Diffusion-limited aggregation (Witten and Sander, 1981)

Diffusion-limited aggregation (DLA) is a way for a cluster to grow: particles wander at random, and each one stops for good where it first touches the cluster. The physicists Tom Witten and Len Sander introduced it in 1981 in Physical Review Letters, as a model of "irreversible colloidal aggregation", in a paper that begins "A model for random aggregates is studied by computer simulation."^1, 2

How it works

The rule is short. Paul Bourke gives it for a picture made of pixels:³

1. Start with an empty image. One particle is fixed: "a single black pixel in the center".
2. Let a new particle loose at the edge and move it one step at a time, in a random direction each time, so that its path is a random walk.
3. When it comes next to a fixed particle, it sticks. It belongs to the cluster now, and the next particle starts.

Nothing tells the cluster to branch. It branches because of where wandering particles arrive. In the physicist Thomas C. Halsey's words, "an arriving random walker is far more likely to attach to one of the tips of the cluster" than to get deep into one of its "fjords" without touching something first. The tips grow, and in doing so they shield the gaps behind them, at every scale.²

The result is a fractal: the number of particles grows more slowly than the area the cluster covers. In two dimensions and without a grid, its fractal dimension is about 1.71, more than a line (1) and less than a filled surface (2).^2, 4 The same rule also works in three dimensions, as Len Sander points out; his review of 2000 covers the model at length.⁵

New drawing

Where it appears

The same branching shows up wherever growth is fed by particles that arrive by diffusion: copper deposited from a solution by an electric current, mineral deposits, fluid pushed into a thin gap between two plates, and the paths of electrical breakdown in an insulator.^4, 6 Bourke adds coral growth and "the path taken by lightning".³

Changing the rule

Small changes to the rule change the form. If a particle sticks only with a certain probability, and otherwise walks on, it gets further into the gaps; Bourke finds that the form then becomes "more hairy and solid".³ Computer drawings of this kind were popular as computer art in the 1990s, often under the name Brownian trees.⁴ In DLA, as in Conway's Game of Life, only the rule is written; the form is left to evolve. See also Generative Art.

Paul Bourke, whose page on DLA this note draws on, read it in October 2026 and suggested one wording, which we have used.