Curve fractal/Koch

FIGURE 1 Two examples of linear fractals Koch curve (a) and self-avoiding random walk (b), representing a deterministic fractal and statistical fractal, respectively. 

Pig. 2.2 First three steps in the construction of the THadic Koch Curve. The fractal curve is obtained in the limit N 00 and has a fractal dimension Dfractal = In 4/ln 3 1.26. 

Figure 2. (a) A deterministic self-similar fractal, i.e., the triadic Koch curve, generated by the similarity transformation with the scaling ratio r = 1/3 and (b) a deterministic self-affine fractal generated by the affine transformation with the scaling ratio vector r = (1/4, 1/2). 

For example, the fractal dimension of the Koch curve is 1.2619 since four (m = 4) identical objects are observed (cf. levels i = 0 and i = 1 in Figure 1.1) when the length scale is reduced by a factor r = 3, i.e., dj- = In4/ln3 1.2619. What does this noninteger value mean The Koch curve is neither a line nor an area since its (fractal) dimension lies between the Euclidean dimensions, 1 for lines and 2 for areas. Due to the extremely ramified structure of the Koch curve, it covers a portion of a 2-dimensional plane and not all of it and therefore its dimension is higher than 1 but smaller than 2. 

Fractals in electrochemistry — Figure. A von Koch curve of Df = 1.5. Note that no characteristic length of the structures can be identified -this is associated with the fact that the size-distribution of the features of the curves is a power-law function... 

Fractals are self-similar objects, e.g., Koch curve, Menger sponge, or Devil s staircase. The self-similarity of fractal objects is exact at every spatial scale of their construction (e.g., Avnir, 1989). Mathematically constructed fractal porous media, e.g., the Devil s staircase, can approximate the structures of metallic catalysts, which are considered to be disordered compact aggregates composed of imperfect crystallites with broken faces, steps, and kinks (Mougin et al., 1996). 

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. 

Calculate the fractal dimension of the Koch curve in Fig. 1.27 with the center third of each segment replaced by three sides of a square (instead of two sides of a triangle as discussed in Section 1.4). 

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

But when we try to apply this definition to fractals, we quickly run into paradoxes. Consider the von Koch curve, defined recursively in Figure 11.3.1. 

A fractal possesses a dilation symmetry, that is, it retains a self-similarity under scale transformations. In other words, if we magnify part of the structure, the enlarged portion looks just like the original. Figure 5.15 shows a fractal shape, the Koch curve. If we magnify by three the part of the Koch curve in the interval (0, 1/3), it becomes exactly identical to the whole shape. The same is true if the part in (0,1/9) is enlarged... 

In order to get more experience with the newly proposed index ( ) we will consider the leading eigenvalue X of D/D matrices for several well-defined mathematical curves. We should emphasize that this approach is neither restricted to curves (chains) embedded on regular lattices, nor restricted to lattices on a plane. However, the examples that we will consider correspond to mathematical curves embedded on the simple square lattice associated with the Cartesian coordinates system in the plane, or a trigonal lattice. The selected curves show visibly distinct spatial properties. Some of the curves considered apparently are more and more folded as they grow. They illustrate the self-similarity that characterizes fractals. " A small portion of such curve has the appearance of the same curve in an earlier stage of the evolution. For illustration, we selected the Koch curve, the Hubert curve, the Sierpinski curve and a portion of another Sierpinski curve, and the Dragon curve. These are compared to a spiral, a double spiral, and a worm-curve. 

The calculated A. and < ) for the selected curves are sumarized in Table 12. The index n in (t indicates the length of the chain (or the size of the D/D matrix). We see that as n increases, (]) decreases, as expected. A comparison of different curves again gives plausible results. Among the fractals considered the Koch curve appears the least... 

This example brings to the fore another important feature of the problem, namely, that prior to an investigation, the minimum or threshold size of a representative sample usually is unknown. We may have good reasons for expecting the scale to exhibit such a threshold value but in most cases its precise value must be determined empirically. In this respect, the behaviors of self-similar systems of physical interest are expected to be more complicated than those of the deterministic fractals, of which the previously mentioned Koch curve is one example. 

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] ... 

Figure 6.1 (a) Examples of three curves Em passing through minima of adsorption potential. Curves have heen generated for molecnles of three different sizes Correlation between the theoretical fractal dimension D i and a fractal dimension that has heen evalnated for the curves passing throngh the minima of the adsorption potential Resnlts shown refer to diadic and triadic Koch curves. Reproduced hy permission of the American Chemical Society [26]. 

Figure 13.5. Different stages in the growth of an exact fractal, the triadic koch curves.

Thus, the Koch curve has a dimension between a line (Z) = 1) and the plane (D = 2). Since D is non-integer, such stmctures are called fractals. An approximate value of D can be obtained for such structures by drawing concentric circles of different radii (R) and then counting the number of branches in each circle. A plot of N(R) against log (R) yields a straight line, the slope of which yields the fractal dimension D. In the same way if we look at the branched structure within a circle of different radii, they all look alike. 

Figure 7.6. Fractal curves of infinite length, constructed by leaving out a middle part of a line segment and replacing it by two segments of the remaining length (the factor of the original line) as shown. This process is iterated on each new line segment and the line turns into the Koch curve. The size of the part replaced determines the dimension of the final curve or its space-filling capacity.

Processes in nature often result in fractal-like forms that differ from the mathematical fractals such as the Koch curve in two ways (a) the self-similarity is not exact but is a congruence in a statistical sense and (b) the number of repeated splittings is finite and random fractals have an upper and a lower cutoff length. A spatial example of a random fractal is the colloidal gold particle agglomerate shown earlier in Figure 7.4. 

The Koch curve illustrates many features usually shown by fractals. Because the iteration is carried out to infinity generating triangles on straight lines at every stage, the... 

The one-dimensional lines, two-dimensional surfaces and three-dimensional solids of Euclidean geometry are concepts so familiar to us that we tend to regard them as common sense . Fractals involve less familiar concepts, such as curves between two points, which have infinite length, and surfaces with infinite surface area. Fractals can be characterized by a dimension, but the dimension is fractional. For example, the triadic Koch curve (see Fractals Fig. 1) has a dimension of approximately 1.26186 and the surface with an infinite succession of regular tetrahedral asperities (ibid. Figure 4), a dimension of approximately 2.58496. In this article, the meaning and calculation of fractal dimension are discussed. Texts on the mathematics of fractals introduce different kinds of fractal dimension, which are beyond the scope of this article. The dimension discussed here is strictly the self-similarity dimension. Further aspects of fractal dimensions are considered in Fractals and surface roughness. 

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