Diffusion-Limited Aggregation 3D
Particles random-walk through three-dimensional space and stick on contact. The same fractal dendrites that grow in the 2D version now branch in every direction — building crystal structures you can rotate with your mouse. Snowflakes, mineral veins, and electrodeposition in full spatial glory.
Df ≈ 2.50 (3D DLA fractal dimension)
DLA in three dimensions
Diffusion-limited aggregation (DLA) was introduced by Thomas Witten and Leonard Sander in 1981. The 2D version produces flat dendritic fractals with dimension ~1.71. In three dimensions, the fractal dimension rises to approximately Df ≈ 2.50 — the cluster is more space-filling, but still branched and porous at every scale.
How the simulation works
A single seed particle sits at the origin. New particles are released from a spherical shell surrounding the growing cluster and execute random walks in 3D — stepping randomly along the six cardinal directions (or 26 neighbors) at each tick. When a walker lands adjacent to an occupied site, it sticks with the configured probability. If it bounces (low stickiness), it continues walking, which lets it penetrate deeper into the cluster and produces denser growth.
The screening effect in 3D
Just as in 2D, tips and promontories intercept incoming walkers before they can reach interior regions. This screening effect is why DLA clusters branch rather than growing into compact spheres. In 3D the effect is slightly weaker (the fractal dimension is higher), but the dendritic, tree-like morphology persists. The structure is closely related to Laplacian growth: the probability field of arriving walkers satisfies the Laplace equation, and growth concentrates where the gradient is steepest.
Real-world 3D dendrites
Three-dimensional DLA-like structures appear in electrodeposition (metal dendrites growing from solution), snowflake formation (ice crystals growing from water vapor), lightning (stepped leaders branching through air), and viscous fingering (when a less viscous fluid pushes into a more viscous one in a 3D porous medium). The 3D structure is typically inaccessible experimentally — which is why simulation is so valuable for understanding the geometry of diffusion-limited growth.