Diffusion Limited Aggregation - Fractals from Random Walks

Random walkers stick when they touch the cluster → branching fractal emerges

Controls

Parameters

Target particles 3000

Stickiness (%) 100

Walkers at once 50

Statistics

Particles 1

Cluster radius 1

Fractal dim. (est.) -

Theoretical dim. ≈ 1.71

The Process

1. Seed particle at center
2. Random walker from edge
3. If touches cluster → stick
4. Repeat → fractal tree grows!

Color Scheme

Witten & Sander (1981)

Physicists Thomas Witten and Leonard Sander introduced DLA to model aggregation processes. Their Physical Review Letters paper sparked decades of research. The surprising result: purely random motion creates highly organized branching structures!

Brownian Trees

DLA clusters are called "Brownian trees" because they grow from random walks (Brownian motion). The branching occurs because outer tips are more likely to catch new walkers—inner regions are "screened" by the branches above them.

Nature's DLA

DLA appears everywhere: lightning bolts, frost on windows, electrodeposition, mineral dendrites, river networks, blood vessels, plant roots, coral growth, and even city sprawl! The same random-walk-plus-sticking rule underlies all these patterns.

Fractal Dimension ≈ 1.71

DLA clusters have fractal dimension about 1.71—more than a line (1D) but less than a filled circle (2D). This non-integer dimension captures how the cluster fills space: branches spread out but leave gaps. The screening effect creates this precise value!