
The HausdorffBesicovitch dimension is the mathematical expression of the
dimension of objects. The generalized approach of Hausdorff makes this
definition useful for natual objects.
 SelfSimilarity Method
Take a line, a square and a cube, each with a unary length.
If you divide that length by two, keeping the same space, you finally
end up with two segments of halflength, four squares with each onefourth
of the original surface and nine cubes with each oneninth of the original
volume.
Again, dividing the unary length segment by three yields, respectively,
3 lines, 8 squares and 27 cubes.
Summarizing this into a table, we have:
Object  Divider  Object Property 
Line  1  1 
 2  2 
 3  3 
Square  1  1 
 2  4 
 3  9 
Cube  1  1 
 2  8 
 3  27 
Note: : Original Data Table corrected by John White (Sep. 2012)
Expressed mathematically, the property of the object, its length, area or
volume, noted N, is related to its dimension, noted D, by:
N = r^{D}
Taking the logarithm on both sides of the equation and solving for D, we
finally obtain the expression of the HausdorffBesicovitch dimension:
D = log (N) / log ( r)
Intuitively, N can be considered as the number of selfsimilar objects
created from the original object when it is divided by r.
Taking the cube as an example, replace N by 27 and r by 3, and you got
the dimension of the cube:
D = log (27) / log ( 3) = 3
Replacing the members N and r of the equation by the adequate values
and one can compute the HausdorffBesicovitch or fractal dimension of any
object.
 BoxCounting Method
The BoxCounting method is useful when the fractal curve fits the boxes of a
simple grid.
To use the BoxCounting method, draw a fractal curve into a square area.
Divide that area in small boxes so that the selfsimilarity drawings fit the
size of the small boxes.
As an example, the Squares Curve is drawn
on such a grid.
The number of boxes covered by the fractal curve is then determined and
the HausdorffBesicovitch or fractal dimension is then calculated using the
fundamental formula of the boxcounting method:
N = 1 / d^{D}
where N: number of boxes covered by the fractal curve
where d: size of the grid box (see below)
where D: fractal dimension of the curve.
This formula relates the fractal dimension with the size of the small boxes
that made the overall grid.
Solving for D, we got: d^{D} = 1 / N
Taking the logarithm on both sides gives:
log d * D =  log N
and then D = log N / log d
To simply the calculations, one can take the side length of the grid as a
unit value and define r as the number of small boxes along one side of that
grid.
The formula becomes: D = log N / log (1/r)
which, finally, gives: D = log N / log r
Woops, it's the selfsimilarity formula showed
above !!!
The number of boxes, noted N, is then recorded for different values of r.
Plotting log(N) on the yaxis against log(r) on the xaxis should give
a straigth line, whose slope is equal to the fractal dimension.

