Iterated function system
Iterated function systems (IFSs) are a simple way to build fractals that look the same at many scales. An IFS is a finite set of contraction mappings on a complete metric space (usually the plane). There is a unique nonempty compact set S that is invariant under the system: S = f1(S) ∪ f2(S) ∪ ... ∪ fk(S). This fixed set S is the fractal produced by the IFS.
You can obtain S by starting with any nonempty compact set S0 and repeatedly applying the Hutchinson operator F, which maps A to the union of its images under each fi: Sn+1 = ∪i fi(Sn). The limit S = closure of lim Sn exists and is unique because the functions are contractions.
A key idea is self-similarity: S is built from several smaller copies of itself, each copy obtained by applying one of the functions. The functions can be linear or affine, but IFSs can also use nonlinear maps, including projective or Möbius transformations. Most IFS fractals are drawn in 2D, though they can exist in higher dimensions. Classic examples include the Cantor set and the Sierpiński triangle; nonlinear examples include fractal flames.
The most common way to draw an IFS fractal is the chaos game: start from a random point and repeatedly apply a randomly chosen fi to move to the next point. The collection of visited points fills out S. Another method is to apply all sequences of functions up to a certain length and plot the results.
In practice, software often requires only that the system be contractive on average. Partitioned or local IFSs (PIFS) can give surprisingly good image compression, storing the rules rather than every pixel. Fractal compression can be fast and storage-light, though solving the inverse problem (finding IFS parameters that reproduce a given image) is hard in general.
IFSs also provide models for natural forms that display self-similarity, such as certain plants and leaves. The concept was introduced by John E. Hutchinson in 1981 and popularized by Michael Barnsley.
This page was last edited on 3 February 2026, at 05:41 (CET).