What determines the 6-fold symmetrical structure of a snowflake?

The answer to this lies all the way down with a single molecule of water.

We’ve all been taught that water is two Hydrogens and one Oxygen. What you might not have learned is that Water molecules have a very specific shape. Both Hydrogens bond to one side of the Oxygen with an angle of 104.4 degrees between them. This shape is what ultimately gives snowflakes 6 sides.

Image Credit: Wikipedia

(sidebar) I would like to say that water molecules look a little bit like Mickey Mouse ears, but I won’t because Mr. Mouse’s ears are patented, trademarked, and defended by lawyers that are rumored to be some odd human/pit-bull/gila monster chimera. (end sidebar)

Each atom is made of a positively charged nucleus and negatively charged electron(s). These electrical charges attract and repel each other. Positives repel positives, negatives repel negatives, and opposites attract. These charges form molecular bonds between the atoms and give molecules their shape.

In water’s molecular bonds the Oxygen shares two electrons with each Hydrogen. The O and H have joint custody of these electrons, but it isn’t fair. Oxygen’s nucleus is much bigger and the electrons spend most of their time hanging around it. The Hydrogen side of the water molecule misses the electrons and has a tiny positive charge. The Oxygen side has a corresponding tiny negative charge from the electrons hanging around. These charges are very weak compared to the atomic bonds, but they are strong enough to help arrange the molecules as water freezes to ice.

Print out a bunch of these and try arranging them. You will find that the weak charges naturally arrange water molecules into 6-sided shapes.

That is exactly what happens as water freezes. As you remove heat from the water molecules the weak charges help arrange them in a specific pattern and they get locked into a hexagonal shape. Here are some fantastic examples from Earth’s South Pole.

Image credit Crystal Faceting

These 6-sided crystals are the heart of a snowflake and the answer to your question. The snowflake has 6 points because it grows from a 6 sided crystal.

The step of growing these crystals is called Crystal Faceting. There is one more step in turning ice crystals to snowflakes. The other step is called branching.

In Branching, water molecules attach to the ice crystal in a random way. To attach, a water molecule has to come close enough to the crystal to be attracted and be cool enough not to bounce right off. As each face of the crystal grows it develops tiny bumps on it where these new molecules attach. These tiny bumps stick out further than the main body of the crystal. This causes a looping pattern:

The further the bumps on the crystal stick out, the more likely they are to catch another water molecule.
The more water molecules that stick to the bump, the further the bump sticks out.
(Repeat)

This loop continues over and over until the snowflake is too heavy for the wind to keep it aloft and it falls to the earth.

There is a fantastic snowflake reference page from some people at Caltech: Snowflakes and Snow Crystals.

They have developed procedures to grow “designer” snowflakes in their lab. The results are quite beautiful.