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.

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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.

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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.

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

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

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

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