26 may. 2011

Blogs and Fractals

Have you ever made a mistake in a blog? I did it in my own blog. But this mistake allowed me to think in the possibility of fractals in internet, also I though in the Pink Panther.
Why?

See my blog carefully (I am also my follower) and let me explain a little bit about fractals.
Fractals are geometric objects in a fractional dimension…

Weird? Yes

Let me continue. We all know that our dimension is 3. We live in a tridimensional world (3D). Imagine a cube, it has 3 dimensions: long, width and high. If you consider only its base (the square), you have an object of 2 dimensions (2D). And if you take only one edge, you have an object of 1 dimension (1D). Well, if you consider only one point, you have the 0 dimension!
It is possible to obtain the volume of the 3D objects, the area of 2D objects and the long for 1D object.

But with fractals the situation is very different.

However, believe it or not, fractals are not imaginary things; they actually come from the real world!

The Polish mathematician Benoit Mandelbrot invented the name fractal in 1975. He used to say:
Clouds are not spheres, mountains are not cones, littorals are not circulars and lightning do not travel in straight lines.
We all know that, but let us think more carefully this situation.
Have you ever though how long is a littoral for example? Take a 1-meter rule and measure the littoral of a small island. Well at least imagine that. The error in the measurement will be related with our measurement device (rule). We will not be able to measure small curves in the littoral of lengths less than our 1-meter rule. If we use a smaller rule, we can measure those curves but not the smaller ones. On the opposite if we use greater rules (e.g. 1-kilometer rule), the mistake will be different. In fact, for every rule, we would obtain different longs.

Take a moment to see this beautiful real life fractal.
Smaller structures are similar than bigger ones for a broccoli and a tree (and for my blog and my smaller myself follower)

Let us back to the fractal dimension.
If you take the logarithm of island perimeter over the logarithm of the scale (the rule dimension), there is a constant relation between them. This is called the ‘Richardson effect’.

Maybe it is clearer to show you the development of a fractal in order to understand this.

Take a line of 1-unit length (this unit could be a meter, a decimeter or any measure that you like). Divide it in two halves. The number of segments is 2 and the rate of the number of segments divided by the original unit is 2.
Take the 1-unit line and divide it in three parts. The number of segments is 3 and the rate of the number of segments divided by the original unit is 3.
The dimension of the line could be calculated using the following formula that involves logarithms (in base 10):
Log (number of segments) / Log (rate)
For the division of halves would be Log(2) / Log(2) = 1
For the division in three parts would be Log(3) / Log(3) = 1
Always 1, no matter how many divisions we input in the formula.

Now, we will try similar calculations with a fractal known as Koch curve.
You can generate this fractal in this way:
Take a 1-unit line and divide it in three parts. Remove the middle part and put instead the segment, two sides of an equilateral triangle (without the base). You have 4 segments in total. And the rate of the number of segments divided by the original unit is 3.
Repeat this process for every segment again. This is the first level of the Koch curve. The fractal is, in fact, an infinite object, as you can guess.

In the first level, the Koch curve has 16 segments and the rate of the number of segments divided by the original unit is 9.
If we apply the formula for calculating the dimension with this numbers, we have:
Log(4) / Log(3) = 1.26
Log(16) / Log(9) = 1.26
Without having our neurons burned, le us try another level. The number of segments is 64 (do you want to count them?) and we divided the line in 1/27, it means the rate is 27. Substituting these values in the formula, we have:
Log(64) / Log(27) = 1.26
Well, this is the dimension f the Koch curve, a fractal dimension of 1.26. It means that Koch curve lives in a world higher than a line but lower than a plane!

My mistake let me thinking in the Pink Panther as well. Of course the problem is different because is related with relativity, but that is another story.


 

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