Fractal dimension Sierpinski

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],... 

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-... 

Figure 4.9. The A = 123 Sierpinski gasket, a two-dimensional uncountable set with zero measure and Hausdorff (fractal) dimension 3/ 2 = 1.584962... the companion Euclidean lattice referred to in the text is a space filling triangular lattice of (interior valency v = 6.

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. 

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. 

This subsection is devoted to the Sierpinski gasket d = 2) and its corresponding sponge d = 3), further on called Sierpinski triangular lattices. This fractal is characterized by a mass fractal dimension ds = ln(d -I- l)/ln2, which depends on the embedding spatial dimension d (see Fig. 2 for examples in d = 2 and d = 3). Note that for Sierpinski lattices in general, the Euclidean distance r between two lattice sites scales as the topological distance , i r, so that there is only one mass fractal dimension ds, M. ... 

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

Configurations that are illustrated in Fig. 66 can be of help in the elucidation of structures of percolation clusters. We can see that an infinite cluster contains holes of all sizes, similar to the Sierpinski chain of Gasket Island. The fractal dimension, d, describes again how the average mass, M, of the cluster (within a sphere of radii, r) scales, i.e., following the simple power... 

From the scaling properties of G(x, t) one can derive that S = const(d, 0)T)ft/d with d = 2dj(2 + 9) the spectral dimension of the fractal. The growth of the cluster s sizes goes on until l L where L is the whole system s size. The further growth of clusters and accumulation of particles stop because the same quantity L is the characteristic scale of a pair of different particles created in the system according to [91] there is no accumulation effect when particles are created by pairs on fractals of the Sierpinski gasket type. 

Another well-known fract,al is the Sierpinsky gasket (Fig. 2.10). The Sierpinsky gasket consists of three congruent pieces. Magnified by 2 they are identical with the whole fractal. Therefore, the dimension of the Sierpinsky gasket is d = ln(3)/ln(2) 1.59. 

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. 

In the case of regular mathematical fractals such as the Cantor set, the Koch curves and Sierpinski gaskets constructed by recurrent procedures, the Renyi dimension d does not depend on q but [16] ... 

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.

We close this section by noting that the relations between the scaling exponent and the spectral dimension are very general. We will meet them again in Sect. 9.3, in the study of regular hyperbranched fractals. It is also noticeable that the inclusion of hydrodynamic interactions into the dynamic picture leads to the loss of scaling for Sierpinski-type polymers in the intermediate regime [116,117]. 

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