In mathematics, the term chaos game referred to a method of creating a fractal, using a polygon and an initial point selected at random inside it.
In mathematics, the term chaos game originally referred to a method of creating a fractal, using a polygon and an initial point selected at random inside it.
The fractal is created by iteratively creating a sequence of points, starting with the initial random point, in which each point in the sequence is a given fraction of the distance between the previous point and one of the vertices of the polygon; the vertex is chosen at random in each iteration. Repeating this iterative process a large number of times, selecting the vertex at random on each iteration, and throwing out the first few points in the sequence, will often (but not always) produce a fractal shape.
The term has been generalized to refer to a method of generating the attractor, or the fixed point, of any iterated function systems (IFS). Starting with any point $x_0$, successive iterations are formed as $x_k+1 = f_r(x_k)$, 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 $x_0$ belongs to the attractor of the IFS, all iterations $x_k$ stay inside the attractor and, with probability $1$, form a dense set in the latter.