What you're seeing
A grid that starts almost empty: just a seed cell (or a row of them, or a scatter). Over and over, a single particle is released far out and made to random-walk — a drunkard's stagger, one step at a time in a random direction. It wanders until it happens to step right next to the growing cluster, and there it sticks, becoming part of the aggregate forever. Then the next particle is released. No particle has a plan; each just diffuses and freezes on contact. Yet from this the cluster grows a delicate, branching, tree-like shape — open and feathery, never a solid blob. The same rule, run long enough, always makes the same kind of object: a fractal.
The rule
The rule is almost nothing: random walkers stick where they first touch. The interesting question is why that produces branches instead of a filled disk. The answer is a feedback loop called screening. A walker arriving from far away is overwhelmingly likely to bump into a protruding tip before it can thread its way into a sheltered cove of the interior — the tips stick out into the flux of incoming walkers and intercept them first. So tips capture more particles than valleys, tips therefore grow faster, and growing faster makes them stick out even further. Any bump that gets ahead gets further ahead. This instability — the rich get richer, geometrically — is what fractures a smooth front into ever-finer branches. The interior is starved of walkers (it is "screened" by the tips) and stays open. That competition between protruding fingers, repeated at every scale, is the entire engine of the pattern.
In this simulation a contact in any of the eight surrounding directions counts, and you can tune the stickiness: the probability that a touch actually freezes the walker. Lower stickiness lets walkers brush past the tips and penetrate deeper before sticking, so the cluster fills in and grows denser and bushier; stickiness of 1 gives the sparsest, most tip-dominated dendrites.
Why it matters
Diffusion-limited aggregation, introduced by Witten and Sander in 1981, was a revelation because a rule a child could state produces a scale-invariant fractal with a definite, reproducible fractal dimension — about 1.71 for off-lattice DLA in two dimensions. Zoom into a branch and it looks statistically like the whole; the cluster has no characteristic length scale between the particle size and the cluster size. That a purely kinetic, far-from-equilibrium growth process should settle on a sharp, universal number put it alongside critical phenomena in physics — the paper's subtitle calls it "a kinetic critical phenomenon." DLA became one of the canonical models of pattern formation away from equilibrium, a touchstone for how complex morphology arises from simple stochastic growth.
In the wild
The DLA shape is everywhere a substance grows by capturing diffusing material at its surface:
- Mineral dendrites in rock — the black fern-like fans of manganese oxide on the bedding planes of limestone and "moss agate," often mistaken for fossil plants. They grow as solutions percolate and deposit, and their statistics match DLA closely.
- Electrodeposition — metal plating out of solution onto an electrode forms branching fractal aggregates, one of the cleanest laboratory realizations of DLA.
- Viscous fingering — push a low-viscosity fluid into a more viscous one between two glass plates (a Hele-Shaw cell) and the interface fingers and branches; in the right limit this maps onto the same mathematics as DLA.
- Lightning and Lichtenberg figures — the branching of a spark. Here we should be honest: lightning is not literal DLA. It is described by the closely related dielectric-breakdown model (Niemeyer, Pietronero & Wiesmann, 1984), in which the growth probability follows the local electric field rather than a diffusing particle flux. DLA is the special case of that family with a particular exponent; lightning, surface flashovers, and the fern-like Lichtenberg figures burned into wood or acrylic are members of the same branching-instability family, not the identical model.
Try this
- Lower the stickiness toward 0.1 and watch the dendrite thicken into a bushy, space-filling cluster — the walkers now slip past the tips and fill the interior.
- Switch the seed to Bottom line and grow a frost forest: dendrites competing upward from a floor, exactly as in electrodeposition fronts and window frost.
- Use grow 2000 repeatedly and watch the cluster radius readout climb while the shape stays statistically the same — that constancy of form across scale is the fractal.
References
- Witten, T. A. & Sander, L. M. (1981). "Diffusion-Limited Aggregation, a Kinetic Critical Phenomenon." Physical Review Letters 47(19), 1400–1403.
- Witten, T. A. & Sander, L. M. (1983). "Diffusion-limited aggregation." Physical Review B 27(9), 5686–5697. (The fuller treatment, including the fractal dimension.)
- Niemeyer, L., Pietronero, L. & Wiesmann, H. J. (1984). "Fractal Dimension of Dielectric Breakdown." Physical Review Letters 52(12), 1033–1036. (The lightning / dielectric-breakdown connection.)
- Meakin, P. (1998). Fractals, Scaling and Growth Far from Equilibrium. Cambridge University Press.