Fractals Given-Mandelbrot

Figure 3. The recursive construction of the Given-Mandelbrot fractal for 6 = 3. (a) The graph of first order triangle.(b) the graph of a (r + 1) order triangle, formed by joining 6(6 + l)/2 r-th order triangles shown as shaded triangles here (c) graph of the 2nd order triangle.

The properties of a polymer chain with self-attraction on a fractal were first studied by Klein and Seitz [34]. They used the self-avoiding walks on the Sierpinski gasket, which is the 6 = 2 member of the Given-Mandelbrot family. We consider below the case of 3-simplex, which is somewhat simpler to treat. 

From the most general point of view, the theory of fractals (Mandelbrot [1977]), one-, two-, three-, m-dimensional figures are only borderline cases. Only a straight line is strictly one-dimensional, an even area strictly two-dimensional, and so on. Curves such as in Fig. 3.11 may have a fractal dimension of about 1.1 to 1.3 according to the principles of fractals areas such as in Fig. 3.12b may have a fractal dimension of about 2.2 to 2.4 and the figure given in Fig. 3.14 drawn by one line may have a dimension of about 1.9 (Mandelbrot [1977]). Fractal dimensions in analytical chemistry may be of importance in materials characterization and problems of sample homogeneity (Danzer and Kuchler [1977]). 

The properties characteristic to fractal objects were mentioned first by Leonardo da Vinci, but the term fractal dimension appeared in 1919 in a publication by Felix Hausdorff [197], a more poetic description of fractals was given by Lewis Richardson in 1922 [198] (cited by [199]), but the systematic study was performed by Benoit B. Mandelbrot [196], Mandelbrot transformed pathological monsters by Hausdorff into the scientific instrument, which is widely used in materials science and engineering [200-202]. Geometrical self-similarity means, for example, that it is not possible to discriminate between two photographs of the same object taken with two very different scales. 

Since Mandelbrot s original description (1977,1983), fractal geometry has found relevance in a number of scientific disciplines including aerosol technology and science [e.g., see Lovejoy (1982), Meakin (1983), Kaye (1984), Sheaffer (1987), Reist et al. (1989)]. Many applications are covered in some detail by Kaye (1989), so only a brief description of fractals is given here. 

In recent years much attention has been given to the application of fractal analysis to surface science. The early work of Mandelbrot (1975) explored the replication of structure on an increasingly finer scale, i.e. the quality of self-similarity. As applied to physisorption, fractal analysis appears to provide a generalized link between the monolayer capacity and the molecular area without the requirement of an absolute surface area. In principle, this approach is attractive, although in practice it is dependent on the validity of the derived value of monolayer capacity and the tacit assumption that the physisorption mechanism remains the same over the molecular range studied. In the context of physisorption, the future success of fractal analysis will depend on its application to well-defined non-porous adsorbents and to porous solids with pores of uniform size and shape. 

Recent advances in percolation theory and fractal geometry have demonstrated that Dc is not a constant when diffusion occurs as a result of fractional Brownian motion, i.e., anomalous diffusion (Sahimi, 1993). The time-dependent diffusion coefficient, D(t), for anomalous diffusion in two-dimensional free space is given by (Mandelbrot Van Ness, 1968),... 

Mandelbrot (24) has suggested a different approach to surface irregularity by using fractal dimensions. A recent proliferation of studies has substantiated the hypothesis of self similarity for a number of natural systems ( 71-73V By this approach, surface irregularity scaling is given by the fractal dimension D, whose range is defined as 2 < D < 3 and which is related to the surface area by the proportionality... 

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. 

The porosity and pore systems in cement-based composites may be one example of fractality, which is characteristic for these materials. Fractal quantity depends on the scale used to measure it for example, the fracture area of a concrete element cannot be determined in an unambiguons way if the method and scale of its determination are not given (cf. Chapter 10.5). The classification and analysis of a pore system and all quantitative results derived depend among other things on the method of observation and magnification of microscopic images (Mandelbrot 1982 Guyon 1988). 

Another very useful mathematical model has been proposed to specifically reveal the fractal characteristics of signals. A detailed description of this technique, also called rescaled range analysis or the R/S technique [where R or R(, s) stands for the sequential range of the data point increments for a given lag s and time t, and S or S(/,s) stands for the square root of the sample sequential variance], can be found in Fan et al. " Hurst and later Mandelbrot and Wallis have proposed that the ratio R(/,s)/S(t,s) is itself a random function with a scaling property described by relation (7.8), where the scaling... 

---