🌿 IFS Fractals — Iterated Function Systems
An Iterated Function System is a finite set of contractive affine maps. The chaos game algorithm — start anywhere, repeatedly apply a randomly chosen map with its given probability — reveals the IFS attractor: a fractal whose self-similar structure emerges from pure iteration. The Barnsley fern uses just 4 maps with carefully chosen probabilities to produce a photorealistic leaf.
Preset IFS
Rendering
Stats
x₀ arbitrary
xₙ₊₁ = Wᵢ(xₙ)
where i ~ p(Wᵢ)
Affine map:
W(x,y) = [a b; c d][x;y] + [e;f]
About Iterated Function Systems
The Collage Theorem (Barnsley & Demko, 1986) guarantees that any IFS with contractive maps has a unique compact attractor A satisfying A = ∪ Wᵢ(A). The Hausdorff dimension of the attractor can be estimated from the contraction ratios. IFS are the foundation of fractal image compression: any image can be approximated as an IFS attractor. The chaos game converges to the attractor regardless of the starting point — a consequence of the Banach fixed-point theorem.