# Snowflakes

I have always loved snowflakes.  On a chairlift, with your head down into a biting wind, one is allowed about twenty minutes to study them settling on your clothes. They have two classes of symmetry; one is rotational and the other is reflectional.

In Group Theory seemingly disparate things can be classified together by their symmetries.

Snowflakes belong to Dihedral Group $$D_6$$, of order $$12$$.  They have six rotational symmetries and six reflectional symmetries.

 The Cayley Table

Fractals is another area of mathematics where snowflakes arise.

We have the Koch Snowflake. This is a finite area bounded by an infinitely long perimeter.

Conversely, we have the Gosper Curve, also known as the flowsnake, which is a plane-filling function.

Our young snowflakes must decide whether they want to settle on a firm base of older and wiser heads

or whether they wish to create an avalanche which brings everything crashing down.