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The science
A snow crystal grows because water vapour diffuses to it from supersaturated air. Away from the crystal the vapour density ρ obeys the diffusion equation ∂ρ/∂t = Dv ∇²ρ (Dv about 2×10-5 m²/s in air); at the surface it is absorbed with a rate that depends on the crystal face and temperature. A bump that sticks out of a flat face reaches into fresher vapour and grows faster, which is the Mullins-Sekerka instability that turns a hexagonal plate into a branched dendrite.
The model is a Reiter-type cellular automaton on a hexagonal lattice. Every cell holds diffusing vapour u and, if it touches the crystal ("receptive"), non-diffusing mass v. One step is: (1) diffusion u ← u + D (mean6 neighbours(u) - u) with D = 1/2, giving a mean-square spread D n per n steps; (2) receptive boundary cells convert vapour to crystal mass and gain a little constant vapour γ; (3) a boundary cell with two or fewer frozen neighbours freezes when v ≥ β, one with three or more freezes when v ≥ 1, or when v ≥ α and the local vapour is below θ; (4) the other boundary cells lose a fraction μ of v back to vapour. Noise perturbs the vapour. Low β and strong melting produce side branches, large β and weak melting give compact hexagons.
Temperature and supersaturation enter through the Nakaya diagram: plates near 0 to -4 °C, needles near -5, hollow columns to -10, sector plates around -11, stellar dendrites near -15 (fastest, most branched at high supersaturation), plates again to -22 and columns below. The temperature sets the branching bias and the ratio of c-axis (thickness) to basal growth, so the extruded prism height of each cell grows with its age at rate proportional to that ratio: thin plates get thin, columns and needles get tall. The "journey" mode drifts the temperature during growth, which is how real crystals acquire sector-plate arms on a needle core.
Simplifications, stated honestly. The lattice is two-dimensional; the third dimension is a thickness law, not a solved diffusion field. A cell that becomes receptive absorbs its free vapour (the CA is not mass-conserving at the surface), so the parameters ρ, β, α, μ, γ are phenomenological functions of temperature and supersaturation tuned to reproduce the habit sequence, not measured kinetic coefficients. Model steps are not calibrated to seconds, surface-energy anisotropy, attachment kinetics and latent-heat warming are omitted, and the colours are artistic.