Watch random-walking particles build branching fractal crystals by sticking to a growing aggregate — the mathematics behind snowflakes, lightning, and mineral deposits.
Witten & Sander (1981) model. Fractal dimension D ≈ 1.71 in 2D. Blue = core, red = tips. Symmetry modes impose n-fold mirror axes.
random walk (4 directions, equal probability).A DLA cluster is a fractal: it exhibits self-similar branching at all length scales. The fractal dimension D ≈ 1.71 in 2D, measured by the box-counting or sandbox method. Because D < 2, the cluster occupies a vanishingly small fraction of the enclosing disc as it grows — the tips branch before the interior can fill.
The screening effect drives the branching: a random walker must pass through the outer tips to reach the interior, but the probability of doing so decreases exponentially with depth. Branch tips grow faster than gaps, amplifying any small fluctuation into a permanent branch.
D ≈ 1.71 in 2D (Witten-Sander universality class). N(r) ~ r^D: cluster mass grows sublinearly with radius.
Outer tips receive far more incoming walkers than interior concavities. Branching is inevitable and self-amplifying.
p=1 → classic sparse DLA (D≈1.71). p→0 → dense Eden model (D→2). Fractal dimension tunes continuously.
Imposing 4-fold or 6-fold symmetry mirrors each stuck particle, creating crystal-like snowflakes.
| Phenomenon | DLA connection | D (approx.) |
|---|---|---|
| Snowflake/ice crystal arms | Water vapour diffuses to ice surface — growth limited by diffusion through vapour | 1.7–1.9 |
| Lightning discharge | Charge carries diffuse and discharge channel grows tip-first by dielectric breakdown | ≈ 1.75 |
| Electrodeposition (zinc) | Metal ions diffuse in solution and deposit on electrode tip-first | 1.66–1.75 |
| Mineral dendrites (manganese) | Mineral ions percolate through rock fractures and deposit diffusion-limited patterns | ≈ 1.7 |
| Viscous fingering (Hele-Shaw) | Less-viscous fluid invades more-viscous fluid in porous medium — finger instability | ≈ 1.7 |
| Level | Topics illustrated |
|---|---|
| GCSE / A-Level | Fractals, random walks, probability, diffusion, crystal growth |
| IB / A-Level Further | Fractal dimension, box-counting, chaos and self-similarity |
| University Year 1–2 | Brownian motion, stochastic processes, Monte Carlo methods, PDE diffusion |
| Research / Extension | DLA universality class, Laplacian growth, harmonic measure, multifractal analysis |
The growth time scales as N·ln(r_max) because each new particle must perform a random walk of average length proportional to r_max to reach the cluster. The simulation runs multiple walkers in parallel and processes 80 steps per animation frame for speed.
The 6-fold symmetry mode mirrors each newly stuck particle to 5 additional positions rotated by 60°, 120°, 180°, 240°, and 300°. This enforces the same hexagonal symmetry as real snowflakes, which grow under the thermodynamic 6-fold symmetry of crystalline ice lattice.
Yes. Adding a drift velocity breaks the statistical isotropy of the random walk, causing the cluster to grow preferentially upward into a needle-like shape with a fractal dimension closer to 1.0 (a nearly 1D structure). This models directed growth like frost on a cold surface or anisotropic crystal growth.