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
selfsimilar pieces with magnification factor N into which the figure
may be broken.
For example, a square may be broken into N^2 selfsimilar
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 selfsimilar
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 halfdimensions
of the original triangle. So at this step of the construction, the Sierpinski
triangle is composed of three selfsimilar 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 selfsimilar
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.