There have been many questions and excellent answers in this community about the symmetry of snowflakes, e.g., here and here. There is however one aspect of snowflakes that does not seem clearly addressed: Why are snowflakes flat?
- One reason could be the crystallization conditions during the snowflake formation, which favor certain crystalline phases. Somewhat similarly to how carbon can crystallize into graphite, i.e., weakly connected graphene sheets ("a snowflake phase") and diamond (and a bunch of more complex forms), depending on the pressure, temperature, etc.
- Another possibility is a selection of flat forms as the snowflakes move through the atmosphere, while the rest are destroyed.
- Finally, the snowflakes may be really having a 3D structure upon formation, but polished (flattened) while moving through the atmosphere.
I am looking for answers based on credible scientific sources.
Update
As the diagram given in the @JohnDumancic answer below demonstrates, snow crystals grow in the form of plates at temperatures from 0 to $-3^\circ$C or below $-10^\circ$. The question is thus essentially reduced to the typical conditions (temperature and water vapour saturation) that exist in the snow clouds.
As noted in the comments, snow crystals galling from the sky are not always flat - so information on whether flat snowflakes are specific only to certain geographic regions could be also relevant.
