But it turns out that as soon as one tries to make a more complete model, there are immediately an immense number of issues that arise, and it is difficult to know which are really important and which are not. At a basic level, one knows that snowflakes are formed when water vapor in a cloud freezes into ice, and that the structure of a given snowflake is determined by the temperature and humidity of the environment in which it grows, and the length of time it spends there.

The growth inhibition mentioned above is a result of the fact that when water or water vapor freezes into ice, it releases a certain amount of latent heat—as the reverse of the phenomenon that when ice is warmed to 0°C it still needs heat applied before it will actually melt.

But there are also many effects. The freezing temperature, for example, effectively varies with the curvature of the surface. The rate of heat conduction differs in different directions on the hexagonal grid. Convection currents develop in the water vapor around the snowflake. Mechanical stresses are produced in the crystal as it grows.

Various models of snowflake growth exist in the standard scientific literature, typically focusing on one or two of these effects. But in most cases the models have at some basic level been rather unsuccessful. For being based on traditional mathematical equations they have tended to be able to deal only with what amount to fairly simple smooth shapes—and so have never really been able to address the kind of intricate structure that is so striking in real snowflakes.

But with models based on simple programs such as cellular automata, there is no problem in dealing with more complicated shapes, and indeed, as we have seen, it is actually quite easy to reproduce the basic features of the overall behavior that occurs in real snowflakes.

So what about other types of crystals?

In nature a variety of forms are seen. And as the pictures on the facing page demonstrate, the same is true even in cellular automata with very simple rules. Indeed, much as in nature, the diversity of behavior is striking. Sometimes simple faceted forms are produced. But in other cases there are needle-like forms, tree-like or dendritic forms, as well as rounded forms, and forms that seem in many respects random.

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From Stephen Wolfram: A New Kind of Science [citation]