About IFS Encyclopedia
What is an Iterated Function System?
An Iterated Function System (IFS) is A finite set of contraction mappings $\{f_1, f_2, \ldots, f_n\}$ on a complete metric space $(X, d)$. Each mapping fi satisfies a contractivity condition: there exists 0 ≤ c < 1 such that d(fi(x), fi(y)) ≤ c·d(x,y) for all x, y in X.
The Attractor
By the Banach fixed-point theorem, every IFS has a unique non-empty compact attractor (also called the invariant set) A satisfying:
$$A = \bigcup_{i=1}^{n} f_i(A)$$
Starting from any compact set, repeated application of all mappings simultaneously converges to this attractor in the Hausdorff metric.
Affine Form
The most common IFS mappings are affine contractions of the form:
$$f_i(\mathbf{x}) = A_i \mathbf{x} + \mathbf{b}_i$$
where Ai is a 2×2 (or n×n) matrix and bi is a translation vector. The contractivity condition requires that the spectral radius of Ai is less than 1.
Generalized IFS (GIFS)
A Generalized IFS (GIFS) defines multiple interleaved attractor sets whose equations refer to each other. Instead of a single equation A = ⋃ fi(A), a GIFS with attractors A1, …, Ak satisfies a system:
$$A_j = \bigcup_{i} f_{ij}(A_{\sigma(i,j)})$$
where each map fij sends one attractor into another according to a dependency graph. Under suitable expansiveness conditions the system has a unique tuple of non-empty compact sets (A1, …, Ak) satisfying all equations simultaneously.
GIFS naturally arise in substitution tilings: a tiling with multiple prototile types (e.g. the CAP / hat monotile with four metatile types H, T, P, F, or the Robinson triangle with two types) can be described as a GIFS where each attractor is the shape of one prototile and the maps encode the substitution rule.
Further Reading
- Mekhontsev, D. (2019). An algebraic framework for finding and analyzing self-affine tiles and fractals . Doctoral thesis, Ernst-Moritz-Arndt-Universität Greifswald. Establishes the algebraic foundations of the IFStile algorithms.
- AIFS Language Reference — algebraic representation used to define all fractals in this encyclopedia.
About This Site
This encyclopedia is an open reference for IFS attractors.
Live rendering
Every fractal image you see — in the catalog, on each entry page, and on the landing page —
is rendered live in your browser. There are no pre-generated images stored
on the server. Each attractor is computed on demand by
ifslib,
a WebAssembly module compiled from the IFStile source. The AIFS program embedded in each
entry is passed directly to ifslib_init / ifslib_render running
inside a Web Worker, and the resulting pixel buffer is drawn onto the canvas — all
client-side, with no server involvement.
Progressive rendering is used: the canvas first shows a quick low-resolution preview (32 px, 64 px, …) that sharpens up to the full resolution in the background. Clicking any canvas opens the attractor in the full IFStile web app for interactive exploration.
Mathematical formulas are rendered with KaTeX.
Content is licensed under CC BY 4.0 . Contributions welcome on GitHub .