Thought Toys · Exhibit 22
A single seed sits in the middle. Specks drift in from the edges on aimless, drunken walks, and the instant one bumps the cluster, it freezes there. No plan, no blueprint — yet what grows is a branching tree of coral, frost, a bolt of lightning. The reason is quietly profound: the reaching tips catch the wanderers first, so the inside never gets a chance to fill.
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Each pale speck is a particle taking a random walk — a step north, south, east or west at random, over and over, like a mote of dust jittering in still air (that aimless drift is diffusion). It starts out near the rim and wanders until, by chance, it lands next to the frozen cluster. There it sticks, and a new walker is released. That's the whole rule.
From it grows a fractal: a shape that branches at every scale, the same feathery reaching whether you look at the whole dish or one twig. The cluster colours from a cool core to bright tips, so you can read its history — the centre froze first, the tips last. Crucially, it grows outward and stays mostly empty inside. Why? A wanderer coming from far away is overwhelmingly likely to brush a tip — a part that pokes out into the open — long before it can thread the narrow gaps down into the interior. The tips, by sticking out, cast a kind of shadow over the bays behind them. The rich get richer; the protrusions grow faster and the hollows stay hollow. (Physicists call it Laplacian screening — the same maths governs where lightning forks and how minerals dendrite.)
Now drag stickiness. At p = 1 a walker freezes on first touch, so only the exposed tips ever grow — wispy, lacy, mostly air (its fractal dimension is about 1.7, between a line and a filled disc). Lower it and a walker can graze the cluster many times without sticking, so it has time to wander into the bays before it finally catches — and the cluster fills in, growing rounder and denser, toward a solid blob. Same drifting particles, same dish; one slider slides the shape from coral to stone.
improve/verify/22-dla.js): on a 221² lattice the mass–radius dimension is
D ≈ 1.68 (well under 2), and lowering p from 1 to 0.2 shrinks the radius
of gyration at equal mass — the cluster genuinely densifies. Counter-example:
the Eden model — grow by occupying a random perimeter cell, i.e. growth with no
diffusion and no shielding — makes a compact blob with D ≈ 2; so it is the
wandering-plus-shielding, not mere random growth, that branches.
← the cabinet · Thought Toys — a cabinet of explorable explanations. Exhibit 22.