Snow Crystal Growth: How DLA Builds a Six-Armed Fractal

Growth at the diffusion limit

A snow crystal is not sculpted top-down; it accretes. Water vapour molecules drift through humid air by random walk (Brownian motion) until, by chance, one bumps into the crystal and freezes onto it permanently. This is the physical basis of Diffusion-Limited Aggregation (DLA), a model introduced by Witten and Sander in 1981: a seed particle sits fixed, and new particles are released one at a time from far away, wandering randomly until they touch the growing cluster and stick irreversibly.

The key physical idea is that growth is limited by the diffusion of material to the surface, not by the attachment step itself. Wherever the random walkers arrive first — typically the tips and outer branches, because they are statistically easier for a wandering particle to reach before threading deep into a crevice — growth accelerates. Corners and tips race ahead of the concave regions behind them, which starve. This positive feedback between exposure and growth rate is what produces DLA's signature fractal, branch-within-branch structure with a fractal dimension around 1.71 in two dimensions, well below the 2 of a solid disk.

Why six arms, not a random blob

Water freezes into a hexagonal crystal lattice: each water molecule's two hydrogen bonds and two lone pairs arrange into a tetrahedral pattern that, stacked in ice Ih, produces six-fold rotational symmetry. On a hexagonal simulation grid, a particle can only stick to one of six neighbour directions, so every arm that begins to grow inherits the same six preferred directions as every other arm. The result is a crystal whose branches all point the same six ways, no matter how chaotic the random walk that built each one — order from symmetry breaking, not from a director choosing the shape.

seed a single hexagonal cell at the centre
loop:
  release a walker at a random point on a large bounding circle
  step it with a random hexagonal-lattice random walk
  if walker is adjacent to the crystal:
      with probability p_stick: freeze it onto the lattice, done
      else: keep wandering (it may re-enter open space and be discarded if it drifts too far)
  else: continue random walk (reflect or discard if it strays outside the circle)

Temperature, supersaturation and the Nakaya diagram

Real snowflakes are far more varied than a single DLA run because temperature and supersaturation (how much excess water vapour is available beyond equilibrium) both change the growth mode. Ukichiro Nakaya catalogued this in the 1930s by growing crystals in a cold chamber and mapping shape against temperature and vapour pressure, producing the now-famous Nakaya diagram: thin plates near 0 °C and below −22 °C, needles and hollow columns around −5 °C, and the classic lacy dendritic stars around −15 °C where supersaturation is highest. Low supersaturation favours simple, compact, faceted crystals because growth is slow enough for the crystal to stay close to its equilibrium hexagonal shape; high supersaturation favours fast, unstable tip growth and elaborate branching — exactly the diffusion-limited regime DLA captures.

In the simulation, the sticking probability plays the role of this same knob. A sticking probability near 1 means a walker freezes the instant it touches the crystal, so growth is dominated purely by which arrival point a random walk reaches first — maximally diffusion-limited, maximally branchy. Lowering the sticking probability lets a walker bounce along the surface for a while before freezing, giving it a chance to find a smoother, more energetically favourable site and producing denser, more compact crystals, mimicking the effect of a lower supersaturation gradient in real ice growth.

No two are alike — and why that is inevitable

The oft-repeated claim that no two snowflakes are identical is, statistically, simply true: a single crystal accretes on the order of 10¹⁸ molecules along a path shaped by millions of independent random-walk arrivals, so the exact microscopic history is essentially unique every time. What is shared is the rule, not the outcome — six-fold symmetry from the lattice, fractal branching from the diffusion limit, and shape family from temperature. DLA is a good teaching model precisely because it separates these: the same stochastic process, replayed, never produces the same crystal twice, yet every replay obeys the same six-fold, branch-favouring statistics.

Frequently asked questions

Try it live

Everything above runs in your browser — open Snow Crystal — DLA Growth and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.