Universal Self Similarity

As recently shown (Boeyens, 2009) the Bode -Titius law, which hints at some harmonious regular organization of planetary motion in the solar system, is dictated by a more general self-similar symmetry that applies from subatomic systems to galactic spirals. The common parameter is the golden ratio, r = 0.61803. Any such cosmic symmetry should be dictated by a successful cosmological model. 

The golden ratio is superimposed on a logarithmic spiral, r = fP, by setting fj, = to produce the golden spiral, r = that leads to the 

It is now conjectured that the Godel compass of inertia has the same structure with hve involutions, which divide the universe into ten segments with alternating chirality. The topological structure remains the same as for the single involution considered before, but the hve-fold self-similarity is now imprinted on all space. 

The consequences of rotating space with periodically alternating chirality have never been contemplated before. If it has some dynamo effect it could explain the appearance of Einstein s cosmological constant A, which stabilizes Godel s solution. 

Bergmann, P.G. (1976) Introduction to the theory of relativity, Dover edition, NY. 

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The estimates of the climatic mean annual parameters of the MRC in the sections normal to the northeastern coast of the Black Sea presented in [17] yielded a distance of its core from the shore about 40 km, a full width of the current (with respect to velocity values of 0.02 m s-1) of 75 km, a penetration depth of 275 m, a maximal geostrophic velocity of 0.31 ms-1, and a volume transport of 1.3 x 106 m3 s x. These estimates are of the same order of magnitude as shown in Fig. 4a within the velocity interval to 0.20 m s x. This allows us to suggest a certain geographical universality (self-similarity) of the MRC profile normal to the coast. 

If we now consider two velocity profiles, one at time /, and the other at time /2, with % > i> we might expect them to reduce to a single universal (self-similar) profile if we were to scale the distance from the wal I v with the length scale = (/ ) of the diffusion process, that is,... 

FIGURE 5.2.5 Universal self-similarity in coalescing liquid-liquid dispersions (neutrally buoyant benzene-carbon tetrachloride mixture with water). Data of Wright and Ramkrishna (1994). (Reproduced with permission of the American Institute of Chemical Engineers. Copyright 1994 AIChE. All rights reserved.)... 

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]. 

The cornerstone of LES methodology is the self-similarity theory of Kolmogorov stating that even though the large structures of a turbulenf flow depend on fhe boundary and initial conditions, the finer scales have a universal... 

J. des Cloizeaux claims that A is a universal number which depends only on the dimension of space (A = A (d). Thus C (x) is not only given by a self - similar expression as was pointed out by P.G. de Gennes [143], it is also completely determined. 

The four different periodic tables account for the observed elemental diversity and provide compelling evidence that the properties of atomic matter are intimately related to the local properties of space-time, conditioned by the golden parameter r = l/. The appearance of r in the geometrical description of the very small (atomic nuclei) and the very large (spiral galaxies) emphasizes its universal importance and implies the symmetry relationship of self-similarity between all states of matter. This property is vividly illustrated by the formulation of r as a continued fraction ... 

Significant new evidence is the self-similarity of sub-atomic, atomic, biological, planetary and galactic structures, all related to the golden section. The astronomical structures are assumed to trace out the shape of local space. The obvious conclusion is that all of space has a uniform non-zero characteristic curvature, conditioned by the universal constants tt and r. Space, in this sense, is to be interpreted as equivalent to the three-dimensional sub-space of the Robertson-Walker metric [104]. In standard cosmology this sub-space... 

This space-time model is a conjecture that has been described in detail [28] and will be reconsidered in chapter 7. A new aspect thereof, which derives from number theory, is that the general curvature of this space-time manifold [26, 29] relates to the golden mean. This postulate is required to rationalize the self-similar growth pattern that occurs at many levels throughout the observable universe. 

Earlier speculations about the effect of the curvature of space on elemental synthesis and the stability of nuclides (2.4.1) are consistent with the interface model. The absolute curvature of the closed double cover of projective space, and the Hubble radius of the universe, together define the golden mean as a universal shape factor [233], characteristic of intergalactic space. This factor regulates the proton neutron ratio of stable nuclides and the detail of elemental periodicity. The self-similarity between material structures at different levels of size, such as elementary particles, atomic nuclei, chemical... 

Barenblatt G1 (1996) Scaling, Self Similarity, and Intermediate Asymptotics. Cambridge Cambridge University Press... 

We have demonstrated miscible blends of PET-PHB/PEI can be formed by rapid solvent casting from the mixed solvent of phenol and tetrachloroethane. The miscibility was confirmed by the systematic movement of Tg in the DSC studies. However, the blend is unstable and undergoes thermally induced phase separation with a miscibility window reminiscent of LCST. The dynamics of spinodal decomposition is non-linear in character and obeys the power law with kinetic exponents of -1/3 and 1 in accordance with the cluster dynamics of Binder and Stauffer as well as of Furukawa. In the temporal scaling analysis, the structure function exhibits universality with time, suggesting temporal self-similarity of the system. 

These results point towards a universality in the particle radial distribution function from r = 0 to 2R. This is shown in Fig. 8 where the first contact peaks, which are self-similar, are collapsed for all packing fractions onto the master Gaussian curve ... 

A solution, such as u(y, t) in the present problem, that depends on a single dimensionless variable < instead of y and / separately is said to be self-similar, and this designation is also the source of the names similarity variable and similarity transformation, 17 The basic idea of self-similarity is that the series of profiles u(y, t) for various fixed times t will collapse into a single, universal form when u is plotted as a function of rj rather than as a function ofy. 

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