THE GOLDEN BOWL STRUCTURE:
THE PLATONIC LINE, FIBONACCI, AND FEIGENBAUM

 

The Difference

The eight certain digits of the fine-structure constant are

0.00729735. . .

In the infinite pentagonal arrangement of ‘golden bowls’at the φ-level of the structure of existence itself, the square of the diameter of the rim of the ‘defining bowl’ -- at once the square of tenfold the constant ratio of the base diameter of a bowl to the rim diameter of the next smaller in the infinite pentagonal series of bowls -- at once the square of tenfold the defining locus of the center of the logarithmic spiral -- is

137.082039324994. . .

Where φ is the ‘golden section’ and 8.2944 is the first factor of the number uniquely comparable to unity qua integral product, and itself a function of φ-geometry,

[137.082039324994 - (8.2944φ²)⁻¹]⁻¹ =

0.00729735 314. . .

that is,

[20φ⁴ - (8.2944φ²)⁻¹]⁻¹ = 0.00729735 314. . .

 

---

 

Let there be the line of which Plato speaks in Book VI of the Republic, divided into two unequal segments, each further divided into two pieces according to the ratio of the initial division:

Although Plato does not specify the ratio of division beyond stipulating the inequality of the parts, let the disequilibrium of the double bifurcation of the line be captured in the equilibrium which is the ‘divine proportion’ of the ancients, where ED = 1, DC = φ⁻¹, CB = φ⁻¹, and BA = φ^-2, so that EC + CA = φ + 1 = φ², and EC²/(EA·CA) = ED²/(EC·DC) = CB²/(CA·BA) = 1.¹ Then let the segments of the initial division, EC and CA, be squared in such a way that the diagonally divided halves of each square have EC and CA, respectively, as their common sides, and arranged so as to preserve the order of the original proportion:

Let the diagram be further modified to make explicit the distorted mirror-inversion, MNOP : MNC :: IJKL : KLC:

then the two triangles, ICO and JCP, emerge, together with the triangles interior to them, respectively, ICM and JCN, which latter two have a combined area of 2, while the former two triangles, of which the latter are a part, have a combined area of 5^1/2 + 1, or 2φ, so that the ratio of the inner to the outer triangles is φ⁻¹, identically the ratio of CA to EC. But 2 : 2φ :: 2/φ : 2 as ED : EC :: CB : CA. Let, therefore, the two analogous exterior triangles whose combined area equals 2/φ emerge by extending EA by FE, squaring FC and CA as above, and folding at C and along MN so that the triangle MCN lies in the place of KCL, while the latter removes to KC₁L where with KCL/MNC it forms the decisive interior base of the triangles IC₁C-JC₁C, the combined area of which is 2, while, further, the line segments, ML, BD, and NK, hidden from sight in the fold, are replaced by JH, EF, and IG (the defining limits of the trapezoid GHIJ) --- first unfolded:

then folded:

---so folded, the triangles GKO-HLP emerge, the combined area of which is 5^1/2 - 1, or 2/φ, so that finally the proportion, CB : CA :: ED : EC, equals, in the form of the ‘absolutely specified’ Platonic line, GKO-HLP : IC₁C-JC₁C :: ICM-JCN : ICO-JCP, both proportions instantiating the ‘golden rectangle’, where x : y :: y : (x + y).

The product of this fold of the ‘divine proportion’-form of the ‘absolutely specified’ Platonic line, the ‘bowl’ which manifestly embodies the beautiful proportion of the ‘golden section’ -- this ‘golden bowl’ -- is related to the bowl which is ‘the foundational measure of the absolute structure of existence² -- the ‘logical bowl’ -- as the construction the base of which (= 2) is the root of the square which is one of the four exploded square segments of the central square of that square which, in turn, ‘perfectly embodies the Absolute Proportion of Absolute Thought’,³ as here illustrated:

whence it is clear that there exists an infinite interlocking and interweaving of ‘golden’ and ‘logical’ bowls in the absolute structure of existence itself, the necessity of which is an infinite interchanging, i.e., a sharing, of the identities of the respective infinite series, i.e., of the respective linear multipliers, φ² and 12, a meta-identity of the two different multiples, which is -- with a difference of less than one in 10,000 -- 10π, the truth (qua rational product, אמאת, "truth," = 10) identified with the circumference of the circle whose diameter is Unity. Indeed, the sum of the differing rational products of the proportion, GKO-HLP : IC₁C-JC₁C :: ICM-JCN : ICO-JCP, that is, the sum of the rational product when the ‘absolutely specified’ Platonic line produces the ‘golden bowl’ and of the rational product when it produces the ‘logical bowl’, respectively, φ^-2 + , is -- with the identical difference of less than one in 10,000 -- .826410456, the volume-surface ratio of the breakthrough superconductor, YBa₂Cu₃O_6+x,⁴ at once -- with a difference of less than one in 100 -- the difference between the roots of the Feigenbaum numbers, i.e., 4.669201609102990^1/ - 2.502907875095892^1/ = .82888166439835.

