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'What Shape is a Snowflake?' is the record of a personal journey, one that I have been making ever since I was born. It's not a chronological record, though: more a reorganization of a collection of scientific ideas. You'll find more than just snowflakes here, because the science and mathematics that explain snowflakes were derived from, and are inextricably linked to, more general questions about how nature makes patterns. But I promise that I will answer the riddle of the snowflake by the time we reach the end of the journey.       
        I've been interested in Nature's patterns for as long as I can remember. The earliest occasion I can recall happened when I was six, and a friend showed me some curious little five-pointed stars that he had found on the beach. They were parts of the stem of a fossilized sea-lily. I spent weeks looking for more, but I never found them. However, I did find some spiral fossils, ammonites. Those had fascinating patterns, too.       
        It took a few years more before anyone told me that these patterns had a mathematical basis. It took even longer before I understood that Nature shows mathematical regularities because the underlying physical laws are mathematical. And I had reached middle age before it occurred to me that this is only half an explanation, because the laws are mathematical on the level of atoms or galaxies, whereas the patterns that we see all around us are on the human scale.
        What is the link, and how does it work? That's a subtle question.
        Snowflakes have an unusual and puzzling combination of features. On the one hand, they are (often) sixfold symmetric, like the mathematician's hexagon but much fancier. They have tree-like branches, and—so the cliche goes—every snowflake is different. How does this strange mixture of regularity and infinite variety work? If the regularity is the result of mathematical laws, where does the variety come from? If the variety is the result of the complexities of stormclouds, or the rest of the universe, where does the regularity come from?
        Our world, even confined to the human level on the surface of this rather small and ordinary planet, is incredibly rich. There are patterns everywhere—rainbows, splashes, feathers, snail shells, grains of sand. And there are non-patterns—irregular, unpredictable things—everywhere too: weather, waterfalls, houseflies, mountains, cats.
        What does this astonishing mixture of such wildly different things signify?
        When we go to other scales, the problem becomes, if anything, even greater. The microscope shows us that there is as much variety in a drop of pond water as in a forest. The telescope shows us that there are cosmic patterns on the grandest of scales—the stately spiral of a galaxy, even the shape of the universe itself.
        Today's science and mathematics is beginning to uncover some of the mechanisms that lie behind the patterns of Nature. The most fundamental concept of all is symmetry. Symmetry characterizes patterns, not by how or why they arise, but by what sort of pattern they are. Symmetry alone is too rigid to explain all of Nature's regularities, but it can be combined with other concepts, such as chaos and complexity, to provide a unified framework for an amazing range of natural regularities. And, crucially, for things that look like irregularities, too.
        To mathematicians, their subject is one of great beauty and intellectual content. To many people it is anything but: a sterile world of pointless 'sums' and confusing symbols. What I want to do here is show you the beauty while skipping the sums completely. They have their own beauty too, but only to the acquired tastes of the specialists. The beauties of mathematical patterns, though, we can all appreciate.