Fibonacci Numbers and the Golden Ratio

Fibonacci (whose real name was Leonardo Pisano) found this sequence as the number of pairs of rabbits n months after a single pair begins breeding, assuming that the rabbits produce offspring when they are two months old. As n 00, the ratios of successive Fibonacci numbers Fn+ilFn approaches a limit designated 0  

FIGURE 1.4 The golden ratio and da Vinci s Mona Lisa. The frame of the picture, as well as the rectangle outlining her face, has divine proportions. 

The positive root, p = (1 + V5)/2 = 1.6180, is known as the golden ratio. According to the ancient Greeks, this was supposed to represent the most aesthetically pleasing proportions for a rectangle, as shown in Fig. 1.4. Leonardo da Vinci also referred to it as the divine proportion.  

The differential dxdy represents an element of area in cartesian coordinates, with the domain of integration extending over the entire xy-plane. 

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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. 

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