Sierpinski fractal

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],... 

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

As a continued effort to prepared hexameric systems based on the self-assembly of directed bw-terpyridine monomers, several interesting in families, e.g. 43, of iron and ruthenium connectivity have appeared <06DMP413, 06DT3518>. But in the assembly process, the creation of a three-step procedure to the novel first nondendritic fractal 44 entitled the "Sierpinski hexagonal gasket" was reported <06MI1782>. 

For the contiguous fractal with dy < 2.0, Pajkossy and Nyikos gave the first experimental evidence of the validity of the generalized Cottrell equation.121 They prepared two kinds of partially active electrodes a regular fractal pattern with cly = (log 8)/(log 3) = 1.893 and a Sierpinski gasket1 with dF =... 

In order to extend the analytical equations to a fractal lattice, we will need the radial distribution function rdf(r) of the Sierpinski gasket, rdf(r) dr being the average number of sites with distance between r and r + dr from a given site. For fractal lattices one has... 

Sometimes, as in the case of particle segregation on fractals (e.g., the planar Sierpinski gasket discussed in Section 6.1) this effect indeed is self-evident [88-90]. Its analytical treatment for particle accumulation was presented in [91, 92] we reproduce here simple mesoscopic estimates following these papers. Particle concentrations obey the kinetic equations... 

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. 

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... 

Another example of a regular fractal is a Sierpinski gasket shown in Fig. 1.13. Start with a filled equilateral triangle [Fig. 1.13(a)], draw the... 

Polymers are random fractals, quite different from Koch curves and Sierpinski gaskets, which are examples of regular fractals. Consider, for example, a single conformation of an ideal chain, shown in Fig. 1.14. As will be discussed in detail in Chapter 2, the mean-square end-to-end distance of an ideal chain is proportional to its degree of polymerization. 

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... 

Figure 1 Generation of perfect fractals (the first four generations are shown). (A) The von Koch curve. (B) The Sierpinski gasket.

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. 

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