Hausdorff-Besicovitch dimension

The Hausdorff-Besicovitch dimension is now defined as the critical value do at which jumps from 0 to oo, i.e. 

The definition (8.2.2) of the Hausdorff-Besicovitch dimension can be applied to any set, fractal or nonfractal, scaling or nonscaling. Applied to our fractal A however, we obtain a paradox. It turns out that the... 

Hausdorff-Besicovitch dimension of A is 2, although its area in phase space converges to zero as the number of kicks approaches infinity. In order to resolve this paradox, the definition (8.2.2) has to be extended by including logarithmic corrections (Hausdorff (1919), Umberger et al. (1986)). The idea is to retain the general structure of (8.2.1), but to admit a larger class of functions than to counterbalance the proliferation of the number of boxes B e) for e 0. We define... 

If the limit (67) exists, then df is called the Hausdorff, or Hausdorff-Besicovitch, dimension. When covering the Cantor set with segments lif, the covered area (Cantor set measure) is equal to... 

The main feature of the definition in Eq. (76) is that max ln —> 0. Whence in general the Hausdorff-Besicovitch dimension is a local characteristic. For deterministic self-similar sets, the local Hausdorff-Besicovitch dimension coincides with the dimension of the set itself. For statistically homogeneous sets, however, the local Hausdorff-Besicovitch dimension may not coincide with the dimension of the whole set. 

The non empty limited set E C O is called a self-similar set if it may be represented as the union of a limited number of two by two nonoverlapping subsets Ei,i = 1, n (n > 1), such that E is similar to E with coefficient k. An arbitrary segment, the Sierpinski carpet and sponge are examples of self-similar sets. For fractal sets, the Hausdorff-Besicovitch dimension coincides with the self-similar dimension. 

For fractal systems, the Hausdorff-Besicovitch dimension is equal to the similarity dimension, that is, df - d . We consider the triangular Sierpinski carpet as an example (Fig. 4). The iteration process means that the triangle is replaced by N = 3 triangles diminished with similarity coefficient K = 1 /2. Thus, the fractal dimension and the triangular Sierpinski carpet similarity dimension are given by... 

Fractals are self-similar objects invariant with respect to local dilatations, i.e., objects that reproduce the same shape during observation at various magnifications. The concept of fractals as self-similar ensembles was introduced by Mandelbrot [1], who defined a fractal as a set for which the Hausdorff-Besicovitch dimension always exceeds the topological dimension. The fractal dimension D of an object inserted in a d-dimensional Euclidean space varies from 1 to d. Fractal objects are the natural filling of the gap between known Euclidean objects with integer dimensions 0,1, 2, 3. .. The majority of naturally existing objects proved to be fractal, which is the main reason for the vigorous development of the methods of fractal analysis. 

Fractal or Hausdorff-Besicovitch dimension of a pattern Diameter of an object... 

Fractal structures are self-similar in that the two-point density-density correlation function and their essential geometric properties are independent of the length scale [59,61-63]. In d-dimensional space, they can be characterized by fractal or Hausdorff-Besicovitch dimension Df [61,63,64]. The... 

For instance, the first iteration step of the Koch curve employs N = 4 elements of length L = 1/3, while the second one uses A= 16 elementary lines of length L = 1/9, etc. The exponent df in Eq. (4.7) takes the value 1.26 it is called Hausdorff-Besicovitch dimension or just/ractaZ dimension. 

Fractal Dimension Measure of a geometric object that can have fractional values. It refers to the measure of how fast the length, area, or volume of an object increases with a decrease in scale. Fractal dimension can be calculated by box counting or by evaluating the information dimension of an object. Generator Collection of scaled copies of an initiator. Hausdorff-Besicovitch Dimension Mathematical statement used to obtain a dimension that is not a whole number, commonly written as d = log (N)/ log (r). 

The complete fractal A as displayed in Fig. 8.2, was never completely analysed. Hillermeier et al. (1992) studied this problem for one embedding dimension defined by considering a horizontal cut through the fractal at n = 1.1. The reduced fractal, then, has a Hausdorff-Besicovitch... 

The important feature of all geometric shapes examined in this chapter is their self-similarity, that is, scaling invariance. The dimensions of such geometric objects can be defined using the Hausdorff-Besicovitch measure. 

In 1919, German mathematician Felix Hausdorff developed the concept of fractional dimension, a measure theory originated by Greek mathematician Constantin Caratheodory in 1914. Russian mathematician Abram Samoilovitch Besicovitch developed the idea of fractional dimension between 1929 and 1934. Taken together, the concepts dehned by Hausdorff and Besicovitch were used by Mandelbrot to... 

---