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Symmetry Of Snowflakes

Written by Ian Stewart, Emeritus Professor of Mathematics and Digital Media Fellow

Published September 2010

How closely do we look at the snowflakes that can cause such disruption and yet bring so much beauty? Snowflakes, famously, are six-sided but they also have six-fold symmetry. Ian Stewart, Emeritus Professor of Mathematics, explains how the formation of ice crystals in clouds results in infinitely symmetrical snowflakes.

Children in snowWhen I was very small, in the days before central heating, I would often wake up on a winter morning to find the window covered in strange fern-like patterns. They were ice, frozen on the inside of the glass. We called them Jack Frost. Ten years later, I would go out into the falling snow with a magnifying glass from my stamp-collecting equipment, and look at another form of ice: snowflakes. And what I saw was a big scientific puzzle.

Snowflakes, famously, are six-sided. Some are featureless hexagons, but many are extraordinarily intricate, Jack Frost in miniature. So intricate that it is often said that no two snowflakes are identical. I’m not sure it’s possible to prove that experimentally—you’d have to examine every snowflake that ever fell. But there are so many different ways to put a snowflake together that it seems entirely likely. Except for the featureless hexagons, of course.

Not only are snowflakes six-sided: they have—to a very good approximation, ignoring damaged or partially melted ones, and so on—sixfold symmetry. The combination of order (symmetry) and chaos (irregular patterns of great diversity) seems difficult to explain. Are snowflakes made by a regular process, or a random one? It seems to be both. And so it is.

Johannes Kepler, famous for discovering that planets move in ellipses, was so intrigued by snowflakes that in 1611 he wrote a book about them: The Six-Cornered Snowflake. By pure thought, he was led to the suggestion that the hexagonal character of snowflakes arises because, on some microscopic level, they are made by packing lots of very tiny identical units together. Six identical coins fit exactly around a seventh to form a hexagon. Hexagons pack together like the cells in a honeycomb. Ice, said Kepler, is made from tiny hexagonal patterns, and snowflakes are crystals of ice.

What makes this idea all the more amazing is that in those days the atomic theory of matter was no more than obscure speculation by a few ancient Greek philosophers. We now know that Kepler was right. A snowflake is crystalline water, forming when water molecules pack tightly together. The commonest packing, which forms at normal atmospheric pressure and a temperature just below freezing, is almost a honeycomb.


Large stellar dendrite snowflake, around 7 mm. 8 shots averaged to boost signal-to-noise ratio. Glass background with backlight, additional lens Helios 44M-5, january 2013 by Alexey Kljatov/Flickr

A water molecule is a tetrahedron with an oxygen molecule at its centre, hydrogen atoms at two of the vertices, and nothing at the other two vertices. In one type of ice crystal, these tetrahedra stack together to form layers of hexagonal prisms arranged in a honeycomb pattern. The layers are not quite flat; they have slight dimples.

That explains how the sixfold symmetry might arise, but not the intricate and varied patterns. Those require two further ingredients: the microscopic growth of ice crystals, and the atmospheric conditions that create snow.

An ice crystal grows when molecules of water adhere to its surface. Certain combinations of humidity and temperature create conditions in which flat surfaces are dynamically unstable, an effect called the Mullins-Sekerka instability. In these conditions, if a flat surface accidentally develops a tiny bump, the bump grows faster than other nearby regions, amplifying the irregularity. A big enough bump is nearly flat, and becomes unstable for the same reason, so new smaller bumps proliferate. This process of repeated ‘tip-splitting’ leads to a fernlike pattern known as a dendrite. Dendritic growth causes the enormous variety of shapes seen in snowflakes, because the branching patterns are extremely sensitive to slight changes in humidity and temperature.

Such variations are unavoidable in clouds, where snow is born. Clouds are mainly composed of water vapour, and the atmosphere within them is in constant motion. Clouds form when a mass of warm, moisture-laden air runs into a region of colder air. Within the mixing zone, where the temperature change is most abrupt, excess water vapour condenses to form ice crystals, which can either grow into flakes of snow or dense lumps of hail. These circulate within the cloud, and eventually fall out of the bottom.

Ice crystals form simple hexagonal plates when the temperature is just below freezing and humidity is low; then the growing edge remains stably flat. But as the amount of water vapour increases, the Mullins-Sekerka instability sets in, generating fern-like crystals. Their precise shape depends on the exact sequence of conditions through which the snowflake passes as it makes its erratic journey through the cloud, accreting water molecules as it does so. These changes in humidity and temperature are erratic because the snowflake’s motion in the cloud is chaotic. But the flake is so small that all six corners undergo the same chaotic changes in humidity and temperature, so they all grow in essentially the same way.

Put any shape, however irregular, into a kaleidoscope, and you see a symmetric pattern with irregular details. In effect, this is how nature manages to make snowflakes that have sixfold symmetry but are otherwise enormously diverse. Each flake follows its own trajectory, experiences its own history, and constructs its own tiny crystalline record of its journey through the cloud... six identical copies, one at each corner.

Professor Ian StewartIan Stewart FRS is Emeritus Professor of Mathematics and Digital Media Fellow. His research interests include dynamical systems, bifurcation theory, pattern formation, and biomathematics. He is also a writer of popular science and of science fiction. He was awarded the Royal Society’s Michael Faraday Medal for furthering the public understanding of science, and has also delivered the Royal Institution’s Christmas Lectures.