Ford Circles

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 follows that the two branches of the Fibonacci-Lucas tree describe Ford-circle periodicities that converge respectively to r and 0.5802 as in Figure 4.6. Assuming that the two branches sprout from and respectively, different Ford-circle sequences can be identified to generate the observed periodicities ... 

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... 

Starting with the hemlines of Figure 4.2 mass-number periodicity in line with the Ford-circle model, and with near 2-fold symmetry around zero, is readily identified ... 

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. 

To explore the periodic structure of the set Sk, and hence of the stable nuclides, it is convenient to represent each fraction h/k by its equivalent Ford circle of radius rp = 1/2k2, centred at coordinates h/k, rp. Any unimodular pair of Ford circles are tangent to each other and to the x-axis. If the x-axis is identified with atomic numbers, touching spheres are interpreted to represent the geometric distribution of electrons in contiguous concentric shells. The predicted shell structure of 2k2 electrons per shell is 2, 8, 8, 18, 18, 32, 32, etc., with sub-shells defined by embedded circles, as 8=2+6,... 

The derived Ford circles define the same periodicity as 4. An alternative convergence through Lucas fractions ... 

Such thoughts of an infinite froth need not be confined to mere armchair speculation. In fact, we can construct and explore such a froth that inhabits the realm of abstract geometry. Moreover, the froth we will make is not simply an artistic design created from some arbitrary mathematical recipe, but rather it comes about naturally from the study of the little-known Ford circles, named after L. R. Ford, who published on this topic in 1938. Ford circles provide an infinite treasure chest to explore, and the circles are among the most mind-numbing mathematical constructs to contemplate. In fact, it turns out that they describe the very fabric of our rational number system in an elegant way. ... 

What follows is a mathematical recipe for creating a Ford froth that characterizes the location of rational numbers in our number system. You can use a compass and some graph paper to get started. No complicated mathematics is required for your journey. Let us begin by choosing any two integers, h and k. Draw a circle with radius 1/2 and centered at h/k, /2k ). For example, if you select h = 1 and k = 2, you draw a circle centered at (0.5, 0.125) and with radius 0.125. Note that the larger the denominator of the fraction h/k, the smaller the radius of its Ford circle. Choose another two values for h and k, and draw another circle. Continue placing circles as many times as you like. 

How many neighbor circles touch an individual circle For the mathematically inclined reader, note that two fractions are called adjacent if their Ford circles are tangent. Any fraction has, in this sense, an infinitude of adjacents. Any circle has an infinitude of circles that kiss it. 

In Figures 14.4, 14.5, and 14.6,1 have represented the Ford circles as spheres and displayed them from different viewpoints and magnifications. By using transparency, we can gaze into some of the spheres and see the internal spheres produced by identical h/k ratios for different values of h and k. Color is used to indicate... 

Gary Adamson from California suggests that we do not plot Ford circles for every possible rational number, but rather that we can confine ourselves to a specific subset of fractions. As one example, consider the sequence of fractions that eventually converge to the golden ratio,... 

What would golden ratio Ford circles look like How many tangent neighbors does each circle have How many circles (fraction terms of the sequence) can you represent before the smallest is too small to be seen without further magnifying your picture ... 

Some technical readers may like to see a proof that the Ford circles representing any two different fractipns cannot overlap. (In the extreme case, they may be tangent.) The mathematical argument follows that of Rademacher. 

Let us consider two Ford circles represented by different fractions h/k and H/K. If d is the distance between their two centers, we have... 

In Fractal Milkshakes and Infinite Archery you ll learn about a bubbly froth lurking in the fabric of our number system. The foam is comprised of an infinite regression of circles known as Ford circles. (A graphical representation of the froth is placed on page ix to whet your appetite.)... 

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. 

Figure 5.2 Ford-circle mapping of the periodic table...

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 ... 

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. 

Fig. 11 Mapping of the periodic table of the elements as the reciprocal radii of the a, unimodular Ford circles...

As discussed in the paper on Atomic Structure, touching Ford circles have radii and -coordinates of 1/2 and -coordinates of hi/ki. The resulting map of 4 converts into the periodic table through the reciprocal radii of the numbered circles. Condition (d) is clearly implied. 

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 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. 

Simulation by number theory is the only known procedure that generates the detailed structure of the periodic table without further assumptions or ad hoc corrections. In its simplest form, the simulation is based on the fact that any atomic nucleus consists of integral numbers of protons (Z) and neutrons N), such that the ratio Z/A is a rational fraction. This ratio converges from unity to the golden ratio (t) with increasing atomic number and yields a distribution commensurate with the periodic table. The detailed structure of the periodic function is contained in the Earey sequence 4 of rational fractions and visuaUzed in its Ford-circle mapping [5]. 

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