Topic · Geometry

The Chaos Game

Roll a die, draw a fractal.

Three dots, a die, and one short step — repeat a few thousand times and a perfect Sierpiński triangle materialises out of pure randomness. The Explorer lets you tune the rule live; here we tell the story behind it.

Discover the story → Open the Explorer

First encounter

Three dots. A die. A halving step.

The big idea

Mark three dots on a page — the corners of a triangle. Drop your pen anywhere. Now roll a three-sided die, walk halfway toward the chosen corner, and make a dot. Treat that dot as your new position and repeat. After a few thousand rolls a perfect Sierpiński triangle stares back at you. Order, from pure noise.

A concrete example

Start anywhere inside an equilateral triangle. Each step: pick one of the three vertices uniformly at random, move halfway toward it, drop a dot. After 200 dots you see a scatter cloud. After 5000 you see the gasket — every "forbidden" region (the central hole, then the three sub-holes, then nine sub-sub-holes) is provably empty.

200 dots

5000 dots

Why it matters

This is the iterated function system in disguise. Each "walk halfway to vertex i" is a contracting map on the plane; the set of dots converges onto the unique attractor of those three maps. Swap the maps and you get other fractals — Barnsley's fern grows from exactly the same recipe, just four affine maps instead of three halving steps.

Below: watch the rule run live, then watch the fern grow.

Section 01 · The minimal recipe

Three vertices, a halving step, a random choice

Place three points v₁, v₂, v₃ at the corners of a triangle. Pick any starting position p₀ in the plane. At each step: roll a fair three-sided die to choose an index i ∈ {1, 2, 3}, then move halfway from your current position toward the chosen vertex — pₙ₊₁ = (pₙ + vᵢ) / 2 — and mark pₙ₊₁ as a dot. Repeat forever. That is the entire chaos game: two ingredients, three vertices, one fixed jump ratio of one-half. No memory, no plan, no global rule. The picture builds itself.

Section 02 · Why it works

Fixed points of three contracting maps

Each available move is a function fᵢ(x) = (x + vᵢ) / 2 — a contraction by factor 1/2 toward vertex vᵢ. The fixed point of fᵢ is vᵢ itself: walking halfway toward a vertex over and over takes you to the vertex. Composing the three maps in every possible order generates a self-similar set: the Sierpiński triangle. The chaos game samples that set with probability one. Each step is a random projection toward one of the three fixed points, and because every map contracts, the influence of where you started decays exponentially — after a short burn-in your trajectory lives entirely on the attractor.

Three contracting maps

f₁(x) = (x + v₁)/2 · f₂(x) = (x + v₂)/2 · f₃(x) = (x + v₃)/2

Each map contracts the plane by factor 1/2 toward its fixed point vᵢ. The attractor of the IFS {f₁, f₂, f₃} is the Sierpiński triangle.

Interactive · the chaos game running live

Vertices3

Jump ratio r0.500

No-repeat rule

0 points

Section 03 · Barnsley's collage theorem

Any compact set has an IFS that converges to it

Michael Barnsley's 1988 book Fractals Everywhere brought the collage theorem to a wide audience: every compact subset of the plane can be approximated arbitrarily well by the attractor of some finite iterated function system. Reverse the question — pick a target shape, then find a few affine maps whose images glue together (a "collage") to cover it. The attractor of that IFS is the shape. The chaos game then samples it with a single random orbit. Coastlines, ferns, mountains, clouds: every fractal in nature is, in principle, a few contractions in disguise.

Section 04 · Beyond triangles

When halving fails — and the rule that fixes it

Try four vertices at the corners of a square with the same r = 1/2 step. The result is no fractal at all: the dots fill the interior uniformly. The reason is that the four halving maps on a square cover the whole square — their attractor is the filled square. The fix is a restriction rule. Forbid the next vertex from equalling the previous one ("no repeat") and a delicate Sierpiński-carpet relative appears. Each n-gon has its own magic jump ratio rₙ = 1 − sₙ that yields a clean self-similar attractor without restrictions. For n = 3 this is 1/2; for n = 5 it is 1/φ ≈ 0.618; for n = 6, 2/3; and so on.

Magic ratio · rₙ = 1 − sₙ

n	rₙ	shape
3	0.5000	Sierpiński triangle
4	0.5000	1/2 (fills the square)
5	0.6180	1/φ (pentaflake)
6	0.6667	2/3 (hexaflake)
7	0.6920	heptagon
8	0.7071	1/√2 (octagon)

Section 05 · Barnsley's fern

Four affine maps. One photograph of a fern.

Drop the vertices entirely. Pick four affine transformations of the plane f₁ … f₄ and four probabilities (0.01, 0.85, 0.07, 0.07). At each step, choose one of the four maps with those probabilities and apply it to your current point. f₁ — the rare 1% — collapses everything onto the stem; f₂ — the dominant 85% — copies the fern onto itself, rotated and shrunk; f₃ and f₄ peel off the left and right leaflets. After a hundred thousand iterations the cloud of points is indistinguishable from a photograph of a real fern. The numbers are not magic: change a coefficient and the fern bends; change a probability and the leaflets thicken. The recipe is editable.

Interactive · Barnsley's fern grows

f₁ · p = 0.01 · stem

f₂ · p = 0.85 · main spiral

f₃ · p = 0.07 · left leaflet

f₄ · p = 0.07 · right leaflet

0 points

Section 06 · Deeper

Chaos, ergodicity, and the SRB measure

The chaos game is more than a drawing trick. The random orbit is a Markov chain on the attractor; the invariant distribution it samples — the Hutchinson measure — is the unique probability measure that splits into weighted pieces under the IFS. In smooth ergodic theory this is a sibling of the SRB measure (Sinai–Ruelle–Bowen) on a hyperbolic attractor: the physical measure that almost every initial condition picks out. The picture in front of you is a finite sample of a probability distribution that has support on a fractal of dimension log 3 / log 2 ≈ 1.585. Geometry, probability, and dynamics meet in a die roll.

Hausdorff dimension

log 3 / log 2 ≈ 1.585

The Sierpiński triangle has fractal dimension between a curve (1) and a region (2). Three copies, each scaled by 1/2, so the dimension d satisfies 3·(1/2)ᵈ = 1.

Take it further

Roll the dice.

The Explorer lets you tune the vertex count, the jump ratio, the restriction rule and the colour — and switch in a heartbeat to the four-map Barnsley fern. Everything you just read, in your hands.

→ Open the Explorer

Where you meet it

How and where this technique lives in the world today.

Bioinformatics

Chaos Game Representation (CGR) is a standard way to visualise DNA sequences — each nucleotide steers a point toward one of four corners of a square; species cluster into recognisable fractal signatures.

Fractal compression

Barnsley's fractal image compression encodes images as a small set of contractive maps recovered via the chaos game.

Procedural texturing

Chaos game outputs (Barnsley fern variants) are widely used for procedural foliage and stylised brush strokes.