How do frost, lightning, and mineral dendrites get their branching fractal shape — and how fractal is it?
▶ Launch the interactive simulationGrow a cluster from a seed by releasing random walkers that stick on first contact; measure D from the mass–radius scaling N(r) ∝ rᴰ over its scaling region, and — since one small cluster is a noisy sample — average D over many regrown clusters.
a branching, screened fractal with 1 < D < 2 (between a 1-D arm and a 2-D filled disk); ⟨D⟩ ≈ 1.65 (cluster-to-cluster spread ±0.06), stable across the cluster sizes reached, below the off-lattice canonical 1.71 — a square-lattice, cumulative-N(r) estimate honestly reads ~1.6–1.7 rather than exactly 1.71