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Jingle flakes
by Citronella
2006-12-20 08:19:39
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We're getting close to that time of the year again, which means the big question is if we're going to have a White Christmas - after all, artificial snow can't do the magic. And it is not without good reason. Have you ever watched a snowflake drifting so gracefully through the cold winter air?

Now, this is a close-up of a snowflake. And if the picture is impressive, what's more impressive is the fact that this chaos isn't so unpredictable.

In fact, we even have the mathematical equation to create it. Let's see how it works.

Remember our napkin collections when we were kids? We took a napkin, fold it again and again and again and then cut little pieces from the corners. When we finally unfold it, we admired our piece of handmade art.

Now, we have a physical barrier here; we can fold but a limited number of times. However, maths exists exactly to question and overcome such barriers.

We'll try to "fold" many, many times - even infinitely.

Let's take a triangle. Imagine we divide each side in three equal parts, cut the middle part and fill the gap with a smaller triangle. What we get is almost our old triangle, only now it looks more like a star.

Since maths is all about asking questions, we wonder how it'd look like if we repeated our little experiment. The result is even more impressive.
And with one iteration more, it's breathtaking – and started reminding us less of a star and more of a snowflake.
We can continue doing this “cut and paste” as many times as we like – cutting lines and pasting triangles. We'll get a new, more complicated star each time and since we're adding lines, the perimeter will be longer each time. And since we can go on infinitely, its perimeter will tend to be infinite. And yet it encloses a finite area!

This line is the Koch Snowflake, named so after the Swedish mathematician Helge von Koch who described it appeared in a 1904 paper entitled "Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire". And it was the earliest fractal curves to have been described.

Now, if we want to get more technical, here are the equations as promised. The Koch snowflake can be completely described as a Lindenmayer System using the following definition:

Angle: π/3 (60°)
Axiom: F++F++F
Rules: F → F-F++F-F

So, with or without equations, what we do know is that no two snowflakes are alike – but they may derive from the same set of rules.

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