The Feigenbaum numbers are those fundamental constants of nature which universally govern the periodic doubling of the ‘bifurcation tree’⁵ which -- beginning at the critical value of about 3.6 for the constant a in the equation x_y+1 = ax_y(1-x_y) [a transformation of the logistic equation, N_y+1 = N_y(a-bN_y), where x = aN/b and has a value between 0 and 1] -- eventuates in the infinite branching of values known as chaotic motion, the approach to which is here illustrated:⁶

where the first bifurcation occurs at a = 3 (qua rational products, נקדת מועא, "point of departure," and שבוש, "disruption"/"breakdown"/"disorder," = 3), the second at a = 1+ 6^1/2, where the Feigenbaum numbers first come into play (, qua rational products, נקדת מועא, "point of departure"/שבוש, "disruption" times  פרוק לגורמים, "resolution into factors"), while chaotic motion begins at a = 3.5699 (the sum of the Feigenbaum constants, 4.669201609102990 + 2.502907875095892 = 7.17210948419888, 3.5699 × 2 , qua rational products,  פרט, "to divide into small parts," times כאוס, "chaos," i.e., 3.6 × 2), and becomes complete ‘beyond the tracery of bifurcations’ at a = 4 (=, qua rational products, נקדת מועא, "point of departure"/שבוש, "disruption" times

אין מספר, "innumerable,” i.e., 3 × = 4, = מסגרת, "limit"/"border"/"framework," = מעט מן המעט, "bare minimum," = מקצת מן המקעת, "tiniest fraction" [‘absolute’ "tiniest fraction" = 8.2944E20, i.e., ‘absolute’ נקרת האפם של תנופה, "dead center," identical with the very identity of ברא, "to create," 8.2944E22 × .01, the ‘absolute’ form of the latter, 4, reciprocally the very identity of מסגרת, "framework," מעט מן המעט, "bare minimum," and מקצת מן המקעת, "tiniest fraction"]).⁷  The radical relation of the Feigenbaum numbers to the ‘golden section’ is observed when it is noted, first, that the product divided by the addition of those numbers (~11.7/~7.2) φ, and, second, that on the φ-level of the infinite interlocking and interweaving of ‘golden’ and ‘logical’ bowls in the absolute structure of existence itself’ there exists a series of infinite bifurcations of ‘golden bowls’ -- minus the characteristic of the science of ‘chaos’, viz., the identification of ‘the division into small parts’ and ‘chaos’ -- as here illustrated:

where corresponding to the ‘chaos’ constants, 4.669201609102990 and 2.502907875095892, is the constant φ², which is the ratio both of the distance between the successive levels of ‘golden bowls’ and of the successive diameters of their rims, so that, quite literally, the conjunction of the latter constants, φ² + φ², minus the critical value of a at which chaotic motion begins ([φ² + φ²] - ~3.6) identified with the conjunction of the Feigenbaum constants ([(φ² + φ²) - ~3.6] × [4.669201609102990 + 2.502907875095892]), 11.7 (the product of the latter constants not reduced to that conjunction of themselves equivalent to the identification of ‘to divide in small parts’ and ‘chaos’), at once 11.70820394, the diameter of the rim of the ‘golden bowl’ next larger than the entry bowl of the series (the base diameter of which last = 2, while its rim diameter = 4.472135955 [if the contrary to fact condition were assumed that the order illustrated immediately above were small-to-large then the base of the latter would be touching the rim of the larger as illustrated; see, however, immediately below]), the base diameter of which next larger bowl, 5.236067979, defines -- in an inversion of the large-to-small order -- that pentagon in the infinite pentagonal arrangement of ‘golden bowls’ which immediately encloses the entry bowl together with an infinite number of smaller bowls, as here illustrated:

