The Mathematics of SnowFlakes.

Living in Alaska, I see snow flakes so much of the winter.  I see them when they fall gently from the sky or when the wind rips it across my face.  I see them piled all over my porch after a night of snow fall and I swear when I have to shovel it off the porch.  We all know that no two snow flakes are alike but that still doesn't mean there is no math involved when talking about them.

By definition, a snowflake is an ice crystal that follows a specific pattern as it grows. It begins with a single ice crystal in the center with smaller crystals growing on the sides eventually creating a more complex shape.


Snowflakes are beautiful, mathematically speaking due to patterns, symmetry, and breaking of symmetry.  If you define a pattern as a repeated design, then snowflakes meet the definition.  If you've ever had the chance to examine them, you'll see beautiful patterns.

One man, Wilson Bentley, aka the Snowflake Man, took pictures of snowflakes through out his life and examined each one until he died.  This man made the comment that no two snowflakes are alike. He based the comment on his examination of thousands of snowflakes.  Everyone who has tried to find a counter example to disprove the comment have not succeeded.

Snowflakes range in shape from small hexagonal prisms, to needles, to columns, to what we think of as snowflakes with their hexagonal shape like in the picture.  These more complex shapes grow from the hexagonal prisms.  In fact, mathematicians see snowflakes as fractals.  The most famous one is the Koch Snowflake created by Helge Von Koch.

Most large snowflakes display symmetry in their hexagonal shape.  The cool thing about this particular type of symmetry is that if you rotate the snowflake by 60n degrees, it is exactly the same.  In addition, if you look carefully, most snowflakes have multiple lines of symmetry, usually a 6 fold symmetry.

It is possible to perform computer modeling which produces a fairly accurate picture of naturally produced snowflakes.  For instance, Reiter's Model creates shapes found in nature as certain variables are changed.  Gravner and Griffeath refined Reiter's model so that it looks at whether the cell is frozen, semi liquid, ice or vapor so there are more control variables and the computer produced dendrites and plates look like those found in nature.

If you'd like to see more of the computer modeling paper on Reiter's Model and the second one, look here.  Keep your eyes on here because in the near future, I'll discuss Koch's snowflake in more detail and how to draw your own, using equilateral triangles.

Let me know what you think, I'd love to hear.  Have a great day.