the attractor. This is in contrast to how most fractals are drawn, which is one pixel at a time, in order on the screen. Instead the image of the attractor emerges from the many dots plotted by the algorithm.
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