Diffusion-Limited Aggregation
One rule, and frost grows. A wanderer drifts in, bumps the cluster, and freezes on the spot. A few thousand more, and a fractal coral falls out of nothing else. Pick a seed, set the stickiness, or draw your own and let it bloom.
low = walkers slip deep before freezing = denser
how many wanderers freeze each frame
A seed and a handful of wanderers. Nothing looks like much yet. Give it a few seconds — the tips have to find each other before the shape commits.
Same one rule every time. The only things that change are where the seed starts and how readily a wanderer agrees to freeze.
The reframe
You just watched a fractal grow with no fractal anywhere in the rule. A particle that can only do one dumb thing — wander, then stick — builds a structure no one designed, every single time, and the structure has a measurable fractal dimension near 1.71 that the universe keeps landing on whether the particles are atoms, ions, soot, or pixels.
That is why this exact branching shows up everywhere the same physics runs: frost feathering across a cold window, lightning and the Lichtenberg figures it burns into wood, the black dendrites of manganese inside moss agate, copper and zinc plating out in an electrolysis cell, soot, coral, bacterial colonies starved for food, the deltas of rivers, and the branching of your own capillaries and airways.
The lesson the cluster keeps teaching: complex, organic-looking form does not need a complex cause. It often needs only randomness, a rule about contact, and the patience to let the tips outrun the bays.
The history
The model is young. Thomas Witten and Leonard Sander defined diffusion-limited aggregation in a 1981 paper, looking for the simplest thing that could explain how smoke particles and metal clumps grow into the same wispy, branched shapes. Their answer — wander, then stick — turned out to produce a fractal with a dimension around 1.71 in two dimensions, a number nobody has yet derived from first principles. The branching itself was photographed long before anyone had the math: Georg Christoph Lichtenberg captured electrical discharge figures in dust in 1777, and mineralogists had been puzzling over the “fossil ferns” of dendritic agate, which are not fossils at all but manganese DLA, for centuries. Today the same model is run on viscous fingering in a Hele-Shaw cell, electrodeposition, dielectric breakdown, and tumor-growth fronts. One rule, and the whole catalog falls out.