MEASURE OF SUPERCONDUCTIVE YBa₂Cu₃O_6+x

The 1987 breakthrough in the work on High Temperature Superconductivity was achieved with the production and determination of the structure of the synthetic stone, YBa₂Cu₃O_6+x ("1-2-3"), here illustrated:¹

a triple cuboid unit cell the middle center of which is occupied by an yttrium metal atom, the top and bottom centers each by a barium metal atom, while the corners of the cuboid sub-units are occupied by copper atoms, between which are located oxygen atoms in sites, as illustrated, either fully or partially occupied (the last in the case of the topmost and bottommost planes of the tetragonal phase), or unoccupied (as in the case of the ends of the front-to-back axis of the topmost and bottommost planes of the orthorhombic phase, as well as in the central planes of both orthorhombic and tetragonal phases).

If the relationship of the metal elements constituting this superconductive ‘synthetic rock’ is set out in the following proportion (the atomic weights proportional to the number of atoms of each metal in the "1-2-3" crystal):

Y (88.9059) : Cu₃ (190.638) :: Cu₃ (190.638) : Ba₂ (274.68),

then, the rational product of this proportion 2φ/10 (= [5^1/2 + 1]/10), the tenth part of the rational product of the proportion of the elements of the first smallest subsquare of the 100-square grid of the unum-founded Fibonacci sequence (composed of its diagonal poles, *F₁ [= 1]   *F₁₂ [= 233] and *F₂ [= 2]   *F₁₁ [= 144], ordered according to their LAU functions, i.e., according to the order, 0 : 1 :: 1 : ).²

With a difference less than 1 in 10 billion -- if it actually exists at all -- the rational product of the proportion constituted of all the elements of the compound "1-2-3," where x = .24,

Y (88.9059)/Ba₂ (274.68) × Cu₃ (190.638)/O_6+x (99.8391), = φ⁻¹,

while, likewise, that of the ‘minimum’ (4−1-2-3) order³of this proportion,

O_6+x (99.8391)/Y (88.9059) × Ba₂ (274.68)/Cu₃ (190.638), = φ.

The rational product of the ‘logical’ (1-2-4-3) order of the elements of this proportion,

Y (88.9059/Ba₂ (274.68) × O_6+x (99.8391)/Cu₃ (190.638), ^φ = (4/3 × 1/4)^φ,

the inherent formal prediction of the effective interchangeability here of oxygen and copper in the last place of the proportion as the base (4th : 3rd :: 1st : 4th) of φ, an interchangeability likewise formally anticipated in the foundational atomic weight ratio of the former to the latter, viz., ~.25.⁴  The intelligibility of YBa₂Cu₃O_6+x, and, by inference, of all other superconductive compounds, in terms of φ-proportionality and the unum-founded Fibonacci sequence is manifest.⁵

Let there be the circumscribed regular decagon embodying φ-proportionality:

Inspection shows that the ratio CD/CB the ratio of the height to the diagonal of the triple-cuboid structure of YBa₂Cu₃O_6+x, revealing that the tenth part of the inscribed regular decagon is the fundamental template of the "1-2-3" structure, while the ratio ACB/(AB+AE) the volume/surface ratio of the breakthrough superconductor. In addition to these purely rational coincidences of the φ section and "1-2-3" structures, there exist ratios of the elements of the structure of the former which predict the pattern of metric values found in the latter: for example, the ratio of the circumference of the circle circumscribing the regular decagon to the line B, i.e., 2πr/DB, the angstrom-unit length of the six-fold diagonal of the cuboid sub-unit of the "1-2-3" crystal, while the ratio 2πr/ACB the angstrom-unit length of the average linear sub-section of the total ‘perimeter’ of the structure. Furthermore, the angstrom length of the ‘perimeter’ itself ten times the ratio 2πr/AD, while ten times the ratio 2πr/BCDB the angstrom surface area of the superconductor’s triple-cuboid side. (When n = 1, CD and CB , respectively, the ‘retrospective’ ‘square’ and ‘linear’ values of *F₁,⁶ each, in turn, one-tenth, respectively, the angstrom height and the angstrom diagonal of the YBa₂Cu₃O_6+x triple-cuboid structure.  The notion that the use of the metric system or a base ten measure is explained by referring it to mere convention, effectively the refusal to transcend the Copernican principle, is here, as it is in other ways throughout the text, called radically into question by the significant coincidence of rational and metric relations which exists in the relationship of YBa₂Cu₃O_6+x and the one-tenth section of the regular decagon embodying φ-proportionality. The fact that the decagonal embodiment of the ‘golden section’ predicts the pattern of metric relations in "1-2-3" is evidence that if, hitherto, all the world had agreed that the magnitude of the meter was to be either more or less than the distance light travels in vacuum in 1/299792458th of a second, then, in that case, it would have been necessary now to redefine the meter to that magnitude which, however, has been in fact agreed upon. There is high historical irony in the fact that the light measure of the meter is a refinement of the measure of the 1/10,000,000th part of the polar quadrant of the Earth. But the transcendent irony is that the substitution of the light-measure of the meter for its Earth-measure neither is nor can be a fundamental shift, for the simple reason that the Earth-light-metric itself turns out to be a root function the ‘golden section’. Let be the mean square metric radius of Earth divided by a square megameter, ~(6.37E6 m/1E6 m)² ( 40.561349431 [the megametric value of the average of the polar and equatorial radii of Earth (~6.37) the base factor of the square-constant of the 10,000-square of the unum-founded Fibonacci sequence ( 6.37E2089)⁷]), then,

