The Golden Spiral

The independent variable angle 9 may have any value so that the curve is unlimited in length. 

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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. 

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

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. ... 

We conclude that all of the results described in this volume can be generated from two related constructs—the golden spiral and projective geometry. Remarkably, models of exactly these two things have recently appeared on the science campus Im Neuenheimer Eeld of the Ruprecht-Karls University of Heidelberg where a large part of this work was done. These are pictured with one of us in Fig. 5. 

Going to the solar spiral stairway, ascend the stairs, feeling the dry, warm breeze blowing up behind you. Walk up the entire stairway, back to the landing of Da ath. From there, enter the Turret Room and, after pulling the purple cord, walk up the golden stairs and onto the battlements. Go to stand upon the window inset into the floor of the battlements. You... 

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. 

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 ... 

To demonstrate the importance of the golden ratio it is assumed that protons and neutrons occur in the nucleus on three-dimensional spirals of opposite chirality, and balanced in the ratio Z/N = r, about a central point. The overall ratio for all nuclides, invariably bigger than r, means that a number of protons, equal to Z — Nt, will be left over when all neutrons are in place on the neutron spiral. These excess protons form a sheath around the central spiral region, analogous to the valence-electron mantle around the atomic core. The neutron spiral is sufficient to moderate the coulomb repulsion while the surface layer of protons enhances the attraction on the extranuclear electrons. 

Recognition of space-time curvature as the decisive parameter that regulates nuclear stability as a function of the ratio, Z/N, with unity and the golden mean, r, as its upper and lower limits, leads to a consistent model for nucleogenesis, based on the addition of -par tides in an equilibrium chain reaction. This model is also consistent with the limitations imposed by the number spiral. 

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...

The spiral Snake is composed of the metal of the foam-bom Aphrodite she stepped in Cyprus on land Aphrodite is the Love, which entwines Persephone (the Soul) like a snake However, this snake must wind itself upward, instead of winding itself downward, to become the snake of wisdom, as the two snakes on Hermes Staff wind themselves upwards, up to the golden-winged sun (The Uraeus - the snake on the forehead of the Egyptians, as a symbol of divine wisdom). 

The close link of elemental periodicity with the golden ratio, which can scarcely be accidental, suggests an intimate relationship between atoms, botanical structures and spiral galaxies. The intermediate position of the solar system in this self-similar chain prompted a re-investigation of the commonality between atoms and celestial structures, previously invoked by Nagaoka and Bohr to formulate their atomic models. 

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. 

Figure 7.3 A set of logarithmic spirals, such as the golden, planetary spiral with divergence angle 2t, may serve as a model of Godd s compass of inertia, going through an odd number of involutions...

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