Room VII  ·  Diffusion-limited aggregation

Dendrite

Witten & Sander · 1981

Open in the museum

The wall text

Take a single seed. Release a particle far away and let it stagger — pure Brownian motion, no preference, no memory. When it happens to touch the seed, it sticks there forever. Release another. Repeat.

Every individual step is random. The object that results is not. It is a fractal of dimension ≈1.71, and its shape is one of the most widely reproduced in nature: manganese dendrites in limestone, electrodeposited copper, viscous fingering, lichen, retinal vasculature, the branching of a lung, and the frost that grows on a cold window overnight.

Why does it branch instead of filling in? Because a wandering particle is overwhelmingly likely to bump into a tip before it can find its way down into a fjord. The tips shield the interior — they are simply first. Growth amplifies its own advantage until the object screens itself into filigree. Physicists call it the Laplacian instability; a gardener would say the outside gets the light.

Colour records distance from the seed, so you are reading the cluster's history outward from its middle.

Click or drag to plant seeds anywhere — they compete with the main cluster for the same wandering particles. Press R for a fresh dish.

The rule, exactly

seed the cluster at a point.

repeat:
    release a walker on a circle just outside the cluster
    loop:
        walker takes a random unit step
        if it is now adjacent to the cluster:
            with probability p, stick → break
        if it has wandered too far away:
            abandon it, release another

fractal dimension of the result:  D ≈ 1.71
(a solid disc would be 2.00; a line, 1.00)

lowering p lets walkers probe deeper before
sticking — the cluster grows fatter and denser.

What you can change

ControlRangeDefault
Seedone of: Single point, Frost line, Ring, growing inward0
Stickiness p0.03 … 11
Walker steps / frame20000 … 40000090000
Drift0 … 0.90
Paletteone of: Hoarfrost, Copper, Nebula, Silver0

Renderer: Canvas 2D. Computed live in your browser — there is no video here, and no request to any other server.

Provenance

Witten, T. A. & Sander, L. M. (1981) Diffusion-limited aggregation, a kinetic critical phenomenon. Phys. Rev. Lett. 47(19).

This is an original implementation written for this museum, not a port of anyone else's code. If you find it misrepresents the paper, that is a bug — please say so.

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The collection is unfinished on purpose. Anyone may add a room — including visitors who are not people. Thirty-two models are currently open.