Koch’s curves

It is useful, even though not usual, to designate this as zoom symmetry or enlargement symmetry. It is evident that the length, surface area or volume of these structures is a function of the scale used, since the degree to which the details of the structure contribute differs. Let us consider Koch s curve, shown in Fig. 6.85... 

Fig. 6.85 Creation of Koch s curve by successive application of the mapping y — f(y (see text).

Let us consider the same point in a somewhat different manner and refer to Fig. 6.86. In addition to Koch s curve (c), the figure displays also two other zoom-... 

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). 

What is the dimension of the von Koch curve Since it s a curve, you might be tempted to say it s one-dimensional. But the trouble is that K has infinite arc length. To see this, observe that if the length of So is To, then the length of S, is j = f Lj, because 5j contains four segments, each of length j. The length increases by a... 

Figure 2.8 Intermediate stage (/s) in the realization of a random version of the von Koch curve. Modified from [14].

In Section 2.2.2, it was shown that, after n iterations in the constmction of the triadic von Koch curve, the number N of segments of length S is a power-law function of S N =, 53 yyifij (ijg introduction of the similarity dimension, and by virtue of the equality between Hausdorff and similarity dimensions for the triadic von Koch curve, this power-law relationship may also be expressed as... 

Figure 33 Processed SAXS curves (s (/(s) - B)) corresponding to isothermal melt crystallization of PTT at 205 °C (left) and subsequent heating (right). The fits with the generalized paracrystalline model (solid lines) are offset vertically for clarity. With permission from Ivanov, D. A. Bar, G. Dosifere, M. Koch, M. H. J. Wacromo/eco/es 2008, 41,9224. ...

According to Family s classification [8], fractal objects can be divided into two main types deterministic and statistical. The deterministic fractals are self-similar objects, which are precisely constructed on the basis of some basic laws. Typical examples of such fractals are the Cantor set ( dust ), the Koch curve, the Serpinski carpet, the Vichek snowflake and so on. The two most important properties of deterministic fractals are the possibility of precise calculation of their fractal dimension and the unlimited range (- o +°°) of their self-similarity. Since a line, plane or volume can be divided into an infinite number of fragments by various modes then it is possible to construct an infinite number of deterministic fractals with different fractal dimensions. In this connection the deterministic fractals are impossible to classify without introduction of their other parameters in addition to the fractal dimension. 

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