Barnsley fern and the chaos game
The Barnsley fern is a celebrated demonstration that randomness can reveal a deterministic attractor. Four affine maps encode the stem, successive leaflets and the overall shape.
Probabilities control density, not geometry
At every step one map is selected with a prescribed probability and applied to the current point. The probabilities determine how often regions are visited and therefore how evenly the picture fills; the transformations determine where the attractor lies.
The four standard maps of the Barnsley fern generate the stem, smaller leaf parts and the dominant organization of the frond. A rarely chosen transformation can be geometrically essential even though it receives few points. Probabilities should therefore not be read as importance of form. They control sampling frequency and are usually chosen so that point density roughly follows the area of transformed attractor pieces.
Why the first points are discarded
The starting point need not belong to the attractor. A short burn-in lets repeated contractions forget that arbitrary start before plotted samples are retained.
The first points still carry a visible influence from the arbitrary initial position. Because every map contracts distances, that influence decays exponentially; after several dozen steps the orbit lies effectively on the attractor. Discarding these points is not cosmetic cleanup but a transient phase, as in other dynamical experiments. Weakly contracting variants may need a much longer burn-in, so the count should not become a magic constant.
A model of appearance, not botany
The image resembles a fern because its affine pieces capture branching at several scales. It is not a simulation of cell growth, vascular transport or plant genetics. Its value is the striking economy with which recursive geometry evokes a living form.
The fern is a remarkable geometric caricature: a few affine rules capture silhouette and layering but simulate neither cell division, light response nor mechanical loading. That reduction is exactly what makes it instructive. It reveals which aspects can arise from repeated scaling and shear—and which biological differences disappear when resemblance of appearance is mistaken for a model of growth.
Sources and further reading
This article summarizes the following specialist sources in original wording. Accessed and editorially reviewed 12 August 2026.
- Iterated Function SystemWolfram MathWorld
- Fractal Image CompressionNotices of the American Mathematical Society
Continue the idea
Iterated function systems
How a small collection of contracting transformations defines a unique fractal attractor.
Trees, ferns and branching plants
How repeated branching and developmental constraints create approximate self-similarity in plants.
Self-similarity and scaling
Understand exact, approximate, statistical and self-affine similarity—and why natural patterns repeat only across limited scales.