Fractals carpet

Here, dg is a spatial fractal dimension of the point set where relaxing units are interacting with the surroundings. R is the size of a sample volume section where movement of one relaxing unit occurs. Rq is the cutoff size of the scaling in the space or the size of the cooperative domain. G is a geometrical coefficient of order unity, which depends on the shape of the system heterogeneity. For example, the well-known two-dimensional recurrent fractal Sierpinski carpet has dG = ln(8)/ ln(3) 1.89, G = V3/4 0.43 [213],... 

Fig. 1.4. The successive initial stages of constructing a Sierpinsky carpet. The fractal is obtained at the infinite limit of this construction process. The empty squares are shaded. At each step of iteration, the linear dimension of the carpet is increased by a factor 3, while its mass increases by a factor 8.

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... 

Determine the fractal dimension of a Sierpinski carpet (see Fig. 1.28), constructed by dividing solid squares into 3x3 arrays and removing their centers. 

Calculate the fractal dimension of a Menger sponge (see Fig. 1.29), a three-dimensional version of the Sierpinski carpet. A solid cube is divided into 3x3x3 cubes and the body-center cube along with the six face-center cubes... 

Betti numbers can be applied to prefractal systems. For example, Fig. 3-7 shows two deterministic Sierpinski carpets with the same mass fractal dimension, dm = 1.896 and Euler-Poincare number, En = 0. The two constructions are topo-... 

Wheatcraft et al. (1991) considered flow and solute transport in a medium composed of high and low sat distributed according to a Sierpinski carpet fractal, reminiscent of low permeability pebbles distributed in a high permeability matrix. A multigrid solver was used to compute the flow field (Fig. 3 1B) and a particletracking algorithm was used to determine the tracer motion. No diffusion was considered. They found that dispersion increased with the scale of the simulation faster than could be predicted with other models. 

Garrison, J.R., Jr., W.C. Peam, and D.U. Rosenberg. 1992. The fractal Menger sponge and Sierpinski carpet as models for reservoir rock/pore systems I. Theory and image analysis of Sierpinski carpets. In Situ 16 351-406. 

At the same time that he coined the term fractal , Mandelbrot [4] pointed out that fractal dimensions would not suffice to provide a satisfactory description of the geometry of lacunar fractals, and that at least one other parameter, which he termed lacunarity , would be necessary. The key reason for this requirement is vividly illustrated by the fact that Sierpinski carpets (Figure 2.15) with greatly different appearances can have precisely the same fractal (similarity) dimension. Therefore, the fractal dimension alone is not a very reliable diagnostic of the geometry and properties of lacunar fractals. For physical objects, such as porous media, where the geometry of interstices and pores influences a wide range of properties, this means that any attempt to find a unique relationship between the fractal dimension of these objects and, for example, their transport or dielectric properties is most probably doomed to failure, unless one also takes lacunarity explicitly into account. 

Figure 2.15 Examples of 2nd iterate prefractals of Sierpinski carpets having different appearances, but identical fractal (similarity) dimension. In all three cases, the iterative construction process consists of dividing the initiator in 7 x 7 = 49 squares, and removing 3x3 = 9 squares. This process is associated with a similarity dimension = ln(49 — 9)/ In 7 = 1.8957.

Of course there are many unsolved problems, and possible directions for further research in this area. The most interesting problem would be to try to extend these exact solutions to some fractals with infinite ramification index. There are some studies of statistical physics models of interacting degrees of freedom on Sierpinski carpets, using Monte Carlo simulations, or approximate renormalization group using bond-moving, or other ad-hoc approximations. An exactly soluble case would be very instructive here. 

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. 

Self-similarity, highlighted by renormalization, represents the essence of fractals. Sierpinski Let us consider what is called the Sierpinski carpet (Fig. I5.6.a). 

Fig. 15.6. Fractals, (a) SierpMski carpet, (b) Mandelbrot set. Note that the incredibly complex (and beautiful) set exhibits some features of self-similarity, e.g., the central turtle is repeated many times in different scales and variations, as does the fantasy creature in the form of an S. On top of this, the system resembles the complexity of the Universe using more and more powerful magnifying glasses, we encounter ever new elements that resemble (but not just copy) those we have already seen. From J. Gleick, Chaos , Viking, New York, 1988, reproduced by permission of the author.

It was found subsequently that, although fractal geometry produces CPE behavior, in practice there is no relation between the CPE exponent and fractal dimensions [333, 334]. Qualitatively, however, higher fractal dimensions lead to smaller values of different type of fractals like Cantor bars [335-338] or Sierpihski carpets [339-341], for which different relations hold. This means that the impedance technique does not allow for the determination of the surface fractal dimension. Such information can be obtained by the analysis of current-time curves in the presence of diffusion to the surface [323, 324,342-344]. 

To account for the cathode CL (PEM) transport properties changes induced by carbon corrosion (ionomer degradation) we can use, for example, spatially-averaged fractal representations of the CLs to describe the impact of carbon (ionomer) mass loss on the microstructural properties changes, such as the evolution of the carbon instantaneous surface area or effective diffusion coefficients in the CL. We have used this approach for example ini d68,i79 relate the temporal evolution of the cathode thickness and carbon surface area with the carbon corrosion kinetics, by representing the carbon phase as a two-dimensional Sierpinski carpet projected in the cathode thickness direction (Fig. 11.11). 

The following equation was obtained for exponent r at simulation of fractal structure as Serpinsky carpet [1] ... 

The following equation for the exponent q was obtained at a fractal structure simulation as Serpinsky carpet [75] ... 

Another way in which the concepts of fractal geometry have been applied to the description of a porous body is to use what is known as a Sierpinski fractal. Mandelbrot, in his development of the concepts of fractal geometry, drew attention to some mathematical models originally developed by Sierpinski (accessible discussion of the original work of Sierpinski is to be found in Mandelbrot s book 41]. (See also reference 31.) Sierpinski s work dealt with the structure of porous bodies and the mathematical curve which is known as the Sierpinski s carpet. The basic concepts involved in draw-... 

Figure 9.11. A mathematicaJ curve known as a Sierpinski Carpet can be used as a model for important characteristics of a porous body, a) Repeated application of the algorithm for creating a Sierpinski carpet results in a figure with no area remaining. Shown are the first three steps in the construction, b) Determination of the Sierpinski fractal fi om the size distribution of the holes to stage three, c) Carpet of (a) with its holes randomized, d) The carpet of (c) disappears more slowly than (a) because some of the holes overlap.

In Figure 5 the construction algorithm for a fractal curve known as the "Sierpinski Carpet is shown. By varying the construction algorithm various carpets can be constructed and they are characterized by... 

P = residual area of the carpet after holes to size a are removed 6s = Sierpinski fractal dimension... 

According to Family s classification [8], fractal objects can be divided into two main types deterministic and statistical. The deterministic fractals are self-similar objects, which are precisely constructed on the basis of some basic laws. Typical examples of such fractals are the Cantor set ( dust ), the Koch curve, the Serpinski carpet, the Vichek snowflake and so on. The two most important properties of deterministic fractals are the possibility of precise calculation of their fractal dimension and the unlimited range (- o +°°) of their self-similarity. Since a line, plane or volume can be divided into an infinite number of fragments by various modes then it is possible to construct an infinite number of deterministic fractals with different fractal dimensions. In this connection the deterministic fractals are impossible to classify without introduction of their other parameters in addition to the fractal dimension. 

For regular mathematical fractals of Cantor sets, Koch curves and Serpinski carpets, constructed by recurrent procedures, the Renie dimension d does not depend on q, but on [14] ... 

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