Golden Geometry

Suppose that the angle of rotation is (j . Each point of the disc x is seen to be mapped to some unique point, x, which is the image of no other point. There is only one point, at the centre, that maps to itself for a rotation oi j) 2nn. For a similar rotation of an annulus, or any surface, not home-omorphic to a disc, the fixed-point theorem states that in this case no hxed point occurs. Such surfaces lack a special point. They have the alternative property that hair on such a surface can all be brushed to lie in the same direction, unlike the hair on a disc, a sphere, or a human head which develops a crown. This is a striking property of a Mobius band, showing that all points in the surface are quivalent and any of these can be considered to be the central point. 

One of the topics addressed by Euclid in the Elements is the golden ratio. 

The symbols and r occur indiscriminately in the literature to represent the golden ratio and/or its reciprocal. We shall follow the convention defined 

To complete the construction circles of radius centred at A and E intersect the unit circles centred at E and A respectively, in points D and B. Point C, unit distance from B and D completes the pentagon ABODE. 

By folding AAae, and the other four corresponding triangles, about oe, etc., so that ABODE meet in T, as shown, a pyramid of height OT = h is formed, such that h/r =.  

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The crystal structures of the Ru(Et2Dtc)3G (268) and Ru(Me2Dtc)3I3 (435) complexes have been determined. On both, the Ru(IV) coordination geometry is pentagonal bipyramidal (Fig. 39) (Table XX). The ruthenium atoms in the ethyl derivative are pendant on infinite chains of iodine atoms in the lattice. This observation and the unusual golden color of the complex suggest that the crystals of this compound may possess interesting electrical properties. 

Covalent interaction in diatomic molecules depends on the golden mean t, the interatomic distance d and the radius ratio x r /r2 of the constituent atoms, as summarized in Figure 5.6. The golden mean is a universal constant that matches the geometry and topology of space-time, the radius ratio is a known function of atomic number and dl relates to the optimal wave-mechanical distribution of valence-electron density in the diatomic system. 

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. 

One number theoretic concept with an obvious link to geometry and to the topology of space-time is the golden ratio, obtained as a root of the quadraric equation... 

Some alchemists even considered gold as condensed sunbeams [1] and used to represent it by the symbol of a circle, the hallmark of mathematical perfection. Let us recall in this respect, e.g., the Golden Section in geometry, the concept of a gold number in colloidal chemistry after R. A. Zsigmondy (Z. Anal. Chem. XL, 697 (1901)) and J. B. Richter and M. Faraday (Phil. Trans. CXLVII, 145 (1857), and the Golden Rule in quantum theory. 

The main building blocks of the proposed new model are the relationship between geometry, numbers and space the theory of relativity and the periodicity of atomic matter. Taken together, these considerations indicate a cosmic symmetry that defines a harmonious holistic system that embraces all objects from the subatomic to extragalactic scales. The common geometrical factor is the ubiquitous golden parameter, r = 0.61803... 

Mihcakan, I.M. "The Effects of Polymer Augmented Alkaline Flooding and Core Geometry on Ultimate Oil Recovery," M. Sc. Thesis, T-2902, Colorado School of Mines, Golden, Colorado, May 1985. 

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