Fibonacci fractions

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

Figure 2.11 Low-rank fractions in the Farey sequence between the Fibonacci fractions 1/1 and 2/3...

Fig. 10 A plot of 1 -modular Farey sequences as a function of the natural numbers defines a set of infinite festoons that resembles the arrangement of nuclides in Figs. 5 and 7. The segment, obtained as a subset defined by limiting Fibonacci fractions that converge from 1 to r and subject to the condition A(mod4) = 0 —> 3, corresponds to the observed field of nuclide stability...

The characteristic values of Z/N = x and of 0.58 for observed and wave-mechanical periodicities are the limits of converging Fibonacci fractions around 3/5. The segmentation of the table into groups of 2 and 8 and of periods 2,8,18,32 summarizes the observed periodicity as a subset of nuclide periodicity. The sublevel structure, despite formal resemblance to the wave-mechanical H solution, emerges from number theory without reference to atomic structure. 

As a first trial, we consider a series of Fibonacci fractions in the range 1/1 to 3/5 to simulate bond orders between 0 and A d = 1, x). The unimodular sequence that converges to 1, i.e. 

Starting from still higher Fibonacci fractions, the same pattern persists, but gaps appear in the sequence of quantum numbers. The infinite sequence between 1 and r is inferred to have the exact bond-order sequence, with large quantum numbers, embedded within it. 

This simulation confirms the results of Table 3 in detail. As a matter of interest, all of these bond orders are approximated in 34, which relates to the Fibonacci fraction 21/34, with a few gaps ... 

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. 

The irrational number, known as the golden ratio, is said to be the most irrational of them all. Like other irrationals, it also occurs as the limit of a regular series of rational fractions, in this case the Fibonacci fractions. In nature, it occurs as the convergence limit of the mass fractions of stable nuclides, Z/(y4 — Z). As a clue to its physical meaning, it is noted that the stability of nuclides depends on their space-time environment [4]. In regions where space-time curvature approaches infinity, the mass ratio Z/ A — Z) 1. In the hypothetical situation of zero curvature, matter does not exist. It is inferred that in an intermediate situation of curvature, conducive to the development of biological life, the mass ratio Z/(A — Z) z. 

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

The stability problem is solved on noting that allowed fractions at small atomic number begin at unity and approach r with increasing Z. This trend should, by definition follow Farey fractions determined by Fibonacci numbers. The first few Fibonacci numbers are 0,1,1,2,3,5,8,13,21, etc. The ra-... 

Because the range of nuclidic stability is bounded by fractions that derive from Fibonacci numbers, it probably means that nuclear stability relates directly to the golden mean. To demonstrate this relationship it is noted that the plot of A vs Z, shown in figure 13 for the A(mod4) = 0 series of nuclides, separates into linear sections of constant neutron excess (A — 2Z) and slope 2. Each section terminates at both ends in a radioactive nuclide. The range of stability for each section follows directly from... 

There is an obvious convergence of Ford circles of diminishing size around the central circle at x = 0,1. Self-similar convergence occurs aroimd each of the smaller circles. Of particular importance is the convergence around the circle at x = 3/5, shown in Figure 5.3. On one side it follows the unimodular fractions defined by the Fibonacci series ... 

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