Self-similarity

3 The idea of self similarity. One of the two figures shown is the geometrically rescaled part of the other. It is really hard to say, which is which I 

So we can see that the word fractal merely means a self-similar object. 

The concept of self similarity is akin to Coleridge s principle of beauty as this unity in multaety . 

Certain structures, when examined on different scales from small to large, always appear exactly the same. Such structures are said to be self-similar or to be endowed with the symmetry of self-similarity. Self-similar structures are found to be invariably associated with the geometrical relationship known as the golden ratio, or what Johannes Kepler (1571 - 1630) referred to as the divine proportion , adding  

1 believe that this geometrical proportion served as idea to the Creator when He introduced the creation of likeness out of likeness, which also continues indefinitely . 

Kepler s obsession with the golden ratio kept him occupied in an effort to demonstrate the self-similar structure of the solar system and the music of the spheres. His biggest problem probably was insufficient data. 

A later generation of German astronomers, including Titius of Wittenberg (1729 - 1796) and Johaim Bode (1747 - 1826) discovered a numerical regularity that interrelates the mean orbital distances of the planets from the sun. Their formula 

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The above recipe can be repeated indefinitely, and the mathematical result would be what is called a self-similar profile. That is, successive magnifications of a section (in this case by factors of 3) would give magnified jagged lines which could be superimposed exactly on the original one. In this limit, one finds for this case that... 

An analogous procedure can be applied to a plane surface. The surface can be roughened by the successive application of one or another recipe, just as was done for the line in Fig. VII-6. One now has a fractal or self-similar surface, and in the limit Eq. VII-20 again applies, or... 

Many of the adsorbents used have rough surfaces they may consist of clusters of very small particles, for example. It appears that the concept of self-similarity or fractal geometry (see Section VII-4C) may be applicable [210,211]. In the case of quenching of emission by a coadsorbed species, Q, some fraction of Q may be hidden from the emitter if Q is a small molecule that can fit into surface regions not accessible to the emitter [211]. 

Figure XVI-1 and the related discussion first appeared in 1960 [1], and since then a very useful mathematical approach to irregular surfaces has been applied to the matter of surface area measurement. Figure XVI-1 suggests that a coastline might appear similar under successive magnifications, and one now proceeds to assume that this similarity is exact. The result, as discussed in Section VII-4C and illustrated in Fig. VII-6, is a self-similar line, or in the present case, a self-similar surface. Equation VII-21 now applies and may be written in the form...

The chemical reactivity of a self-similar surface should vary with its fractional dimension. Consider a reactive molecule that is approaching a surface to make a hit. Taking Fig. VII-6d as an illustration, it is evident that such a molecule can see only a fraction of the surface. The rate of dissolving of quartz in HF, for example, is proportional to where Dr, the reactive... 

Coarse-grained models have a longstanding history in polymer science. Long-chain molecules share many common mesoscopic characteristics which are independent of the atomistic stmcture of the chemical repeat units [4, 5 and 6]. The self-similar stmcture [7, 8, 9 and 10] on large length scales is only characterized by a single length scale, the chain extension R. 

Weitz D A and Huang J S 1984 Self-similar structures and the kinetics of aggregation of gold colloids Kinetics of Aggregation and Geiationed F Family and D P Landau (Amsterdam North-Holland) pp 19-28... 

Fig. 16. Self-similar crack propagation in an isotropic material. The crack propagates in a direction perpendicular to the cycHc loading axis (mode I loading).

Wool [32] has considered the fractal nature of polymer-metal and of polymer-polymer surfaces. He argues that diffusion processes often lead to fractal interfaces. Although the concentration profile varies smoothly with the dimension of depth, the interface, considered in two or three dimensions is extremely rough [72]. Theoretical predictions, supported by practical measurements, suggest that the two-dimensional profile through such a surface is a self-similar fractal, that is one which appears similar at all scales of magnification. Interfaces of this kind can occur in polymer-polymer and in polymer-metal systems. 

In proposing an arrangement of the atoms in the second layer, we will focus first on the metal coating of Cat). Note that we speak of layers, not shells. The term shell implies self-similarity which, as we will see... 

Note that the structures depicted in Fig. 5 are not self-similar because the angle of rotation of the faces differs for each layer. The layers should, therefore, not be called shells as they are called in the case of pure alkaline earth-metal clusters. With increasing size, the shape of the cluster will converge asymptotically to that of a perfect icosahedron. 

The assumption of a self-similar flow (Reynolds number-independent flow) simplifies full-scale experiments and is also a useful tool in the formulation of simple measuring procedures. This section will show two examples of self-similar flow where the Archimedes number is the only important parameter. 

FIgura 4.1. Overpressure as a function of flame speed for three geometries. The relationships are based on calculations by use of a self-similar solution (Kuhl et al. 1973). 

Self-similarity applies to one-dimensional, time-dependent problems in which dependence on one of two independent variables can be eliminated by nondimen-... 

Guirguis, R. H., M. M. Kamel, and A. K. Oppenheim. 1983. Self-similar blast waves incorporating deflagrations of variable speed. Progess in Astronautics and Aeronautics. 87 121-156, AIAA Inc., New York. 

The Universal Hopkinson-Cranz and Sachs Laws of Blast Scaling have both been verified by experiment. These laws state that self-similar blast (shock) waves are produced at idendcal scaled distances when two explosive charges of similar geometry and the same explosive composition, but of different size, are detonated in the same atmosphere [49]. 

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... 

Patterns of this third class in fact demonstrate a complex form of scale-invariance by their self-similarity, in the infinite time limit, different magnifications observed at the same resolution are indistinguishable. The pattern generated by rule R90, for example, matches that of the successive lines in Pascal s triangle ai t) is given by the coefficient of in the expansion of (1 - - xY modulo-tv/o (see figure 3.2). 

What does the orbit look like for Ooo It is an infinite (and therefore aperiodic) self-similar point set with fractal dimensionality, Dfractai 0.5388 [grass86c]. Figure 4.5 shows the first six stages in the Cantor-set like construction. 

Fig. 4.5 Sets of stable attractors for the first six critical values of a. Note the self-similarity between the boxed subpattern for oe and the entire pattern for 04 appearing two lines above. A Cantor-set-like fractal pattern appears in the limit an-+oo-...

At the risk of oversimplifying, there are essentially three different dynamical regimes of the one-dimensional circle map (we have not yet formed our CML) (I) j A < 1 - for which we find mode-locking within the so-called AmoW Tongues (see section 4.1.5) and the w is irrational (11) k = 1 - for which the non mode-locked w intervals form a self-similar Cantor set of measure zero (111) k > 1 - for which the map becomes noninvertible and the system is, in principle, ripened for chaotic behavior (the real behavior is a bit more complicated since, in this regime, chaotic and nonchaotic behavior is actually densely interwoven in A - w space). 

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