Tool · Playground

IFS designer

Build a new iterated function system from scratch. Each map is an affine contraction (x, y) → (a·x + b·y + e, c·x + d·y + f) applied with weight p. The attractor of any IFS where every map is a contraction (‖A‖ < 1) is rendered by the chaos game in milliseconds.

Presets:

dim ≈ 1.585 (Moran)

Render

Iterations = 80,000

Color by mapAuto-fit view

Affine maps · 3

map #1‖A‖ = 0.500

a = 0.500b = 0.000e (tx) = 0.000c = 0.000d = 0.500f (ty) = 0.000weight = 0.333

map #2‖A‖ = 0.500

a = 0.500b = 0.000e (tx) = 0.500c = 0.000d = 0.500f (ty) = 0.000weight = 0.333

map #3‖A‖ = 0.500

a = 0.500b = 0.000e (tx) = 0.250c = 0.000d = 0.500f (ty) = 0.500weight = 0.333

Design hints

Hutchinson's theorem: if every map is a contraction, the IFS has a unique compact attractor. The contraction column in the UI flags maps with ‖A‖ ≥ 1 in red — those can blow up.
Moran dimension: for similarity IFS (rotation + uniform scale, no shear), the dimension solves Σrᵢᵈ = 1. The estimator here uses the singular value upper bound as a stand-in.
Weights only set colour, not the attractor: the geometric attractor depends only on the set of maps. Weights govern density (how often each map fires in the chaos game).
Two maps make a Cantor set; three at corners of a triangle make Sierpinski; shear a map and ferns/leaves emerge.

See the curated IFS catalogue for named presets, and /fractals/barnsley-fern for the canonical IFS example.

FAQ