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.