Snowflakes, crystals and branching growth
A snow crystal grows from water vapor on a hexagonal ice lattice. Its symmetry is molecular; its branching detail emerges from diffusion, temperature, humidity and growth instability.
Six arms share an atmosphere
The crystal lattice favors sixfold symmetry. All arms experience broadly similar changing conditions as the crystal falls, so their growth histories correlate. Local fluctuations keep them from becoming exact mirror copies.
The six arms of a snow crystal grow through the same moist, cold air, so their broad developmental sequence is related. Molecules arrive stochastically, making fine details non-identical. Changes in temperature and supersaturation during the fall can alternately favor plates, columns and side branches. Symmetry records shared environment; differences record local growth history.
Tips grow faster
Protruding tips intercept more diffusing vapor and can advance faster than recessed regions, amplifying small irregularities. Side branches inherit the competition at smaller scales and can create dendritic, fern-like forms.
Protruding tips extend into a stronger diffusion field and receive more vapor. They advance further while recessed regions become screened, an unstable feedback that encourages branching. Surface kinetics and crystal anisotropy select favored directions. A realistic model therefore needs both transport through air and rules at the ice surface.
Not a Koch snowflake
The mathematical Koch snowflake follows one exact substitution forever and has a fixed dimension. Atmospheric ice crystals change habit with environmental conditions and stop at finite scales. The shared word snowflake describes appearance, not identical geometry.
The Koch snowflake has exact substitution, infinite perimeter and fixed sixty-degree geometry. A snow crystal is a finite three-dimensional growth object with statistical variation. Both offer an intuitive sixfold branching image, but neither dimension nor mechanism must agree. The comparison works as an introduction when its limit is stated directly in the caption.
Sources and further reading
This article summarizes the following specialist sources in original wording. Accessed and editorially reviewed 12 August 2026.
- A New System for Identifying Snow CrystalsAmerican Meteorological Society
- Fractal Geometry: Mathematical Foundations and ApplicationsWiley
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