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The title says it all. Why are snowflakes symmetrical in shape and not a mush of ice?

Is it a property of water freezing or what? Does anyone care to explain it to me? I'm intrigued by this and couldn't find an explanation.

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When water freezes, you get ice. Ice, like many solid materials, forms a crystalline structure. In the case of water, the crystalline structure may be attributed to the hydrogen bond, a special kind of an attractive interaction.

So a big chunk of ice will have a crystalline structure - preferred directions, translational symmetry, and some rotational symmetries.

enter image description here

But what about a snowflake?

A snowflake differs from a big chunk of ice by its being small. Even more importantly, it is in the process of growing. You should think about the process in which the snowflake was created.

At the beginning, it was small. A few atoms formed a small piece of crystal. Such a small piece of crystal almost always has some hexagonal (or different) symmetry. What happens if you wait for a little while and it continues to freeze?

Well, the water molecules are added to the crystal because it's energetically favored: vapor turns into ice - and you need vapor to create snowflakes because liquid water freezes differently.

Nature adds one water molecule at a time. The molecules always try to choose the most energetically favored position on the frozen body. Because these laws of creation of a snowflake are symmetric with respect to the rotational symmetries, it follows that any symmetry that exists at the beginning - a hexagonal symmetry of a small number of molecules in the initial crystal - will be preserved. It's pretty much inevitable that all the arms are growing approximately equally, so the initial symmetry group is preserved and becomes a symmetry of a macroscopic object.

The more difficult question is actually why the snowflakes are so diverse and beautiful. Most likely, it is not an accident that they achieve one shape or another. Changing pressure, temperature, or humidity as a function of time changes the conditions that determine the optimal places where the new vapor molecule should be added. But even if pressure, temperature, and humidity depends on time, the hexagonal symmetry is still maintained.

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By the way, Luboš, we (well the people in charge of Stack Exchange, really) are trying to move away from greetings and sign-offs etc. See I thought I'd let you know so you could omit the "Dear ..." and "Cheers ..." in the future. – David Z Jan 24 '11 at 21:02
Understood and deleted, David. – Luboš Motl Jan 24 '11 at 21:12
I remember a rather simple and small program included in "fractint" (the ultimate freeware package on fractals). In this program a particle diffused around in a box, and clinged to a "seed" in the center. Result of that were forms very similar to snowflakes. "Snowflake" was the programs name, afIr. – Georg Jan 24 '11 at 22:25
Nice programs, Georg, at least to understand that a complexity may emerge - although the detailed agreement between the model and the reality may be completely coincidental, too. – Luboš Motl Jan 26 '11 at 9:12

K Libbrecht has a nice paper that answers your question in considerable detail and has some nice pictures-- his homepage: Scroll down to the article in American Scientist in his publications list "The Formation of Snow Crystals," K. G. Libbrecht, American Scientist 95, 52-59 (2007). View pdf.

the pdf is here

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Not all snowflakes are symmetrical. One can disrupt the symmetry quite easily by introducing impurities or some mechanical artifact. In nature, snowflakes have plenty of time to form and it is more natural for them to form symmetric shapes because of the molecular structure of water.

That is, when there is more time for the molecules to move about and position themselves they will do so in a way that is in accordance with some crystalline structure that the molecule can exhibit.(you can use other things besides water to get "crystals")

The nature of the shape depends on many many factors but to see that such diversity can come from something simple you just have to look at IFS's. Basically you take some very simple rules and generate a huge number of variations by making small changes in the rules.

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""Not all snowflakes are symmetrical."" This wording is not good. Crystals may be of higher/lower, simple/complicated symmetry, but all have some symmetry. If not, the object should'nt be called a crystal. The less beautiful snow crystals look like columns or needles, but even those have some symmetry. – Georg Jan 25 '11 at 12:33

Not quite an answer but the first attempt to explain the shape was published by astronomer Johannes Kepler in 1611, the original is in Latin - "Strena Seu de Nive Sexangula" (A New Year's Gift of Hexagonal Snow). There is an English translation ("The Six-Cornered Snowflake") available at Amazon and elsewhere.

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