Self-similar symmetry

If cosmology is to be consistent with all of science it requires serious revision in order to come into line with chemical evidence pertaining to the periodicity of matter, the cosmic abundance of nuclides and the self-similar symmetry between objects large and small. All of these aspects, either refute, or remain neutral to the provisions of the big-bang model. A valid model, in line with... 

These steps are represented by the index n in Table 5.4. Each value of n represents an allowed orbital distance for a satellite from its parent attractor. The planets have indices of Neptune(O), Uranus(2), Saturn(6), Jupiter(9), Asteroids(12), Mars(15), Earth(18), Venus(21) and Mercury(24). Because of the self-similar symmetry of the golden spiral this progression can be continued indefinitely on a continuously increasing scale. 

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 demonstration that the solar system displays the same self-similar symmetry, also noticed in the image of spiral galaxies and hurricanes, makes it hard to deny that the pattern conveys more than coincidence. It probably is one of the more rehable guides towards a better imderstanding of the natural world. 

Unlike crystals that are packed with identical unit cells in 3D space, aperiodic crystals lack such units. So far, aperiodic crystals include not only quasiperiodic crystals, but also crystals in which incommensurable modulations or intergrowth structures (or composites) occur [14], That is to say, quasiperiodicity is only one of the aperiodicities. So what is quasiperiodicity Simply speaking, a structure is classified to be quasiperiodic if it is aperiodic and exhibits self-similarity upon inflation and deflation by tau (x = 1.618, the golden mean). By this, one recognizes the fact that objects with perfect fivefold symmetry can exist in the 3D space however, no 3D space groups are available to build or to interpret such structures. 

We assume stationarity and radiative equilibrium for the energy balance because the radiative timescales are short in respect to the hydrodynamic timescales soon after the initial increase in luminosity. Spherical symmetry is assumed. According to detailed numerical models (Falk and Arnett, 1977 Muller, personal communication, 1987 Nompto, 1987 Nomoto et aL, 1987) and also analytical solutions for strong shock waves in spherical expanding enveloped (Sedov, 1959) density profiles are taken which are given by the self-similar expansion of an initial structure i.e. 

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 ... 

The relationship between the exponent v, (v = lnp/lnfc), and the fractal dimension Dp of the excitation transfer paths may be derived from the proportionality and scaling relations by assuming that the fractal is isotropic and has spherical symmetry. The number of pores that are located along a segment of length Lj on the jth step of the self-similarity is / , — pi. The total number of pores in the cluster is S nj (pJf, where d is the Euclidean dimension... 

On the fractal lattice, Dv is a constant, and holes exhibit the Gaussian characteristics, The self-similarity of the fractal has dilation symmetry shown in Eq. (8). Using the Fourier-Laplace transformation in time,... 

Increasing time of exposure of the recording film results in the appearance of more and more points throughout space in an eventually dense spacefilling array. All points satisfy the same symmetry and self-similarity properties. 

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... 

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 ... 

Until the parallel between number and cosmos is demonstrated to be an illusion we shall use this idea to model the universe. The power of this approach lies therein that all regularities in the physical world can be reduced to the same mathematical rules as the commensurable relationships in the solar system. The same mathematics that optimizes the distribution of matter in spiral galaxies and solar systems, shapes the growth of nautilus shells and sunflower heads. This ubiquitous symmetry, known as self-similarity is... 

Empirical evidence at variance with standard cosmology is, likewise, totally ignored. Even the most fundamental of empirical observations, known as universal CPT (charge conjugation-parity-time inversion) symmetry, which dictates equal amounts of matter and antimatter in the cosmos, is dismissed out-of-hand. Less well known, but of equal importance, cosmic self-similarity, is not considered at all. 

Mandelbrot [2, 3] systematized and organized mathematical ideas concerning complex structures such as trees, coastlines and non-equilibrium growth processes. He pointed out that such patterns share a central property and symmetry which may be called scale invariance. These objects are invariant under a transformation, which replaces a small part with bigger part that is under a change in a scale of the picture. Scale-invariant structures are called fractals [7]. More recently the relevance of natural and mathematical structure has become clearer with the help of computer simulation. Self-similarity turns out to be a general invariance principle of these structures. 

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