The Hausdorff-Besicovitch dimension is the mathematical expression of the dimension of objects. The generalized approach of Hausdorff makes this definition useful for natual objects.
• Self-Similarity 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 half-length, four squares with each one-fourth of the original surface and nine cubes with each one-ninth 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:

ObjectDividerObject
Property
Line11
22
33
Square11
24
39
Cube11
28
327
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 = rD

Taking the logarithm on both sides of the equation and solving for D, we finally obtain the expression of the Hausdorff-Besicovitch dimension:

D = log (N) / log ( r)

Intuitively, N can be considered as the number of self-similar 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 Hausdorff-Besicovitch or fractal dimension of any object.

• Box-Counting Method

The Box-Counting method is useful when the fractal curve fits the boxes of a simple grid.

To use the Box-Counting method, draw a fractal curve into a square area. Divide that area in small boxes so that the self-similarity 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 Hausdorff-Besicovitch or fractal dimension is then calculated using the fundamental formula of the box-counting method:

N = 1 / dD

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:   dD = 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 self-similarity formula showed above !!!

The number of boxes, noted N, is then recorded for different values of r. Plotting log(N) on the y-axis against log(r) on the x-axis should give a straigth line, whose slope is equal to the fractal dimension.