since

c = 299792458 m/s φ^40.561349431 m/s (φ^{6.37E6 m/1E6 m})² m/s,

c φ m/s,

and,

φ m c^1/ s.

The ‘golden section’ embodied in the circumscribed regular decagon essentially predicts, then, the micrometric value pattern of the atomic structure of YBa₂Cu₃O_6+x, and (on a scale 3.2E15 times that of the superconductor) the f(φ) radical Earth-light macrometric relation. In the Earth-light relation the square megametric value of the mean radius of Earth -- a square on the scale of Earth reduced to a pure number exponent of the ‘golden section’ ratio -- produces the whole number metric value of the velocity of light.⁸  This macrometric relation is at once clear proof of the whole number significance of the micrometric value pattern found in the φ section "1-2-3" decagonal template (3.125E−16 times Earth-scale), which real significance finally depends on the fact that

φ m c^1/ s,

precisely as the definition of the angstrom depends on that of the meter. Here the φ proportion is perceived in essence, i.e., qua foundation of absolute rational unity, to embody the symmetry of an absolutely asymmetric nothing: the symmetry of a nothingless existence: the absolute unity without ‘duality within identity’ which is nothing asymmetric, nothing irrational, i.e., is not irrational unity: absolute unity without duality without identity: perfect unity neither without nor within duality, rational symmetry without an irrational point: in the beginning the absolute asymmetry of unity + nothing (1 + 0 [ex nihilo] = [existence = everything + something]).⁹

Notes

¹ Drawing after R.M. Hazen, The Breakthrough: The Race for the Superconductor (New York, 1988), p. 251.  Cf. "The Woodstock of Physics," Physics News Update, Number 814, March 9, 2007.
² Cf. D.G. Leahy, Foundation: Matter the Body Itself (Albany, 1996), Sections III.1-2.
³ Cf. Leahy, Foundation, Section III.1.
⁴ Cf. ibid., also, Sections III.5 and IV.2.
⁵ The square-planar arrangement of one copper (63.546) atom to four oxygen (15.9994) atoms -- an arrangement essentially predicted in the form of the absolute structure of existence itself (cf. Leahy, Foundation, Section IV.2) -- pertains not only to "1-2-3," but to the superconductivity of the superconductors discovered following the latter’s discovery. Cf. R.M. Hazen, "Perovskites" in Scientific American, June 1988: "Of particular interest is the finding that the copper atoms are not surrounded by the usual octohedral cages. Instead they are mostly in fourfold square-planar or fivefold square-pyramid coordination: they are closely bonded to the four or five surrounding oxygen atoms, which form a square and a square-based pyramid respectively. In both arrangements the copper atoms lie in the same plane as the square. Since 1-2-3 and other high-temperature superconductors are superconducting in directions parallel to the copper planes, many theorists now believe this unusual planar arrangement contributes to the remarkable electronic properties of the materials. Exactly how it does so remains to be clarified, however." Cf. the pyramidal formation of the Trinitarian structure, on this web, Thinking Creation Ex Nihilo. Cf. also, on this web, The Physical Constants: Functions of the ‘Golden Bowl’ Arrangements . . ..
⁶ Cf. Leahy, Foundation, Section III.2.
⁷ On the model of the 100-square grid (cf. ibid).
⁸ The fact that, in effect, the square of the base factor of the mean metric radius of Earth, a part, equates, as exponent of φ, to a whole, the metric measure of the velocity of light, is of the very essence of the geometry of φ, which as mean proportional demonstrates that the whole is the φ² identity of the part identified with the whole. Also, compare φ m ( c^1/ s) to the Planck length, 1.61605E-35 m ( φ × 1E-35 m).
⁹ The logic of the beginning is not to be confused with the attempt to resurrect the past in the form of ‘philosophical geometry’ which posits an equality, qua ‘first geometric elements’, between circle and square, dividing immediately therefore into the two fields of which it understands these figures to be, respectively, the first elements, viz., harmony and geometry (cf. A. VandenBroeck, Philosophical Geometry [Rochester, VT, 1987], pp. 17-20). In the eventuality in fact thought for the first time in the form of the absolute universality of logic, the circle itself is reduced to the square, as volume is to surface, as 3 itself is to 2, as, accordingly, harmony is to geometry (cf. Leahy, Foundation, Sections III.1 and IV.2).

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