The Snowflake

What you're seeing

A single frozen cell sits at the centre of a hexagonal grid of moist air. It is colder than its surroundings, so water vapour condenses onto it and freezes — and the cell grows. But it does not grow into a smooth ball. Wherever the ice pokes out even slightly, that tip reaches into air that hasn't yet been depleted of vapour, so more water arrives at the tips than in the valleys. Tips therefore grow faster, poke out further, and gather still more vapour: a runaway feedback that splits the growing front into branches. Because the underlying lattice is hexagonal, the six directions are equivalent, so all six arms grow alike — and you get a six-pointed dendrite. No part of the crystal is following a blueprint; the snowflake's shape is an emergent consequence of one local freezing rule plus the diffusion of vapour.

The rule

Every cell holds a real number s — the amount of water there. A cell is frozen (part of the crystal) once s ≥ 1. The grid starts at a uniform background vapour level β everywhere, except the single centre seed set to 1. Each tick repeats four steps (this is Clifford Reiter's 2005 model):

• Receptive set. A cell is receptive if it is frozen or touches a frozen neighbour. These are the sites where ice is actively capturing vapour.
• Split. The water in receptive cells is held (it will not diffuse away); the water everywhere else is the freely diffusing field.
• Feed. Add a tiny constant γ of fresh vapour to every receptive cell — the trickle the crystal pulls in from outside.
• Diffuse. Average the free field with its six neighbours at rate α, so vapour flows from rich regions toward the depleted boundary; receptive cells also soak up whatever diffuses into them. Any cell whose total reaches 1 freezes.

That is the entire specification — a purely local rule, no global symmetry enforced by hand. The branching is a diffusion-limited instability: exactly the same mechanism that makes diffusion-limited aggregation and frost ferns branch. Tune β (humidity) and you slide between compact plates and feathery dendrites.

Why it matters

The snowflake is the textbook case of emergent natural form — and of the slogan "same rules, endless variety." Every snow crystal obeys identical physics, yet no two are alike, because each one rides its own meandering path through the cloud, meeting a different sequence of temperatures and humidities on the way down. The shape is a record of that journey. Here, nudging β mid-growth (or hitting randomize) stands in for the crystal drifting into wetter or drier air.

The famous six-fold symmetry is real, and it comes from the hexagonal arrangement of water molecules in ice — the lattice in this simulation is a stand-in for that molecular hexagon. The dependence of shape on temperature and humidity is captured by the Nakaya diagram (Ukichiro Nakaya, who grew the first artificial snow crystals in the 1930s): plates near −2 °C, needles near −5 °C, broad dendritic stars near −15 °C, and so on. The branching itself is a diffusion-limited instability, which is why the Atlas pairs this entry with its stochastic cousin, DLA: DLA grows a branched fractal from random walkers with no symmetry, while the snowflake grows a branched crystal from a deterministic rule on a hexagonal lattice with six-fold symmetry. Same instability, opposite ends of the order/randomness axis.

In the wild

Real snow crystals show all of these forms — plates, sectored plates, stellar dendrites, fern stars — and which one you get is set by the temperature and supersaturation as mapped by Nakaya's morphology diagram. The same diffusion-limited branching drives frost on a cold window, dendritic mineral growth (manganese-oxide "moss agate"), and the dendrites in solidifying metals.

Honesty matters here. Reiter's model is a qualitative caricature, not a physics engine. It genuinely captures the emergence — local rule → branched, six-fold crystal — and it reproduces the plate-vs-dendrite trend with humidity. But:

• The six-fold symmetry here is partly imposed by the chosen lattice. In a real snowflake the six-fold-ness is earned — it comes from ice's molecular structure — but on a perfect hexagonal grid with a single seed, the symmetry is also baked in by construction.
• The arms stay identical only because the environment is perfectly uniform. Real arms match because they share one history, not because anything forces them to agree; a stray fluctuation would break the match.
• The real physics — anisotropic attachment kinetics at the ice surface (Libbrecht) and faithful mesoscopic models (Gravner & Griffeath) — is considerably richer than this single-field rule. Treat the simulation as an intuition pump, not a measurement.

Try this

• Sweep the β (humidity) slider and watch the crystal morph from a compact plate at high humidity toward a fern-like dendrite as you lower it — the model's version of the Nakaya diagram's humidity axis.
• Switch presets: Stellar dendrite for the classic six-armed star, Fern for a fine feathery crystal, Sectored plate for a broad hexagonal plate.
• Hit randomize a few times. Each press re-seeds and jitters the humidity slightly, so you get a fresh, slightly different flake every time — same rules, never the same snowflake.
• Open DLA side by side and compare: both branch by a diffusion-limited instability, but one is a random-walk fractal and the other a symmetric crystal.

References

1. Reiter, C. A. (2005). "A local cellular model for snow crystal growth." Chaos, Solitons & Fractals 23(4), 1111–1119.
2. Gravner, J. & Griffeath, D. (2009). "Modeling snow-crystal growth: A three-dimensional mesoscopic approach." Physical Review E 79, 011601.
3. Libbrecht, K. G. (2005). "The physics of snow crystals." Reports on Progress in Physics 68(4), 855–895.
4. Nakaya, U. (1954). Snow Crystals: Natural and Artificial. Harvard University Press.
5. Witten, T. A. & Sander, L. M. (1981). "Diffusion-Limited Aggregation, a Kinetic Critical Phenomenon." Physical Review Letters 47(19), 1400–1403. (The branching instability's stochastic cousin.)