3D Diffusion-Limited Aggregation
Particles wander randomly through space until they collide with a growing cluster, sticking on contact. The result is a branching fractal structure whose dimension emerges from pure randomness. Watch it grow in three dimensions, slowly rotating to reveal its intricate geometry.
About this lab
Diffusion-limited aggregation (DLA) is a process where particles undergoing Brownian motion cluster together to form intricate branching structures. First described by Witten and Sander in 1981, DLA produces fractal patterns that appear throughout nature — from the branching of lightning bolts and river networks to the dendritic growth of crystals and the structure of certain bacterial colonies. The key insight is that the tips of branches are more likely to capture new particles than the inner regions, creating a self-reinforcing instability that drives the fractal growth.
In three dimensions, DLA clusters have a fractal dimension of approximately 2.5, meaning they fill space in a way that is more than a surface but less than a solid volume. This fractal dimension emerges purely from the random walk statistics and the sticking rule — no complex instructions are needed to generate the elaborate structure. The simulation estimates the fractal dimension by measuring how the number of particles N scales with the maximum radius R of the cluster: if N ~ R^D, then D is the fractal dimension, computed as D = log(N)/log(R).
Reducing the sticking probability below 100% creates denser, more compact clusters. When a particle can bounce off and continue walking before eventually adhering, it penetrates deeper into the fjords of the cluster, filling gaps that would otherwise remain empty. This transition from spindly (high sticking) to compact (low sticking) aggregates demonstrates how a single parameter can dramatically alter the emergent geometry of a self-organized structure.