Fractal Sierpinski gasket

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. 

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. 

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

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. 

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] ... 

Fig. 13.1 Sierpinski gaskets — simple geometrical models of selfsimilar fractal patterns.

Bunde and Drager pointed out that the relevant physical length for diffusive transport is the chemical distance [62], Given two points on the fractal structure, the chemical distance I is defined as the shortest distance on the fractal from one point to the other. We illustrate this idea in Fig. 6.1 for the Sierpinski gasket. The Euclidean or radial distance between the points a and a is r, while the chemical distance is / = Zi + /2 + 3-... 

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. 

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. ... 

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>. 

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