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We get different kinds of snowflakes depending on the temperature and humidity.  The vertical axis here shows the humidity in grams of water per cubic meter.   The curve shows when the air is saturated: it can't hold any more water above that curve.  I'm no expert, but I bet snowflakes gradually shrink below this curve due to sublimation: ice turning to water vapor.  As you can see, there are no fancy snowflakes below this curve.   I don't know why this curve has a peak at -15 °C.

While this chart raises lots of questions, and I don't know the answers, I know the story is even more complicated than this, because as snowflakes fall or get blown upwards by updrafts, they encounter air with changing temperature and humidity!   Their ultimate shape is a record of their whole history.

I got this graph from the Alaska Lake Ice and Snow Observatory Network:

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I don't know much about this, but the ice crystals below the curve are responsible for the beautiful halos that can be observed sometimes when there are cirrus-like clouds (which often contain ice crystals).

A very good source to learn how the shapes of the crystals produce various halos is the website
In addition to the theory, there are galleries of pictures as well as a picture of the day section. And also, there's a halo simulator: depending on the position of the sun and the composition of the clouds, you can simulate what type of halos you may observe.

It's really a lot of fun to play with!
+Romain Brasselet said what I had intended to say :)
I also studied thoroughly when I was trying to understand the optics behind some halos I had seen.

But do we ever experience those prisms and columns on the ground? I can't recall that I have seen them...
+Jonas Neergaard-Nielsen - you can see nice photos of prisms and columns here, along with many other kinds of snowflakes:

Kenneth Libbrecht, who took these photos, says "These show real snow crystals that fell to earth in Northern Ontario, Alaska, Vermont, the Michigan Upper Peninsula, and the Sierra Nevada mountains of California."  So, I guess they do reach the ground.  Maybe only in cold places?
So many pretty crystals! An amazing variety - water is indeed a wondrous element :)

He says the columns are almost too small to see with the naked eye - so I probably just didn't look hard enough.
+Jonas Neergaard-Nielsen , when the temperature is below 0°c and the humidity high, but it's not snowing, then I would guess that the air is filled up with these plates and columns. A very cold fog is probably made of these.

Also, it is known that the occurrence of halos, and thus the presence of plates and columns, is strongly influenced by the latitude: you're unlikely to see halos near the equator (too warm even at high altitudes), and the most beautiful displays of halos are usually taken in Antarctica.
That's funny... just the other day I was wondering how it was snowing tiny needle shaped snowflakes. Now I know why! :)
Yes, the circumhorizon arc remains very rare, even where it's the less rare! Though I tried a lot, I've only seen it once in summer in Spain. It was even more a feat because the closer you are to warm seas or oceans (which is the case in Europe because of the Gulf Stream), the harder it is to observe halos. The sea/ocean probably acts as a heat capacitance that prevents the air to cool down quickly with altitude, and thus prevents the formation of ice crystals, though I'm not sure about this.
Plates and columns can also form by recrystallization within the snowpack in certain conditions, creating weak avalanche-starting layers.
+Romain Brasselet I've seen circumhorizontal arcs once or twice in the US, though its counterpart above the sun, the circumzenithal arc, is much more common.
As the video says, the reason why we get these complex, fractal-like shapes at high enough supersaturations is that instabilities form at the 120 degree corners of the snowflake - the corners have more water vapor available than other parts of the snowflake, so they will grow fastest. This leads to the branching we see in the plot, and the more supersaturation there is, the more complex the branching. But if the supersaturation is low, the corners will suck up all the water vapor available around them which inhibits corner growth and leads to more regular shapes.

Another interesting feature of the plot is that we tend to get plate-like snowflakes at some temperatures, and column-like flakes at others. The reason is that the the sizes of the snowflakes correspond to different faces of the ice crystal: the "basal" face on the top and bottom, and the "prism" face on the six sides of the hexagon. It is known and has been verified in the lab, that the growth rate of these faces is strongly temperature dependent. Curiously, the question of exactly why and how this temperature dependence occurs seems to be an open problem!

As for the snowflakes we usually see in everyday life, they are actually usually not "pristine" flakes like these. A couple of processes create even more variation in the shapes. Firstly, the flakes collide and stick to each other (aggregation). The dendrite-like flakes which form around -17 to -12 C are the most effective at aggregating because their branches interlock mechanically. Secondly, ice clouds often have some supercooled water around, and when the tiny supercooled droplets hit the ice crystals, they freeze immediately into spherical deformations on the crystal surface. 
thanks for the explanation. I just had not gotten that far in my reading yet.
+Jussi Leinonen - Thanks a million for these nice explanations!  I always feel like telling people "if you read my posts when they first come out, make sure to reread them after they get some comments, because the comments are the most interesting part!"  This is a great example.

Where did you learn this stuff?  Do you know a good book or review article on the formation of snowflakes?  It's just as interesting to hear about the open problems as the stuff that's known.  If I didn't have a million other things to do, I wouldn't mind being banished to Hokkaido and working on snowflakes for a few decades.
I don't have a book that is specifically about snowflakes, but for a general introduction to the topic, including atmospheric ice, I recommend Lamb & Verlinde's Physics and Chemistry of Clouds: My own contact to the topic is kind of secondary: my PhD research is/was (mainly) on measuring these things with precipitation radars.

The temperature dependence could be a really interesting problem to crack. It comes down to how fast new ice can attach to the crystal faces - but this is not as easy to model as it sounds, as the surface of an ice crystal is covered with a "quasi-liquid" layer: an area, a few molecules thick, where the organization and density of H2O molecules resembles liquid water more than either vapor or ice. For what it's worth, this layer is apparently the reason why skates glide so well on ice. As far as I understand it (and I might be wrong here), to get the quantitative answer to the temperature dependence  problem, you'd have to be able to model the ice/quasi-liquid/vapor interface properly.
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