Farey fraction

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-... 

It will be shown that, on interpreting Farey fractions as representing the ratio... 

The important Z/N ratio must by definition always be a rational fraction, and an ordered set of nuclides must therefore correspond to a Farey sequence. It is readily demonstrated [7] that a set of A -modular simple Farey fractions... 

Fig. 2 Periodic table of the elements, defined directly by the Ftirey sequence 4. Shaded blocks show how the Farey fractions subdivide the periods into regions that correspond to the s, p,d, f blocks of chemical elements. Blocks of two and eight elements define the compact form of the appended periodic table...

Without assuming an exclusion principle, the Farey fractions subdivide the periodic sets into subsets that correspond to the traditional s, p,d, f energy levels, derived from atomic spectra. The resulting sublevel structure is highlighted by shading in Fig. 2. It is necessary to emphasize that these fractions do not refer to... 

The most convincing derivation of periodic structure, using the concepts of number theory, comes from a comparison with Farey sequences. The Farey scheme is a device to arrange rational fractions in enumerable order. Starting from the end members of the interval [0,1] an infinite tree structure is generated by separate addition of numerators and denominators to produce the Farey sequences ni of order n, where n limits the values of denominators... 

Each rational fraction, h/k, defines a Ford circle with a radius and y-coordinate of 1/(2k2), positioned at an -coordinate h/k. The Ford circles of any unimodular pair are tangent to each other and to the x-axis. The circles, numbered from 1 to 4 in the construction overleaf, represent the Farey sequence of order 4. This sequence has the remarkable property of one-to-one correspondence with the natural numbers ordered in sets of 2k2 and in the same geometrical relationship as the Ford circles of 4. 

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

The principle that governs the periodic properties of atomic matter is the composition of atoms, made up of integral numbers of discrete sub-atomic units - protons, neutrons and electrons. Each nuclide is an atom with a unique ratio of protonsmeutrons, which defines a rational fraction. The numerical function that arranges rational fractions in enumerable order is known as a Farey sequence. A simple unimodular Farey sequence is obtained by arranging the fractions (n/n+1) as a function of n. The set of /c-modular sequences ... 

The equivalence between Sk, the infinite Farey tree structure and the nuclide mapping is shown graphically in Figure 8.4. The stability of a nuclide depends on its neutron imbalance which is defined, either by the ratio Z/N or the relative neutron excess, (N — Z) jZ. When these factors are in balance, Z2 + NZ — N2 = 0, with the solution Z = N(—1 /5)/2 = tN. The minimum (Z/N) = r and hence all stable nuclides are mapped by fractions larger than the golden mean. 

Figure 2.9 Farey sequence of rational fractions. Starting with the first row and reading from right to left 3/1 is identified as the 12th rational fraction.

All fractions beyond 3/4, at the centre of the Farey subset, are further seen to define points of intersection between the curves of constant A — 2Z and the straight line between coordinates of (14/19,0) and (2/3,87), at the intersection with the line 1 —> r. These lines provide the correct stability limits. They have the additional merit of automatically limiting the maximum allowed atomic number of a stable element to 83. 

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

As discrete numbers of nucleons are involved in the constitution of nuclides the periodicity of atomic matter is readily simulated in terms of the elementary number theory of rational fractions, Farey sequences and Ford circles. 

However, the inferred electronic configuration for the elements of periodic groups 1 to 10 and for the lanthanides is not in line with this sequence. It is therefore of interest to note that the observed periodic pattern occurs in the arrangement of Ford circles, as defined by the Farey sequence of rational fractions. 

I developed an interest in the field on noticing that cosmological theories of nucleogenesis are totally out of fine with reality. The important clue came from the observed periodicity of the stable nuclides, which is governed by simple concepts of number theory. Any atomic nucleus consists of integral numbers of protons and neutrons such that the ratio Z/N is always a rational fraction, which can be ordered in Farey sequences and mapped by Ford circles. This ordering predicts a unique periodic function, which is readily demonstrated to predict the correct observed cosmic abundances of the elements. 

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...

Nucleogenesis in the interior of massive stellar objects yields 100 natural elements of composition Zj A - Z) = 1. Because of radioactive decay at reduced pressure in intergalactic space, the stability ratio converges as a function of mass number to a value of t at yl = 267 = (A — Z ) t> = Z. As a result, only 81 stable elements survive in the solar system as a periodic array conditioned by r. The observed periodicity corresponds to a Ford-circle mapping of the fourth-order unimodular Farey sequence of rational fractions. 

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. 

This is the unimodular condition that characterizes neighboring fractions in a Farey... 

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