This part present the calculation of the fractal dimension
of the Sierpinski triangle.
The dimension is simply the exponent of the number of
self-similar pieces with magnification factor N into which the figure
may be broken.
DEFINITION OF THE DIMENSION
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)
With the definition above, we can calculate the fractal
dimension of the Sierpinski triangle.
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.