The tetrahedral shape of the molecules define the 6 sides. I believe that would be a constant.

Temperature, moisture, pressure and maybe more variables are the parameters.

A snowflake is a 3D recursive growth algorithm over time. The macroscopic shapes are inherent in an earlier seed of molecular structure and arrangement of other factors by chance and circumstance. As they form while falling through the air, I imagine the crystalline shape grows in three dimensions, stretched by gravity or air resistance. So I'd say their design, at any moment, is a slice of a fourth-dimensional structure-over-time that can be described by a mathematical function, which ends in a wet puddle on the ground.