Golden ratio

It has been observed that the height of a man from the crown of the head to the sole of the foot is equal to the distance between the tips of the middle fingers of the two hands when extended in a straight line. 

If a line is divided into two segments such that 

A Fibonacci blueprint and the golden ratio appear to be able to describe plant growth and leaf spacing. The reproduction of honeybees, cows, and rabbits appears to be related to the Fibonacci sequence, as is the arrangement of cauliflower and broccoli florets. The ratio of the head-to-toe height in humans to the height from navel to toes also approximates the golden 

FIGURE 7.3.1 Having fun with Fibonacci numbers. (Courtesy of FOXTROT 2009 Bill Amend. Reprinted with permission of Universal Press Syndicate. All rights reserved.) 

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The structure factor for the 104-atom complex with almost perfect icosahedral symmetry determines the intensities of the diffraction maxima, in correspondence with the inverse relationship between intensity in reciprocal space and the atom-pair vectors in real space that was discovered fifty years ago by Patterson.27 The icosahedral nature of the clusters in the cubic crystal explains the appearance of the Fibonacci numbers and the golden ratio. 

Livio, M. (2002) The Golden Ratio The Story of Phi, the World s Most Astonishing Number. Broadway Books, New York. 

The quantity (I + V5)/2 = 1.62 is known as the golden ratio. It appears often in works of an, as for example to determine the approximate ratio of the height to width of a classic painting - and this page. 

The Golden Ratio in the Creations of Nature Arises in the Arehiteeture of Atoms and Ions... 

Keywords Covalent and ionic radii, Golden ratio, hydrogen bond length... 

This paper starts with a brief description of the Golden ratio and the ( )-based crystal ionic radii and is then followed by the (()-based aqueous ionic radii and hydration lengths. The role of in the sizes of the ions in the crystal and in aqueous solutions and their hydration bonds with water can be seen in Fig. 12.3 for Na" and Cl" ions (used as the examples). 

The Golden Ratio, Golden Point and Golden Sections... 

The Golden Ratio, Aqueous Ionic Radii and the Hydration Bonds... 

The Golden Ratio and the Length of the Hydrogen Bond in General... 

Retaining the latest estimates for x and X2, the slope of the line follows more closely the form of the function than in the false position method. The order of convergence can be shown to be 1.61B, the "golden ratio", which we will encounter in Section 2.2.1. The root, however, is not necessarily bracketed, and the next estimate x3 may be far away if the function value ffx ) is close to f(x2>. Therefore we may run into trouble when starting the search in a region where the function is not monotonic. 

Like the periodic table of the elements (Chapter 4) and gaps in the asteroid belt, the spacing of Saturn s rings fits a numerical pattern based on the golden ratio [72]. 

The negative quantity in brackets is an irrational number known as the golden ratio, t = 0.61803. The solution Z = — iV(1.61803...) = — nuclide stability, as defined on both plots, converges to a point on the line r = Nx = r at A 267, the maxinum possible mass number for nuclides, stable against /1-type decay. By definition, this maximum,... 

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. 

Figure 4.10 Atomic-number periodicity derived from, golden-ratio packing of nucleons.

All of the primary and secondary sequences can be traced back to tangent Ford circles. The two independent patterns have common points at the four most significant, generally accepted, magic numbers 2, 50, 82 and 126. The points at which the eleven hem lines intersect the golden ratio line are indicated by arrows. Ford circles from the Farey sequence (2k2 = 50) appear... 

The natural appearance of nuclear magic numbers, and the golden-ratio limitation on nuclear distribution, indicate the development of an excess surface layer of protons, which correlates well with periodic variation of nuclear spin, and which may be an important parameter in the understanding of superconductivity. 

The curves of Figures 5.4 and 5.5 show remarkable dependence on the golden ratio. The attractive and repulsive curves intersect where d = r, at which point the diatomic dissociation energy D = 2r. Again, the limiting curves (marked 1.0 and 0.25(H) in Figure 5.5) are maximally separated at d = 2r,... 

The formulation of spatially separated a and 7r interactions between a pair of atoms is grossly misleading. Critical point compressibility studies show [71] that N2 has essentially the same spherical shape as Xe. A total wave-mechanical model of a diatomic molecule, in which both nuclei and electrons are treated non-classically, is thought to be consistent with this observation. Clamped-nucleus calculations, to derive interatomic distance, should therefore be interpreted as a one-dimensional section through a spherical whole. Like electrons, wave-mechanical nuclei are not point particles. A wave equation defines a diatomic molecule as a spherical distribution of nuclear and electronic density, with a common quantum potential, and pivoted on a central hub, which contains a pith of valence electrons. This valence density is limited simultaneously by the exclusion principle and the golden ratio. 

Interatomic distances are determined by steric factors, of which the most important is the exclusion principle that depends directly on the geometry of space-time, observed as the golden ratio. Bond order depends on the ratio between the number of valence electrons and the number of first neighbours, or ligands, and affects interatomic distances by the screening of internuclear repulsion. 

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