Thursday, January 31, 2008

Beauty of Math - Fractals

This is the second part in a series of posts about this subject - beauty of math. This one is about fractals. You are also advised to read the first part - Algebraic surfaces.
Fractals are very common in our world. In fact most of the things we see can be called fractals, although they are not mathematical fractals.

Fractals were first discovered by Maldenbrot. He was a British mathematician. The discovery was triggered by a rather practical question - how can we measure the coast of Britain effectively?
He reasoned that if we will take a ruler with marks of 25 km long, we will get some length but it will not be correct, because the ruler was too large. So lets take a smaller one, 5 km. But still the measurement is not accurate. There are small parts of the coast line that were overlooked when the measurement was taken. The length we get with this ruler is however larger than with the previous one, because a smaller ruler allows us to get closer to the real coast line.
If by making the ruler smaller and smaller we get better approximations of the length of the coast, in order to get the actual length we need to use a ruler of zero length. But if this happens then the length will be infinite. However, the coast line has a finite length - you can take a tour around all of it.

If the coast line was a mathematical line, the measurement would be easy and independent of the ruler length. It is also not a plane. Thus its dimension is neither 1 or two. Which means that it must have a dimension that is not a whole number - a fractal.

From this thought we get the two main descriptions of a fractal - it is shape that has a fractional dimension. Also as it was with the coast, fractals have self similarity at high magnification.

The following videos are example of fractals. They were generated by a program called XaoS.


video

This one is the so called Koch snowflake

video


video

video

The last one is the Sierpinski carpet.

The quality of this videos is not would I would like it to be. But they give a good example to the beauty of fractals. If you want to see more fractals, you can also check my picasa album: Fractals.

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