In Silico Flurries – Scientific American Blog Network

In “The Chemical Basis of Morphogenesis,” Alan Turing proposed that chemical substances (morphogens) can transform and spread through tissue to yield natural patterns—stripes, spots and spirals—seen on animals. Turing expressed this idea in the language of differential equations, whose initial conditions controlled the evolution of the patterns.

The complexity and detail in the patterns from such reaction-diffusion systems can be truly surprising—a relatively simple process and small number of parameters can yield an endless variety of familiar patterns and shapes. One season-appropriate example of this is snowflakes, known for their variety and complexity. 

Figure 1. An example of a snowflake grown using the Gravner-Griffeath model. Various amounts of ice (encoded by the blue tone) at different parts of the snowflake make up the six-fold radially symmetric shape. You can browse the details about this snowflake in our online snowflake collection.

Credit: Martin Krzywinski

Compellingly realistic snowflakes can be simulated using the Gravner-Griffeath model, which you can run yourself with code made available by the authors (Figure 2). The snowflake’s radial symmetry is inherited from the hexagon grid on which it grows. Three different kinds of mass are simulated at different grid sites: ice, quasi-liquid and vapor. Initially, ρ amount of ice is initialized at a snowflake site and ρ vapor mass everywhere else. The model then transports and transforms a fraction of these masses by diffusion, freezing (controlled by κ) and melting (controlled by μ and γ). For example, during freezing a fraction κ of vapor at a boundary mass becomes ice and the remaining fraction 1–κ becomes quasi-liquid. Boundary sites around the snowflake become part of the snowflake through a process of attachment, which is controlled by the parameters α, β and θ, which act as cutoff values. Once a site becomes part of the snowflake, all of its mass becomes ice that…

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