An alternative algorithm is to generate each possible sequence of functions up to a given maximum length, and then to plot the results of applying each of these sequences of functions to an initial point or shape.Įach of these algorithms provides a global construction which generates points distributed across the whole fractal. It consists of picking a random point in the plane, then iteratively applying one of the functions chosen at random from the function system to transform the point to get a next point. The most common algorithm to compute IFS fractals is called the " chaos game". The Fractal flame is an example of an IFS with nonlinear functions. However, IFSs may also be built from non-linear functions, including projective transformations and Möbius transformations. is required to be a linear, or more generally an affine, transformation, and hence represented by a matrix. This is the source of its self-similar fractal nature.įormally, an iterated function system is a finite set of contraction mappings on a complete metric space. Hence, the shape of an IFS fractal is made up of several possibly-overlapping smaller copies of itself, each of which is also made up of copies of itself, ad infinitum. The functions are normally contractive, which means they bring points closer together and make shapes smaller. The canonical example is the Sierpiński triangle. The fractal is made up of the union of several copies of itself, each copy being transformed by a function (hence "function system"). IFS fractals, as they are normally called, can be of any number of dimensions, but are commonly computed and drawn in 2D. IFS fractals are more related to set theory than fractal geometry. In mathematics, iterated function systems ( IFSs) are a method of constructing fractals the resulting fractals are often self-similar.
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Colored IFS designed using Apophysis software and rendered by the Electric Sheep.
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