Snowflake Crystal Growth

Growing

Crystal Statistics

Frozen cells 1

Timestep 0

Radius 0

Symmetry 6-fold

Crystal types

Low ρ → plates

Med ρ → dendrites

High ρ → fernlike

Vapor density (ρ) 0.40

Freezing threshold (β) 1.00

Simulation speed 5

Adjust ρ (vapor density) and β (freezing threshold) to change crystal morphology · Low vapor → compact plates · High vapor → branching dendrites · Press Reset to grow a new crystal

What’s happening

The Reiter model

This simulation implements a simplified version of Clifford Reiter’s 2005 cellular automaton model for snowflake growth. The grid uses hexagonal cells (implemented on a triangular lattice), where each cell holds a vapor density value. At each timestep, vapor diffuses between non-frozen cells according to a discrete Laplacian. When a non-frozen cell adjacent to the crystal accumulates enough vapor (exceeding the freezing threshold β), it freezes and joins the crystal.

Why six-fold symmetry?

Real snowflakes grow on the hexagonal crystal lattice of ice Ih (ordinary ice). The water molecules arrange in a hexagonal pattern, and crystal faces grow at rates that depend on the local vapor supersaturation. Since the initial seed is a single cell on a hexagonal grid, and the diffusion equation treats all six directions equally, the resulting crystal inherits the lattice’s six-fold rotational symmetry. Each arm “sees” identical conditions, so they all grow the same way — producing the classic snowflake shape.

Morphology diagram

The shape of a snowflake depends primarily on temperature and supersaturation. At low vapor density, growth is slow and produces compact hexagonal plates. As vapor density increases, the crystal tips grow faster than the edges (the Mullins-Sekerka instability), creating branches. At high vapor density, secondary branching occurs, producing the intricate fernlike dendrites. This is why real snowflakes show such variety — they encounter different temperature and humidity conditions as they fall through the atmosphere, and each arm records the history of that journey.

Diffusion-limited growth

The key mechanism is diffusion-limited growth: vapor molecules must diffuse through the surrounding air to reach the crystal surface. Protruding tips are exposed to more vapor than concave regions, so they grow faster. This positive feedback amplifies small perturbations into branches, which then develop sub-branches, creating the fractal structure. The same mechanism produces dendritic patterns in electrodeposition, mineral growth, and bacterial colonies — any system where growth is limited by the transport of a resource to the surface.

The real equation

d(x,t) → d(x,t+1) via diffusion: d' = d + D ∇²d
Freezing: if cell is boundary and d(x) ≥ β, freeze it
Boundary cells: non-frozen cells adjacent to frozen cells

The diffusion constant D = 1/12 for the hex grid ensures stability. The parameter ρ sets the initial and far-field vapor density, while β controls how much vapor a boundary cell needs to freeze. Together, these two parameters span the full morphology diagram from plates to ferns.