Infinite Barnsley Fern | mysimulator.uk

Chaos Game

Points / frame 1200

Point size 1.6

Stem probability 0.01

Points plotted

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Total iterations

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Last transform

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Buffer fill

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How it works

The Barnsley fern is the attractor of an iterated function system (IFS): four affine maps of the plane, each of the form (x, y) → (ax + by + e, cx + dy + f). The "chaos game" picks one map per step at random, weighted by fixed probabilities, and plots the resulting point — never the whole shape, just one pixel at a time. After a few thousand steps the orbit converges onto the fern regardless of the starting point, because the maps are contractions (each shrinks distances).

map	a	b	c	d	f	p	role
f₁	0	0	0	0.16	0	0.01	stem
f₂	0.85	0.04	−0.04	0.85	1.6	0.85	main leaflets
f₃	0.20	−0.26	0.23	0.22	1.6	0.07	left leaflet
f₄	−0.15	0.28	0.26	0.24	0.44	0.07	right leaflet

xₙ₊₁ = a·xₙ + b·yₙ + e
yₙ₊₁ = c·xₙ + d·yₙ + f

Points / frame — how many chaos-game steps run each animation frame; higher fills the fern in faster.
Point size — the on-screen size of each plotted instance.
Stem probability — perturbs f₁'s real 0.01 weight so you can see how rare the stem draw is compared to the 85% main-leaflet map; the other three probabilities rescale to keep the total at 1.
Buffer fill — points are written into a fixed-size instanced buffer; once full, new points overwrite the oldest ones, so the fern keeps regenerating forever — the "infinite" part of an infinite IFS.

This is the same construction Michael Barnsley published in 1988 to show that natural, organic-looking forms can emerge from a handful of contracting linear maps plus randomness — a cornerstone example in fractal geometry.

Stem (f₁) Main leaflets (f₂) Left leaflet (f₃) Right leaflet (f₄)

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Infinite Barnsley Fern Iterated System