Snowflake Maths

The governing equation is the : [ \frac\partial c\partial t = D \nabla^2 c ] where ( c ) = water vapor concentration. On the crystal surface, the Stefan condition applies: [ v_n = D , (\nabla c \cdot \mathbfn) ] with ( v_n ) = normal growth velocity.

At the molecular level, water molecules arrange into a hexagonal lattice due to hydrogen bonding. This dictates the 6-fold rotational symmetry of a snowflake. snowflake maths

| Model | Approach | Produces | |-------|----------|-----------| | | Random walkers stick to cluster | Fractal, noise-driven shapes | | Phase Field | Solves coupled PDEs for phase & concentration | Realistic anisotropy | | Cellular Automaton | Rule-based growth on hexagonal grid | Stellar plates, dendrites | The governing equation is the : [ \frac\partial

This arrangement forms a hexagonal (six-sided) prism. Because the underlying molecular "blueprint" is six-sided, the macroscopic crystal reflects that same symmetry as it grows outward. Rotational Symmetry: A snowflake possesses 60∘60 raised to the composed with power This dictates the 6-fold rotational symmetry of a snowflake

Every snowflake begins as a microscopic dust grain around which water vapor freezes. The primary reason snowflakes have six sides (or points) is rooted in . The Hydrogen Bond: Water molecules ( H2Ocap H sub 2 cap O

She traced a shape on the condensation. "But that one, the big feathery one? That's a dendrite. It grew fast. It fell through a cloud thick with water vapor. The maths says that for every tiny bump that forms, the surface area increases, so it catches more water, so it bumps out further. It's exponential growth, Joey. It’s calculus, but the snow does the homework while it falls."