Golden spiral

My little boat turns and glides above bright shining swirls of mist and a symmetrical melange of intertwined, golden spirals. The elves chuckle. I am moving toward open sea. 

Table 8.1 The orbital distances (astronomical units) at perihelion, mean and aphelion of the planets, compared to scaled intercepts, with divergence angle of 108°(3/10 x 360), on a golden spiral...

Figure 5.8 Double golden spiral with self-similar positions on different scales,...

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. 

Whirlpool galaxy M51 (Courtesy Hubble Heritage Teanty ESA, NASA) Golden spiral superimposed. 

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

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...

Fig. 13 Simulation of Thomas-Fermi and Hartree-Fock electron densities for unit atoms. The calculated points are those predicted by golden-spiral optimization and scaled to match the Thomas-Fermi curve, shown as a solid line. The stippled curve simulates the HF result...

Fig. 14 Simulation of integer bond orders on a golden spiral. The dimensionless distances d = tind r, which represent zero and fourth order, respectively, must, by definition, be sepeirated by a conveigence angle of 90° on a golden spirtil. Conveigence angles for intermediate integer and htilf-integer orders foUow diiectly...

Optimization by a golden spiral predicts the correct distribution of matter in the solar system [31], with the inference that the spiral structure reflects space-time topology. Fractal models of the universe, which imply cosmic self-similarity, would then indicate the same optimization for extranuclear electron density. The resulting wave structure inevitably carries an imprint of the golden ratio. 

Keywords Atomic wave model Electron density Golden-spiral optimization Ionization radius Self-similarity... 

The same periodic function results from optimization on a golden spiral with a variable convergence angle of Art In — 1), which describes a spherical standing wave with nodes at n. Analysis of the wave structure shows that it correctly models the atomic electron distribution for all elements as a function of the golden ratio and the Bohr radius, uq. Normalization of the wave structure into uniform spherical units simulates atomic activation, readily interpreted as the basis of electronegativity and chemical affinity. 

It has been shown that the electronic charge distribution in an atom is readily calculated by the same optimization procedure, based on a golden spiral [6], that correctly predicts all satellite orbits in the solar system [27]. The simulation is... 

Fig. 2 Atomic shell structure as it emerges from electron-density optimization on a golden spiral. The variable convergence angle of An/ In — 1) manifests in the appearance of 2n — 1 additional cycles s, p, d, /) in each interval between Bohr levels n and n — I, shown here as elementary ripples. In contrast to the Bohr-Schrodinger (BS) model, closed shells in the Ford-circle simulation (FC) invariably coincide with noble-gas configurations...

We contend that the shape of large molecules in empty space is affected by the topology of the four-dimensional space-time manifold. Guided by the principle of cosmic self-similarity, it is reasonable to assume that, like many spiral galaxies, extended molecules tend to curve like the surface of a golden spiral. It lies in an... 

Apart from the relative abundance of isotopes, all other basic data usually shown on a periodic table of the elements are predicted by elementary number theory. Based on the periodic table shown as an Appendix, the chemically important parameters of ionization radius and electronegativity have been used, together with the golden section and golden spirals, to derive all essential parameters pertaining to covalent interaction and the optimization of molecular structure by MM. The detailed results are described in the preceding chapters in this volume. 

In all cases where the golden section or the golden spiral correlates with chemical phenomena, convergence to some singularity is observed. The most striking example, shown in Fig. 4, occurs as the composition of stable nuclides, measured as Z/N, converges to the golden ratio as Z 102. At the same time, the hem lines, which define nuclide periodicity of 24, map out the observed periodic table of the elements at Z / A = t. ... 

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