where, if the largest bowl completely within sight is taken to be the entry bowl, the perimeter of the pentagon = 10φ², the diameter of the (undrawn) circle in which it is inscribed = 8.908130915 (*F₁₀/10), and the radius of the circle (the dividing lines of the pentagon) = 4.454065458.⁸  Now the line bisecting the pentagonal section from the center of the pentagon to the mid-point of the side formed of the base of the ‘defining’ bowl 3.6 (3.60341465 [ φ]) the a value in the logistic equation at which chaotic motion begins,⁹ here the length of the distance from the boundary of the ‘golden bowl’ formed of the fold of the φ-form of the ‘absolutely specified’ Platonic line to the end of the infinite descent of progressively smaller bowls. But the finally remarkable confirmation of the fact that the observed structure of the approach to chaotic motion in physics approximates the φ-level of the absolute structure of existence itself is the fact that the actual distance from the center of the side of the perimeter to the base of the entry bowl, the foundational distance separating the bases of the bowls = 3.60341465/φ = 2.227032729 (the reduction of ‘to divide into small parts’ to the ‘golden section’), which, in turn, re-inverted (= .449027977) -- with a difference slightly greater than one in 1000 -- (6^1/2 + 1) - 3 (= .449489743 φ/3.6), that is, the actual increase in the value of a between the first and second bifurcations in the ‘bifurcation tree’ occurring in the approach to chaos. The specifically significant participation of the Feigenbaum numbers in the ‘divine proportionality’ which is the unifying template of the physical constants¹⁰ is also evident in the form of the proportion which embodies the analogy: the ratio of successive Feigenbaum gaps between the occurrences of bifurcations is to the foundational distance (from the base of the entry bowl to that of the ‘defining’ bowl) as the ratio of the successive Feigenbaum gaps between the two values of a bifurcation is to the foundational diameter of the base of the entry bowl, that is, 4.669201609102990 : 2.227032729 :: 2.502907875095892 : 2, the rational product of which proportion,

(4.669201609102990/2.227032729) × 2.502907875095892/2, φ²,

which, differently expressed, is to say that the product of the Feigenbaum numbers reduced to φ² is -- with a difference less than one in 100 -- the product of the respectively related actual measures of the fundamental ‘golden bowl’, or, again, that the product of the Feigenbaum ratios φ²r, the radius of the foundational regular pentagon (= 2 × 2.227032729) times the universal ratio of the infinite series of ‘golden bowls’.¹¹  (It ought not to be overlooked that 4.669201609102990 × 2.502907875095892 φ²r .08542455⁻¹ [= α^−1/2, the inverse of the absolute value of the electromagnetic coupling constant], the latter at once -- with a difference less than one in 100 -- the radius of the circle circumscribing the pentagon which immediately encloses the ‘defining’ bowl of this pentagon [the diameter of the rim of the latter (with a difference slightly more than one in 10,000) .08542455⁻¹, and is in fact exactly 10 times the constant ratio of the diameter of the base of any ‘golden bowl’ to the rim diameter of the bowl next smaller in the infinite series, therefore 10 times the ratio of its base to the rim of the entry bowl],¹² the perimeter of which next larger pentagon 8.2944², the square of the base factor of ‘absolute’ נקדת האפס של תנופה, "dead center,"¹³ with a difference slightly more than one in 10,000 when exactly .08542455⁻¹ [α^−1/2] is substituted for the value of that radius.)¹⁴

The intricate forest begins to surface, where among the trees clear paths are marked out. Let the base of the ‘defining’ bowl of the pentagonal arrangement of the infinite series of ‘golden bowls’ (whose rim diameter α^−1/2) be the side of a regular hexagon, as here illustrated:

where the perimeter of the hexagon -- with a difference slightly more than one in 100,000 -- is 10, while the circumference of the circle in which the hexagon is inscribed = 32.89918539 (the six-fold angstrom diagonal of the cuboid sub-unit of the "1-2-3" superconductor [the tenth part of this circumference -- with a difference less than one in 10,000 -- equals the value of the Rydberg constant, Rc, divided by the exponential ten by which the Earth is larger than YBa₂Cu₃O_6+x]).¹⁵  Feigenbaum speaks as follows:

One has to look for scaling structures--how do big details relate to little details. You look at fluid disturbances, complicated structures in which the complexity has come about by a persistent process. At some level they don’t care very much what the size of the process is--it could be the size of a pea or the size of a basketball. The process doesn’t care where it is, and moreover it doesn’t care how long it’s been going. The only things that can ever be universal, in a sense, are scaling things.

In a way, art is a theory about the way the world looks to human beings. It’s abundantly obvious that one doesn’t know the world around us in detail. What artists have accomplished is realizing that there’s only a small amount of stuff that’s important, and then seeing what it was. So they can do some of my research for me. When you look at early stuff of Van Gogh there are zillions of details that are put into it, there’s always an immense amount of information in his paintings. It obviously occurred to him, what is the irreducible amount of this stuff that you have to put in. Or you can study the horizons in Dutch ink drawings from around 1600, with tiny trees and cows that look very real. If you look closely, the trees have sort of leafy boundaries, but it doesn’t work if that’s all it is--there are also, sticking in it, little pieces of twiglike stuff. There’s a definite interplay between the softer textures and the things with more definite lines. Somehow the combination gives the correct perception. With Ruysdael and Turner, if you look at the way they construct complicated water, it is clearly done in an iterative way. There’s some level of stuff, and then stuff painted on top of that, and then corrections to that. Turbulent fluids for those painters is always something with a scale idea in it.

