Koch curves

Thus, whenever the set A has a manifest self-similarity, so that, like the Cantor set, it can be defined by a recursive geometric construction, Dfractal oan be easily calculated from this relation. The Koch Curve, for example, the first three steps in the construction of which are shown in figure 2.2, has a length L which scales as... 

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). 

Figure 1.1 The first four iterations of the Koch curve.

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). 

Identical reasoning can be applied to any two consecutive generations of the Koch curve, making the exponent V [Eq. (1.10)] valid on all length scales. The self-similar nature of the Koch curve is clear from the fact that if a small piece of the curve is magnified, it looks exactly like the larger piece. 

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. 

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... 

In the next example we confirm our earlier intuition that the von Koch curve should have a dimension between 1 and 2. 

Show that the von Koch curve has a similarity dimension of In4/ln3 1.26. Solution The curve is made up of four equal pieces, each of which is similar to... 

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. 

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