Sierpinski 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.

Fig. 13. Conceptual model of a fault damage zone using a Sierpinski carpet. The fault zone structure can be considered to be characterised by a population series (i.e., the number of faults of different sizes, 1 8 64) and a dimension series (i.e., the size cascade of the faults present, 1, 1/3, 1/9).

Figure 4. Constructing a Sierpinski carpet (a) quadrangle (b) triangle.

Consider a square with side 1. We divide it into nine equal squares and remove the central part. Then we repeat this procedure with each of the eight remaining squares (Fig. 4). Repeating the procedure n times (n —> oo), we obtain a shape that is called the Sierpinski carpet (Fig. 4). 

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... 

Sierpinski carpet) Consider the process shown in Figure 1. The closed unit box is divided into nine equal boxes, and the open central box is deleted. Then this process is repeated for each of the eight remaining sub-boxes, and so on. Figure 1 shows the first two stages. 

FIGURE 8.20 Radiation patterns of a two-iteration Sierpinski carpet array (a) for the simple case and (b) for 45° scanning... 

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. 

Figure 2.5 First stages of the construction of the Sierpinski carpet. 7o is the initiator. Ii is the...

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. 

This Subsection deals with the Sierpinski carpet (d = 2) and its corresponding sponge (d = 3), further on called Sierpinski square lattices (an example in d = 2 is shown in Fig. 6). Note that in this Subsection, we use the identical notation as in the previous Subsection discussing Sierpinski triangular lattices, avoiding a reiteration of Eqs. (17) to (22). Nonetheless, a major difference between the two types of Sierpinski structures is the ramification, as Sierpinski triangular lattices are finitely ramified, whereas Sierpinski square lattices are infinitely ramified. The infinitely ramification renders the application of RG technique elusive and has drastic consequences on the actual values obtained numerically for the exponents. 

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