Farey sequence

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. 

It is of interest to explore the possibility of an independent identification of magic numbers and the neutron spectrum by the relationship between neutron number and Farey sequences as mapped by Ford circles. Such an analysis presupposes the occurrence of periodic sequences of 32, 18, 8 and... 

All of the primary and secondary sequences can be traced back to tangent Ford circles. The two independent patterns have common points at the four most significant, generally accepted, magic numbers 2, 50, 82 and 126. The points at which the eleven hem lines intersect the golden ratio line are indicated by arrows. Ford circles from the Farey sequence (2k2 = 50) appear... 

The orbits from Venus to Ceres are represented by the unimodular series 4. In the outer system the Ford circles of only Uranus and Neptune are tangent, but the likeness to Farey sequences in atomic systems is sufficient to support the self-similarity conjecture. 

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 ... 

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.

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

Conveniently, MMOs are characterized by a symbolic notation where L denotes the number of large and S the number of small oscillations during one period. Thus, the MMOs depicted in Fig. 27 are designated as F , P, and 1 P states. In the notation of the latter state, it is indicated that one period is built up from concatenated principal states. In fact, in the simulations, many such concatenated states were found for example, between the P and the P state, P(P) states with n going from 1 to 10 were observed. These sequences are called Farey sequences because a one-to-one correspondence of successive MMO states and the ordering of the rational numbers, which is conveniently represented in a Farey tree (see Fig. 31), can be established. In general, at low values of the resistance, the sequences of MMOs obey an incomplete Farey arithmetic. 

Equation (16) was originally derived to model the reduction of In " from SCN solution on the HMDE. The bifurcation behavior of this system is summarized in the two-parameter bifurcation diagram in Fig. 29. Most remarkably, the two distinct MMO sequences of the model also show up in the experiment. Farey sequences were observed close to the Hopf bifurcation at low values of the series resistance, whereas at the high resistance end of the oscillatory regime, periodic-chaotic mixed-mode sequences were found. Owing to this good agreement of the bifurcation... 

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. 

Figure 3.5 The Farey sequence F5 as defined by the visible points of a lattice, for convenience shown here with an obtuse angle...

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. 

The periodic table of the elements is a subset of a more general periodic function that relates all natural nuclides in terms of integer numbers of protons and neutrons, the subject of elementary number theory. The entire structure is reproduced in terms of Farey sequences and Ford circles. The periodicity arises from closure of the function that relates nuclear stability to isotopic composition and nucleon number. It is closed in two dimensions with involution that relates matter to antimatter and explains nuclear stability and electronic configuration in terms of space-time curvature. The variability of electronic structure predicts a non-Doppler redshift in galactic and quasar light, not taken into account in standard cosmology. 

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. 

From a chemical perspective the most important cosmological evidence includes the relationship between the periodicity of matter, prime numbers, Farey sequences, other aspects of number theory, cosmic abundance of the elements and nucleogenesis. These emerging periodic patterns are diametrically opposed to accepted explanations based on standard cosmology, but well in line with Veblen s projective relativity theory, Godel s solution of the general relativistic field equations and Segal s chronometric alternative to Hubble s law. 

Oscillations in an Electrochemical System. I. A Farey Sequence Which Does Not Occur on a Torus. 

Obviously, one cannot expect to observe an infinite number of generations on the Farey tree, but Maselko and Swiimey did find that when they were able to adjust their residence time with sufficient precision, they saw the intermediate states predicted by the Farey arithmetic, though after a few cycles the system would drift off to another, higher level state on the tree, presumably because their pump could not maintain the precise flow rate corresponding to the intermediate state. An even more complex and remarkable Farey arithmetic can be formulated for states consisting of sequences of three basic patterns (Maselko and Swinney, 1987). The fact that the mixed-mode oscillations in the BZ system form a Farey sequence places significant constraints on any molecular mechanism or dynamical model formulated to explain this behavior. 

The variation of Z/N with Z, shown in Fig. 7, mirrors the variation of the infinite A -modular sets of Farey sequences, defined by... 

More simply, a plot of the unimodular Farey sequence... 

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 Ford circles that represent the Farey sequence of order 4 represent the periodic table of the elements in complete detail. In particular, they predict the appearance of electron shells with n = 1,6, consisting of 2, 8, 8, 18, 18 and 32 electrons, in this order. 

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... 

The relationship between unimodular Farey sequences and Ford circles [8] enables direct mapping of the periodic function by touching Ford circles, producing a table of the form shown in Fig. 3 [8]. 

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. 

An even better simulation, based on this sequence, is obtained by selecting suitable terms from a single Farey sequence, which means that all terms have the same denominator. This way we derive from the sequence... 

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. 

At some stage, most scientists benefit by the unreasonable effectiveness of mathematical models in science, but too few look for the cause behind this coinsidence. There is a school of thought that detects a parallel between the subjective choice of topics for analysis and the formulation of matching formulae. At the other extreme, numerical systems are considered to have independent existence in the same way as their physical counterparts. It is, for instance, an undisputed fact that the sophistication of cosmological models faithfully follows developments in number theory. While the concept of infinity remains mathematically unresolved the cosmos remains infinite. However, whether number theories are invented or discovered does not decide their utility in science. If nobody understands the remarkable similarity between Farey sequences. Ford circles and the periodicity of matter, it is no excuse not to exploit this consilience to develop powerful new number-theoretic models for chemistry. 

The most important contribution of chemistry to science, the periodic table of the elements, can be derived directly by elementary number theory. The fundamental relationship between Farey sequences and Ford circles is readily demonstrated. 

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