Ice Crystal Growth Lab (3D)

Why snowflakes have six arms

Ice Ih crystallises on a hexagonal lattice, so water molecules attach fastest along six symmetric directions. Vapour diffuses to the crystal, and any bump that sticks out reaches into richer vapour than a flat face: the Mullins-Sekerka instability. Tips outrun faces, side branches nucleate, and because all six arms live in the same air, they stay almost identical.

The Reiter cellular automaton

Each hex cell holds a water mass s. A cell is frozen if s ≥ 1 and receptive if it is frozen or touches a frozen cell. Receptive cells keep their water (v = s + gamma) and cannot diffuse it; the rest is mobile (u = s elsewhere, 0 in receptive cells). One step diffuses the mobile part: u' = u + (alpha/2) (mean of 6 neighbours of u - u), then s' = u' + v. The boundary is held at the background density beta. This is the model of Reiter (2005), using the diffusion-limited-growth rule of Gravner and Griffeath.

What the parameters mean

beta is the vapour density far from the crystal (a proxy for supersaturation): low beta gives thin branching dendrites, high beta gives fat plates. gamma is vapour added to boundary cells each step (a stand-in for the attachment kinetics): large gamma fills the gaps between arms into a plate. alpha sets the diffusion rate. Real crystals respond to temperature and humidity through the Nakaya diagram (plates near -2 C and -15 C, columns and needles near -5 C and below -25 C); this lattice model reproduces the plate/dendrite alternation qualitatively only, and it has no explicit temperature.

Diffusion law check

Each step a unit of mobile water jumps to one of 6 neighbours with total probability alpha/2, so the mean-square displacement grows as <r^2> = (alpha/2) t lattice units squared. The third check releases a lone pulse in a closed domain and compares the measured second moment with that formula. Closed domains also conserve water exactly (first check), and with no noise the frozen set must be invariant under a 60-degree rotation (second check).

Honest simplifications

The model is two-dimensional (real flakes are thin 3D plates with facet growth in the third direction), time is measured in automaton steps rather than seconds, the surface kinetics are collapsed into a single number, latent heat and the quasi-liquid layer are ignored, and the 3D page only extrudes the 2D result with a thickness that falls with cell age. It is a teaching model of pattern formation, not a predictor of any particular snowflake.

What to try

• Run the five presets and compare the Compactness readout: dendrites sit near 0.2-0.3, sectored plates about 0.5, solid plates above 0.8.
• Raise gamma from 0 to 0.05 at beta 0.4 and watch the arm gaps fill in.
• Switch the noise on, restart several times: no two flakes agree, yet each keeps a rough sixfold pattern. Set noise to 0 and the symmetry check returns 100%.
• Look at the vapour profile graph: the depleted vapour halo around the crystal is the reason tips outgrow faces.
• Turn on the age colouring and read the growth history of the flake from centre to tips.