Self-similarity fractal construction

In the following section the power of the fractional derivative technique is demonstrated using as example the derivation of all three known patterns of anomalous, nonexponential dielectric relaxation of an inhomogeneous medium in the time domain. It is explicitly assumed that the fractional derivative is related to the dimension of a temporal fractal ensemble (in the sense that the relaxation times are distributed over a self-similar fractal system). The proposed fractal model of the microstructure of disordered media exhibiting nonexponential dielectric relaxation is constructed by selecting groups of hierarchically subordinated ensembles (subclusters, clusters, superclusters, etc.) from the entire statistical set available. 

A fractal that is not self-similar is constructed as follows. A square region is divided into nine equal squares, and then one of the small squares is selected at random and discarded. Then the process is repeated on each of the eight remaining small squares, and so on. What is the box dimension of the limiting set ... 

What does the orbit look like for Ooo It is an infinite (and therefore aperiodic) self-similar point set with fractal dimensionality, Dfractai 0.5388 [grass86c]. Figure 4.5 shows the first six stages in the Cantor-set like construction. 

The simplest fractals are mathematical constructs that replicate a given structure at all scales, thus forming a scale-invariant structure which is self-similar. Most natural phenomena, such as colloidal aggregates, however, form a statistical self-similarity over a reduced scale of applicability. For example, a colloidal aggregate would not be expected to contain (statistical) self-similarity at a scale smaller than the primary particle size or larger than the size of the aggregate. 

Fractals are self-similar objects, e.g., Koch curve, Menger sponge, or Devil s staircase. The self-similarity of fractal objects is exact at every spatial scale of their construction (e.g., Avnir, 1989). Mathematically constructed fractal porous media, e.g., the Devil s staircase, can approximate the structures of metallic catalysts, which are considered to be disordered compact aggregates composed of imperfect crystallites with broken faces, steps, and kinks (Mougin et al., 1996). 

According to the Family classification [9], fractal objects can be divided into two main types, namely, deterministic and statistical objects. Deterministic fractals are self-similar objects that can be precisely constructed on the basis of several fimdamental laws. 

Typical examples of these fractals are the Cantor set ( dust ), the Koch curve, the Sierpinski gasket, the Vicsek snowflake, etc. Two properties of deterministic fractals are most important, namely, the possibility of exact calculation of the fractal dimension and the infinite range of self-similarity -°° +°°). Since a line, a plane, or a volume can be divided into an infinite number of fragments in different ways, it is possible to construct an infinite number of deterministic fractals with different fractal dimensions. Therefore, deterministic fractals cannot be classified without introducing other parameters, apart from the fractal dimension. 

Let us consider the case of deterministic fractals first, i.e. self-similar substrates which can be constructed according to deterministic rules. Prominent examples are Sierpinski triangular or square lattices, also called gasket or carpet (in d = 2) and sponge (in d = 3), respectively, Mandelbrot-Given fractals, which are models for the backbone of the incipient percolation cluster, and hierachical lattices (see for instance the overview in Ref. [21]). In this chapter, however, we restrict the discussion to the Sierpinski triangular and square lattice for brevity. 

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