Golden Fibonacci sequences

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... 

Throughout history, mathematicians have studied number patterns. Research Pascal s Triangle, Fibonacci and the Fibonacci Sequence, and the Golden Mean that was used by the Greeks in building the Parthenon. 

Let us recall that the Fibonacci sequence is the set of the successive rational ratios Pk/qk, k = 1,. ..oo, obtained when developing the golden... 

Figure 6. Variation with time of the FLI for a set of periodic orbits belonging to the Fibonacci sequence for the standard map with e = 0.9715. The rational ratios Pk/qk, k = 1,. ..7, are written in the figure near the corresponding FLI curve. The FLI curve which grows linearly with time all over the interval 0 < t < 5 106 iterations is obtained for the golden torus.

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... 

Fig. 3 A golden spind inscribed in a golden rectangle defined by a Fibonacci sequence. The increasing size of successive squares is defined by the Fibonacci labels ( ), and the ratio of their side lengths approaches r as —> oo. The inscribed spiral approximates an equiangular logeirithmic spiral [4]...

Not only the golden ratio itself but also its integral powers, r", are embedded as convergent series in the Fibonacci sequence. Comparison of the first few of these reveals a striking pattern ... 

When Archimedes computed % by his approximation of the circle by a sequences of polygons, or when the Sumerians approximated V2 by an incredible numerical scheme, which was much later rediscovered by Newton, they all were well aware of the fact that they are dealing with the unusual numbers [484]. As early as in the year 1202, the population growth was evaluated by the number of immature pairs, i.e., A +i = A + A ./ with Ao=0,Aj = 1, continuing 1,2,3,5,8,13,21,34,55,89,144,... (meaning that the state at time n+1 requires information from the both previous states, n and n-1, known as two-step loops) called the Fibonacci sequence. The ratio of A +i/A is steadily approaching some particular number, i.e., 1,618033988..., which can be equaled to (l+V5)/2) and which is well-known as the famous golden mean ( proportio... 

The frequency of occurrence of letters L and S in this sequence is represented in Table 1.21, and it is easy to recognize that they are identical to the consecutive members (F +i and F ) of the Fibonacci series. The corresponding limit when the number of words, n, approaches infinity is the golden mean, t... 

Fig. 6 A sequence of Fibonacci squares on a scale of 1 2 serves to generate the 21 cm X 13 cm golden rectangle with its inscribed spiral. Direcily measurable radii of n a at convergence angles of 4jr/(2n — 1) terminate at the labeled points...

The apparent quantization of bond order corresponds to the numerators in Farey sequences that converge to the golden ratio. As the limiting Fibonacci fraction n/(n + 1) -> T approaches the golden ratio, the values of quantized bond order, predicted by the Farey sequence +i, approach the simulation of Fig. 4. 

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