Hexagonal symmetry and branching complexity emerge from water molecules attaching one at a time.
A snowflake starts as a tiny hexagonal prism, then water molecules latch on one by one. The symmetry comes from the ice lattice; the complexity comes from the competition between diffusion and surface energy.
The bonding geometry of water favors a hexagonal arrangement. Vapor diffuses inward and freezes on contact with the crystal. Cold temperatures produce branches; warm temperatures produce compact plates.
Figure 1. Crystal growth on a hexagonal grid. Color encodes freezing order from deep blue (early) to pale ice (late). Pause to inspect the boundary; slow the speed to watch molecules attach one at a time.
Tips that poke into the vapor field grow faster because they intercept molecules first. This instability creates branches. Surface energy works against this, trying to keep faces flat.
Figure 2. The diffusion field (toggleable) shows vapor concentration as pale blue: deeper blue = more vapor, near-white = depleted. Protruding tips reach high-concentration regions and grow faster. Raise the supersaturation to sharpen the branching instability.
Each snowflake records the history of temperatures and humidities it encountered as it fell.
Figure 3. Different trajectories produce distinct morphologies: thin plates, stellar dendrites, and fern-like structures.