Golden spiral optimization

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...

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

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. 

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...

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. 

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