Iterated function system

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Colored IFS designed by Chris Ursitti using Apophysis software and rendered by the Electric Sheep.

In mathematics, iterated function systems or IFSs are a method of constructing fractals; the resulting constructions are always 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 Sierpinski gasket. 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.

1 Definition
2 Properties
3 Examples
4 Example: a fractal "fern"
5 History
6 See also
7 External links

[edit] Definition

Formally,

$S = \bigcup_i^N f_i(S),\,$

where

$S \subset \mathbb{R}^n\,$

and

$f_i:\mathbb{R}^n\to\mathbb{R}^n.$

are the functions to be iterated. S is the fixed point of a Hutchinson operator, which is the union of the functions $f i$ .

[edit] Properties

The collection of functions $f i$ together with composition form a monoid. If there are only two such functions, the monoid is the dyadic monoid. Composition can be visualized as an infinite binary tree, where, at each node of the tree, one may compose with the one or the other function (i.e take the left or the right branch). In general, if there are p functions, then one may visualize the composition as a p-adic tree.

Because function composition takes this form, the elements of the monoid can be seen to be isomorphic with the p-adic numbers; that is, each digit of the p-adic number indicates which function is to be composed with.

The automorphism group of the dyadic monoid is the modular group; this can be seen to describe the fractal self-similarity of many fractals, including the Cantor set and the de Rham curves.

[edit] Examples

Construction of an IFS by the chaos game

Sometimes each function $f i$ is required to be linear, or more accurately an affine transformation and hence can be represented by a matrix. However, IFSs may also be built from non-linear functions, including projective transformations and Möbius transformations. The Fractal flame is an example of an IFS with nonlinear functions

The most common algorithm to compute IFS fractals is called the chaos game. It consists of picking a random point in the plane, then iteratively applying one of the functions chosen at random from the function system and drawing the point. 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.

Each of these algorithms provides a global construction which generates points distributed across the whole fractal. If a small area of the fractal is being drawn, many of these points will fall outside of the screen boundaries. This makes zooming into an IFS construction normally impractical. Where a high degree of detail is required in a small area of the fractal, local construction methods based on calculating forward orbits and the fate of individual points may be more efficient (although no software for solving IFS does this).

[edit] Example: a fractal "fern"

Fractal fern using IFS

Here is an example of a fern-like image computed using an iterated function system.

The first point drawn is at the origin (x₀ = 0, y₀ = 0) and then the new points are iteratively computed by randomly applying one of the following four coordinate transformations:

x_{n + 1} = 0

y_{n + 1} = 0.16 y_n.

This coordinate transformation is chosen 1% of the time and maps any point to a point in the line segment shown in green in the figure.

x_{n + 1} = 0.2 x_n − 0.26 y_n

y_{n + 1} = 0.23 x_n + 0.22 y_n + 1.6.

This coordinate transformation is chosen 7% of the time and maps any point inside the black rectangle to a point inside the red rectangle in the figure.

x_{n + 1} = −0.15 x_n + 0.28 y_n

y_{n + 1} = 0.26 x_n + 0.24 y_n + 0.44.

This coordinate transformation is chosen 7% of the time and maps any point inside the black rectangle to a point inside the dark blue rectangle in the figure.

x_{n + 1} = 0.85 x_n + 0.04 y_n

y_{n + 1} = −0.04 x_n + 0.85 y_n + 1.6.

This coordinate transformation is chosen 85% of the time and maps any point inside the black rectangle to a point inside the light blue rectangle in the figure.

The first coordinate transformation draws the stem. The second draws the bottom frond on the left. The third draws the bottom frond on the right. The fourth generates successive copies of the stem and bottom fronds to make the complete fern. The recursive nature of the IFS guarantees that the whole is a larger replica of each frond. Note: The fern is within the range -5 <= x <= 5 and 0 <= y <= 10.

[edit] History

Iterated function systems are a generalisation of the Cantor set, first described in 1884, and of de Rham curves, a type of self-similar curve described by Georges de Rham in 1957. IFS were conceived in their present form by John Hutchinson in 1981 and popularized by Michael Barnsley's book Fractals Everywhere. In 1992 Scott Draves developed the Fractal flame algorithm.