200.

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		1.4.4— More General Fractal Dimensions: The Hausdorff-Besicovitch Dimension

		The Hausdorff-Besicovitch dimension is usually what mathematicians mean when they say "fractal dimension."

		The formal definition of the Hausdorff-Besicovitch dimension is quite technical. Only a brief hint of the those details will be given here.

		The Hausdorff-Besicovitch dimension is similar, but not identical to, the capacity. In the capacity, we count the number of balls N(r) of a given radius r needed to cover an object. The equation d = Log N(r) / Log (1/r), implies that N(r) the number of balls needed to cover the object is proportional to r-d. The capacity dimension d is determined directly from how the number of balls needed to cover the object varies with the radius of the balls.

		In the Hausdorff-Besicovitch dimension, we cover the object with sets. We evaluate the sum of a function applied to the diameter of each covering set. This function is called the gauge function. Analogous to the capacity and self-similarity dimension, the gauge function that is used is to raise the diameter r of each set to the power s. The sum of the diameters of all the sets each raised to the power s is then computed. The behavior of this sum as a function of s is then studied in the limit as the diameter r of the sets approaches 0. As r approaches 0, this sum will grow very large if s is less than a certain number, and it will grow very small if s is greater than a certain number. The value of the number that separates these two types of behavior is called the Hausdorff-Besicovitch dimension.

		We can now see how the Hausdorff-Besicovitch dimension is similar to the capacity. The number of sets needed to cover an object is proportional to r-d, where d is the capacity. The number of sets times the diameter of each raised to the power s is thus equal to rs-d. As r approaches 0, this sum will grow very large if s<d, and it will grow very small if s>d. Thus the boundary between these limits occurs when s = d.

 

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