Fractal dimension

This part present the calculation of the fractal dimension of the Sierpinski triangle.

• DEFINITION OF THE DIMENSION
The dimension is simply the exponent of the number of self-similar pieces with magnification  factor N into which the figure may be broken.
For example, a square may be broken into N^2 self-similar pieces (squares in this particular case) with magnification factor N.  So the dimension of a square is 2.
In conclusion we could define the dimension of an object as follows :

dimension = log(number of self-similar pieces) / log(magnification factor)

• DIMENSION OF THE SIERPINSKI TRIANGLE
With the definition above, we can calculate the fractal dimension of the Sierpinski triangle.
In the previous part of the report, we have said that after  the first iteration, there remains three triangles with half-dimensions of the original triangle. So at this step of the construction, the Sierpinski triangle is composed of three self-similar pieces with magnification factor equals to 2.
If we call S the Sierpinski triangle dimension :

S = log(3) / log(2) = 1.58

To verify the value of the dimension, we can also think that the Sierpinski triangle, after N iterations, breaks into 3^N self-similar pieces each with a magnification factor of 2^N. So in this general case, the fractal dimension is :

S = log(3^N) / log(2^N) =(N log3) / (N log2) = log(3) / log(2) = 1.58

To conclude, we can say that the fractal dimension of the Sierpinski triangle is 1.58, somewhere between 1 and 2 because it is "larger" than a line but not a plane.