Iterated function systems

In mathematics, iterated function systems (IFSs) are a method of constructing fractals; the resulting fractals are often self-similar.

IFS fractals, as they are normally called, can be of any number of dimensions, but are commonly computed and drawn in 2D. The fractal is made up of the union of several copies of itself, each copy being transformed by a function (hence "function system"). The canonical example is the Sierpiński gasket, also called the Sierpiński triangle. The functions are normally contractive, which means they bring points closer together and make shapes smaller. 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. This is the source of its self-similar fractal nature.

Formally, an iterated function system is a finite set of contraction mappings on a complete metric space. Symbolically,

is an iterated function system if each ${\displaystyle f_{i}}$ is a contraction on the complete metric space ${\displaystyle X}$.

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