Manfred comments on Reductionism - Less Wrong
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So this turns out to be a really cool question. Part of what makes snowflakes unique is that each one is grown in a slightly different environment, and over the course of the growth of a snowflake this has a startlingly big impact. There are some cool attempts to model this with nonlinear systems / differential equations, and it does seem to be the case that if you have uniform growth conditions, you can get really different looking snowflakes that are still symmetrical.