And a coral grows from a single pixel.
Drop a seed in the middle of an empty grid. Release a particle at the edge; let it stagger on a random walk. The instant it touches the seed, it freezes there. Release another. Then ten thousand more. A branching, dendritic shape grows — the same fractal coral that copper takes on an electroplating tank, that lichen draws on a wall, that a lightning strike leaves on bare skin.
Put one stuck pixel at the centre. Release walkers at the boundary that drift on pure random walks. The moment a walker touches the cluster, it freezes there forever — and a branching shape grows out from a single seed.
One seed pixel. A few thousand random walkers spawned around the edge of a circle. Each walker staggers — up, down, left, right with equal odds — until it grazes the cluster and locks on. Out comes the coral you see below.
Witten and Sander showed in 1981 that this two-rule game has a universal fractal dimension of ≈ 1.71 — independent of seed or lattice. The same shape returns in electroplated copper, lichen, bacterial colonies, and the Lichtenberg figures left by lightning.
Place a single stuck pixel at the centre of a grid — that is the seed. Spawn walkers at the boundary, each one taking a random step up, down, left or right at every tick. There is no force, no field, no plan; just diffusion.
The instant a walker lands next to any frozen cell, it sticks there and never moves again. That is the entire physics. No surface tension, no curvature term, no temperature — one binary rule, applied to every walker that wanders too close.
Tips grow faster than valleys — branches reach outward and starve the interior.
More walkers · faster growth. Lower stickiness · denser, blobbier cluster: walkers slip past tips before freezing.
A walker drifting in from far away is far more likely to brush a tip sticking out than to slip down into a valley between two arms. Tips grow faster, valleys grow slower, and the gap widens with every frame. The cluster sharpens itself into a coral — geometry alone, with no hidden parameter.
Walkers drift in from far away. Tips that stick out catch them first; valleys between arms see almost nothing.
Witten & Sander, 1981: the Hausdorff dimension of a 2D DLA cluster is about 1.71, somewhere between a curve (D = 1) and a filled disc (D = 2). Niemeyer, Pietronero and Wiesmann (1984) wrapped it in the broader dielectric-breakdown family; Sander (1986) collected the early evidence. The number is universal across seeds, lattices, and walker rules.
The Hausdorff dimension of a 2D DLA cluster. Universal across seed shapes and lattice types.
Electroplated copper deposited too quickly grows DLA-like dendrites. Lichtenberg figures — the branching scars left by lightning or high-voltage discharges in resin — share the same statistics. So do lichen colonies on rock, the tips of bacterial mats on agar, and the dendrites of pyramidal neurons in cortex.
Lower the stickiness and walkers bounce off the cluster a few times before freezing — the dendrite thickens into a denser blob. Swap the centre seed for a horizontal line and you get a frost-on-glass crystal; swap it for a ring and the coral grows inwards. Lift it into three dimensions and the fractal dimension shifts to about 2.5.
The Explorer lets you switch between point, line and ring seeds, push the walker count up to a thousand per frame, slide the stickiness, and watch the dendrite grow in real time. Everything you just read is one click away.
→ Open the ExplorerHow and where this technique lives in the world today.
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