Unimodular fraction

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 ... 

A pair of fractions, adjacent to each other in any sequence, has the property of unimodularity, such that for the pair h/k, l/m the quantity hm — kl = 1. 

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 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 ... 

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,... 

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. 

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. 

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

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