The Chaos Game: Building Fractals from Pure Randomness

A fractal from coin flips

The chaos game is one of the strangest results in mathematics: pick a few fixed points (vertices of a polygon), start anywhere, and repeatedly jump partway toward a randomly chosen vertex. Plot every point you land on. Despite the randomness at every single step, the accumulated dots do not form a formless blur — they trace out a precise, infinitely detailed fractal. For a triangle with a jump ratio of one-half, the result is the Sierpiński triangle, the same shape you get from the fully deterministic, non-random recursive construction of repeatedly removing the middle triangle.

pick 3 vertices: A, B, C
start at any point P
repeat forever:
    choose a vertex V at random (A, B or C)
    P = P + 0.5 * (V - P)      // jump halfway toward V
    plot P

This was popularised by mathematician Michael Barnsley in the 1980s as a vivid, hands-on illustration of an iterated function system (IFS) — a set of contraction maps (here, three "jump halfway toward vertex" rules) whose repeated random application converges to a unique fixed shape called the attractor, regardless of where you start. The starting point only affects the very first few dots; wait long enough and every run converges onto the identical fractal.

Why the middle never fills in

The reason a hole appears at the centre of the triangle is purely combinatorial: jumping halfway toward any one of three vertices can never land you in the central region, no matter which sequence of vertices you choose, because that region is only reachable by paths that the halfway rule forbids. Run the process long enough and the density of dots reveals self-similar copies of the whole triangle inside each of its three corners, and inside each of those, forever — the defining property of a fractal.

Other shapes from the same idea

Changing the jump ratio, the number of vertices, or adding rules (like "never choose the same vertex twice in a row") produces an entire family of related fractals from the same basic engine. A square with a half-ratio jump just fills in solidly (there is no forbidden region), but restricting which vertex can follow which produces intricate patterns again. The Vicsek fractal comes from five points (four corners plus the centre of a square) with a jump ratio of one-third. The famous Barnsley fern — a strikingly realistic fern frond — uses four affine transformations (each a small stretch, rotation and shift) applied with different, carefully tuned probabilities rather than a simple uniform vertex choice, showing that the chaos-game idea generalises far beyond straightforward polygons.

Sierpiński triangle:  3 vertices, ratio 1/2
Vicsek fractal:       5 points (corners + centre), ratio 1/3
Barnsley fern:        4 weighted affine maps, probabilities tuned by hand

The deep idea: order hides inside randomness

The chaos game is a small, visual proof that a process built entirely from independent random choices can still have a completely deterministic, reproducible long-run outcome. This is closely related to why Monte Carlo methods work in physics and finance: individual samples are random, but their statistical distribution converges reliably to a fixed answer. It is also a gentle introduction to ergodic theory, the branch of mathematics that studies when the long-run average behaviour of a random or chaotic system settles into a stable pattern independent of its starting conditions.

Frequently asked questions

Try it live

Everything above runs in your browser — open Chaos Game and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.