Self-similar scale

Under the assumption that the morphology of the self-affine interface has the self-similar scaling property, the apparent selfsimilar fractal dimension d ss of the electrode was calculated... 

Figure 9 demonstrates the dependence of the scaled length SL on the segment size SS obtained from the self-affine fractal profiles in Figure 7 by using the triangulation method for the Euclidean two-dimensional space. The linear relation was clearly observed for all the self-affine fractal curves, which is indicative of the self-similar scaling property of the curves. 

From the above results, it is noted that the self-similar scaling property investigated by the triangulation method can be effectively utilized to analyze the diffusion towards the self-affine fractal interface. This is the first attempt to relate the power dependence of the current transient obtained from the self-affine fractal curve to the self-similar scaling properties of the curve. 

Bearing in mind that diffusing ions move randomly in all directions, it is reasonable to say that the diffusing ions sense selfsimilar scaling property of the electrode surface irrespective of whether the fractal surface has self-similar scaling property or self-affine scaling property. Therefore, it is experimentally justified that the fractal dimension of the self-affine fractal surface determined by using the diffusion-limited electrochemical technique represents the apparent self-similar fractal dimension.43... 

In summary, from the above theoretical and experimental results, it is concluded that ionic diffusion towards self-affine fractal electrode should be described in terms of the apparent selfsimilar fractal dimension rather than the self-affine fractal dimension. In addition, the triangulation method is one of the most effective methods to characterize the self-similar scaling property of the self-affine fractal electrode. 

It seems at the first sight that due to space fluctuations all crumples could penetrate each others with loops, destroying the self-similar scale-invariant crumpled structure described above. 

Consider a set of points on the limited straight fine, L. Any other set of points of limited segment Li is self-similar (scale multiplier)... 

A comparison of the formulas (5.91) and (5.79) might lead one to conclude that use of the Lie differential equation formalism has enabled us to eliminate the uncertainty associated with the choice of the threshold value T for the self-similarity scale. This is not so. The self-similarity threshold, like the period of a periodic function, is an inherent feature of each physical object expected to exhibit self-similarity. By adopting the differential equation formulation of the GPRG theory one implicitly has selected a threshold scale equal to unity, that is, t = 1. 

In Figure 5.9 the inner and outer planets are placed along separate spiral arms on different, but self-similar scales. The outer planets are on a diverging spiral, starting from r, whereas the inner planets are on the complementary converging spiral. 

Figure 5.9 Planetary orbits defined as products of N-numbers and the mean. Jovian orbit. Orbits of the inner planets are shown on a larger, self-similar, scale...

Fractal (self-similar) scaling will be very common in biology, no matter what the level. 

Fig. 12 Simulation of planetary orbits by golden-spiral optimization. With the mean orbital radius of Jupiter as unit, the outer planets are on orbits defined by integral multiples thereof. On the same scale, the asteroid belt is at a distance x from the sun and the inner planets have orbital radii of t/ . For clarity, the inner planets are shown on a larger self-similar scale...

---