Fibonacci

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. 

Fibonacci numbers, where the number sought is the nth number in a series defined in terms of relationships to the n-1, n-2, etc. members of the series. All recursive procedures must have a terminating condition, so that they do not call themselves endlessly. 

The Simplex algorithm and that of Powell s are examples of derivative-free methods (Edgar and Himmelblau, 1988 Seber and Wild, 1989, Powell, 1965). In this chapter only two algorithms will be presented (1) the LJ optimization procedure and (2) the simplex method. The well known golden section and Fibonacci methods for minimizing a function along a line will not be presented. Kowalik and Osborne (1968) and Press et al. (1992) among others discuss these methods in detail. 

One way to solve the problem of unphysically short atomic distances is to project onto the Rpm subspace only those grid points included in a selected strip (gray area), with width of a (cos a + sin a) in the A per subspace. The slope of RPai shown in Fig. 1 is 0.618..., an irrational number related to the golden mean [( /5 + l)/2 = 1.618...]. As a result, the projected ID structure contains two segments (denoted as L and S), and their distribution follows a ID quasiperiodic Fibonacci sequence [2] (c.f. Table 1). From another viewpoint, the ID quasiperiodic structure on the par subspace can be conversely decomposed into periodic components (square lattice) in a (higher) 2D space. The same strip/projection scheme holds for icosahedral QCs, which are truly 3D objects but apparently need a more complex and abstract 6D... 

Table 1 The Fibonacci sequence and its relationship to the golden mean, 1.618...

The Fibonacci sequence can be generated by transformations of L LS and S—>L in each cycle. L/S represents the sequence of ACs that can exist for any QC system. With increasing order, the L/S ratio converges to the golden mean value... 

PHI is the ratio of any two sequential numbers in the Fibonacci (V sequence. If you take the numbers 0 and 1, then create each subsequent number in the sequence by adding the previous two numbers, you get the Fibonacci sequence. For example, 0, 1,1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. Ifyou sum the squares of any series of Fibonacci numbers, they will equal the last Fibonacci number used in the series times (10) the next Fibonacci number. This property results in the Fibonacci spiral seen in everything from seashells to galaxies, and is written mathematically as l2 + l2 + 22 + 32 + 52 = 5 x 8. 

Plants illustrate the Fibonacci series in the numbers of leaves, the arrangement of leaves around the stem, and in the positioning of (IV leaves, sections, and seeds. A sunflower seed illustrates this principal... 

Fibonacci sequence—a progression famous not only because the sum of adjacent terms equaled the next term, but because the quotients of adjacent terms possessed the astonishing property of approaching the number 1.618—PHI ... 

The optimum seeking methods which have been found to be particularly useful are the modified Fibonacci search (search by golden section) for one-dimensional searches and the Hooke-Jeeves search for multi-dimensional searches. Beveridge and Schechter (8) give a complete description of these searches. 

A crystal has both symmetry and long-range order. It also has translational order it can be replicated by small translations. It is possible to have both symmetry and long-range order without translational order. A one-dimensional example is a Fibonacci series that is composed of two segments, A and B. The series consists of terms N such that N = N - + N -2- For example, the series starting with... 

A Fibonacci series in which each element of the series is the sum of the previous two elements is a one-dimensional analog. An example is the series, starting with L and S,... 

Fig. 3. All five Clar structures of a branched benzenoid hydrocarbon and the corresponding five colored Gutman trees. The indicated coloring generates F4, the fourth Fibonacci number...

Indirect methods solve the necessary conditions for an optimum (looking at the shape of the function) directly via iteration. Region elimination techniques such as Fibonacci and Golden Section searches use function evaluations only to delete a portion of the independent variable range at each iteration. Interpolation techniques use polynomial fitting (quadratic or cubic oftentimes) to predict the location of the optimum. 

A one-dimensional search optimization technique, such as the Fibonacci search, is employed to minimize Equation 8-113. A computer program (PROG81) was developed to estimate the equivalent number of ideal tanks N for the given effluent tracer response versus time data. Additionally, the program calculates the mean residence time, variance, dimensionless variance, dispersion number, and the Peclet number. 

The rational fractions defined by successive Fibonacci numbers in the sequence ... 

The convergence follows the Fibonacci fractions which appear in the Farey tree structure that develops between the limits and . 

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