Release a random walker and let it stick where it lands to grow fractal trees that resemble lightning and coral.
Place a seed particle. Release another far away to walk randomly until it bumps the first and sticks. Repeat. The growth is a fractal tree of nested branches, not a round blob.
This is diffusion-limited aggregation (DLA): particles arrive slowly via random walks and stick fast. The arrival/sticking asymmetry produces fractals.
The simulation below places a seed at the center of the grid. Walkers are released from a circle around the cluster. Each walker steps randomly on a lattice: up, down, left, or right. When a walker lands next to an occupied cell, it sticks and becomes part of the cluster. Then a new walker is released.
Figure 1. DLA cluster growing from a single seed. Color encodes arrival order (dark = early, light = late). The fractal dimension is estimated from the radius of gyration as D ≈ log(N) / log(Rg); standard 2D DLA converges to D ≈ 1.71. Drag the speed slider to control how many walkers are processed per frame.
The structure grows one particle at a time from randomness and contact. No blueprint, no symmetry constraint, just the rule: stick where you land.
Why branches and not blobs? Screening. A random walker approaching the cluster is more likely to hit an outer tip than to reach an inner crevice. Tips grow faster: small bumps become big branches, big branches sprout their own bumps. Self-reinforcing.
Figure 2. Click "Show Heatmap" to overlay arrival probability on the cluster boundary. Bright = high probability (exposed tips). Dark = low probability (screened crevices). This gradient is why DLA produces branches, not blobs.
A walker from far away sees a silhouette: tips poke out, crevices hide behind them. The deeper the gap, the less likely a walker finds its way in. The same effect explains why lightning preferentially strikes protruding conductors.
A reduced sticking probability lets a walker bounce near the surface and explore deeper before sticking, defeating the screening effect. High stickiness amplifies branching; low stickiness fills space. Sticking probability controls fractal dimension.
Figure 3. Higher stickiness (left) amplifies screening and branching. Lower stickiness (right) allows walkers to penetrate deeper, producing denser and rounder clusters. The fractal dimension D reports how much each cluster fills space: expect the left to approach ≈ 1.71 while the right climbs toward 2 as stickiness drops. Adjust the slider, then click Reset Both to apply.
Arrival probability at a boundary cell equals the Laplacian field value there. The dielectric breakdown model (DBM) of Niemeyer, Pietronero, and Wiesmann makes this explicit: solve for potential φ with φ = 1 on the cluster and φ = 0 on a far boundary, then grow into an empty neighbor with probability proportional to |∇φ|η.
DBM contains DLA at η = 1. Higher η concentrates growth at the strongest-field tip, producing a needle. Lower η flattens the field bias, fattening the cluster toward a space-filling blob. The lightning bolt is the visible result of a stepped leader doing DBM search at η ≈ 1–2.
Figure 4. A stepped leader grows from top to bottom via DBM. At each step the Laplace equation is relaxed on the empty cells and growth picks a neighbor with weight |∇φ|η. Try the Bushy (η = 0), Natural (η = 1.5), and Needle (η = 4) presets. The background glow visualizes the electric potential concentrating at the tips.
Lightning, river networks, and DLA all produce branching trees because they are all Laplacian growth with different boundary conditions: lightning has a single source branching to a distributed ground, rivers have distributed sources converging to a single outlet, DLA has a radial source and point seed.
Lightning (η = 1, top-down)
River network (η = 0.5, bottom-up)
DLA (η = 1, radial)
Figure 5. Three instances of the same Laplacian growth rule. Only the boundary conditions differ: a line source at top for lightning, a line sink at bottom for rivers, a point seed in the center for DLA.
Lightning and rivers are dual: flip growth direction and the branching inverts, but the underlying Laplacian model is identical. A frozen lightning bolt looks like an inverted river delta because both grow where the gradient is steepest.
With multiple seeds, each cluster competes for incoming walkers. Branches from one cluster screen territory from others, producing Voronoi-like boundaries.
Each cluster captures walkers nearest to its exposed surface — the same pattern as mineral dendrites or frost crystals competing for moisture.
Click to place 2-4 seed points, then watch them compete.
Figure 6. Competing DLA clusters from multiple seeds. Boundaries emerge where clusters screen each other from incoming walkers.
Witten and Sander discovered DLA in 1981. The same growth rule explains electrodeposition, mineral dendrites, bacterial colonies, and dielectric breakdown — any system where particles diffuse and irreversibly attach.