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!
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!
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 ‘
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.