Menger sponge, created by using IFS.

Iterated function systems or IFSs, are a kind of fractal which were conceived in their present form by John Hutchinson in 1981 and popularized by Michael Barnsley's book Fractals Everywhere. Image File history File links Please see the file description page for further information. ... Image File history File links Please see the file description page for further information. ... The Menger sponge is a fractal solid. ... The boundary of the Mandelbrot set is a famous example of a fractal. ... John Hutchinson is the name of a number of notable people: John Hutchinson, an English writer John Hutchinson, a leader in the 17th century Puritan revolt in Britain John Hutchinson, the controversial inventor John Hutchinson from South Dakota, and the namesake of Hutchinson County, South Dakota; John Hutchinson, the Australian... Michael Barnsley is the researcher and entrepreneur who has worked on fractal compression; he holds several patents on the technology. ...

IFS fractals as they are normally called can be of any number of dimensions, but are commonly computed and drawn in 2D. An IFS fractal is a solution to a recursive set equation. 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. The Sierpinski triangle, also called the Sierpinski gasket, is a fractal, named after Waclaw Sierpinski. ... In mathematics, a contraction mapping, or contraction, on a metric space (M,d) is a function f from M to itself, with the property that there is some real number k < 1 such that, for all x and y in M, The smallest such value of k is called the...

Formally, where and

Serpinki gasket created using IFS

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. Image File history File links Serpinski1. ... Image File history File links Serpinski1. ... The word linear comes from the Latin word linearis, which means created by lines. ... In geometry, an affine transformation or affine map (from the Latin, affinis, connected with) between two vector spaces consists of a linear transformation followed by a translation: In the finite-dimensional case each affine transformation is given by a matrix A and a vector b, which can be written as... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, of elements of a ring-like algebraic structure. ... A projective transformation is a transformation used in projective geometry: it is the composition of a pair of perspective projections. ... In mathematics, a Möbius transformation is a bijective conformal mapping of the extended complex plane (i. ...

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. The chaos game or chaosgame is a means of creating a fractal, using a polygon and a random point inside it. ...

Random game IFS from a set using five linear and then two nonlinear, (reversed Julia set, C = [0, 0]), transformations in sequence.

Fractal flames are a generalization and refinement of IFS fractals. Image File history File links IFS from a set of five linear and two nonlinear transformations, a fractal. ... Image File history File links IFS from a set of five linear and two nonlinear transformations, a fractal. ... The word linear comes from the Latin word linearis, which means created by lines. ... To do: 20th century mathematics chaos theory, fractals Lyapunov stability and non-linear control systems non-linear video editing See also: Aleksandr Mikhailovich Lyapunov Dynamical system External links http://www. ... Julia sets, described by Gaston Julia, are fractal shapes defined on the complex number plane. ... Categories: Fractals | Math stubs ...

The inverse problem of finding an IFS that has a given set as IFS-fractal is addressed by the Collage theorem due to Barnsley. Barnsley received several patents on image compression using IFS and created the company Iterated Systems, Inc. to market the technology. The company ultimately found compression with simple IFS impractical, and developed the mathematical theory to support more sophisticated compression techniques.

Example: Computation of a fractal "fern" using an iterated function system

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 4 coordinate transformations: Image File history File links Fractal_fern_explained. ... Image File history File links Fractal_fern_explained. ...

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+0.04 y_n
y_n+1=-0.04 x+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.

See also

• L-system
• Fractal compression

See L-system for information on Lindenmayer systems. ... Fractal compression is a lossy compression method used to compress images using fractals. ...

External links

• A definition of IFS and a Java illustration with several built-in examples at cut-the-knot
• Interactive 3D fractal polyhedra in Java
• Xenodream - a 3d IFS modeling program
• Examples of 3d IFS models created with Xenodream
• IFS Fractal fern and Sierpinski triangle - JAVA applet