I truly do want to know how to describe clouds. But to say there’s a piece over here with that much density, and next to it a piece with this much density--to accumulate that much detailed information, I think is wrong. It’s certainly not how a human being perceives those things, and it’s not how an artist perceives them. Somewhere the business of writing down partial differential equations is not to have done the work on the problem.¹⁶

But not merely a ‘definite interplay’, not merely a definite indefiniteness, a ‘combination’ ‘somehow’ between the indefinite and the definite, which gives the ‘correct perception’, but the beginning of the resistance of the medium at once the very identity of the flow, the beginning of the ‘infinitely shared edge, the existence itself of order’,¹⁷ not merely the Menger sponge¹⁸ with ‘infinite surface area, yet zero volume’, but, for the first time in history the essentially ‘iterative way’ a surface infinitely the very identity of volume:¹⁹ φ, the scaling structure of very existence: the universal scaling thing ‘in the beginning’: the scaling thing essentially first, such that φ not merely "retains an unqualified sameness which makes all and any φ proportion similar to all others.... differ[ing] in scale only...and...tied to the identity of the relation of small part to large,"²⁰ such that φ not merely "opens the possibility of measure, and thereby allies itself with all subsequent measure, particularly with the geometric proportion," such that φ is not merely "but one specific case" of the latter,²¹ but rather such that φ opens the actuality of measure,²² that φ is the ‘irreducible amount’ of the ‘stuff’ of the beginning, the essentially dimensionless number essentially indifferent to scale identically scaling different scales.  φ is not merely either the ‘same’ or the ‘identical’ thing on different scales, but f is the very identity of scale for the first time, the absolute beginning of the body of absolutely dimensionless being, the very inception of absolutely dimensionless body, the very measure of the absolute discontinuity of the continuum. The φ-flatness of the spherical universe manifests itself in the circumscribed six-pointed star:

where each of the twelve sides = the diameter of the rim of the ‘defining’ bowl, ~11.7 ( 11.70820393), at once the inverse of the absolute value of the electromagnetic coupling constant, ~.08542455⁻¹ (= 11.70623667), and the angstrom height of the triple-cuboid structure of YBa₂Cu₃O_6+x, ~11.7, which one-twelfth section of the perimeter of the star -- with a difference of less than one in 10,000 in the case of α^−1/2 -- can be constructed in the clockwise direction (the counter-clockwise sections, mirror images) by dividing the line CD (which drops from the point on the circle perpendicular to the radius AB in the regular decagonal one-tenth [φ-]section of the area of the circle, and [(φ² + φ²) - 3.5699] × [4.669201609102990 + 2.502907875095892])²³ by ~1.0368^1/2 (= the base factor of the one-twelfth linear section of the ‘dead center’ cube,²⁴ at once1.0545887^1/3, the cube root of the base factor of , the geometrical Planck constant) and rotating it through an angle of 24º toward the radius AC.²⁵  The diameter of the circle circumscribing the six-pointed star = 40.55840818, while its circumference = 127.4179972, but the latter, qua base factor, is with a difference less than one in 1000 the diameter of Earth, while the former , the square of the megametric value of the mean radius of Earth,²⁶ at once, qua base factor, with a difference slightly more than one in 100, the circumference of Earth. Here the diameter of Earth is to the circumference of the circle circumscribing the star composed of 12 ‘defining’ bowls as the circumference of Earth is to the diameter of the circle, so that the diameter/circumference relation of Earth is reflected, as in a mirror, in the φ-proportionality of this Star-of-David, with the further twist that the diameter of the starred circle is in itself formed of the square of the megametric value of the mean Earth-radius, that is,

^DE ^{: C}φ ^{:: C}E ^{: D}φ ^{(= )}

(where the φ-Earth complicacy of the last term is the diameter of the circle at once that power of φ which is the metric value of c, the velocity of light²⁷) where, on this surface absolutely indifferent to (any) volume not itself essentially surface, the interchangeability of diameter and circumference is a function of the fact that the a priori φ-relation perfectly measures Earth as the ‘divinely proportional’ root of the intelligible universe.²⁸  Essentially nothing other than the lingering vestige of subjectivity keeps the ‘species-making species’ from creating an absolutely new universe.²⁹

Notes

¹ Cf. D.G. Leahy, Foundation: Matter the Body Itself (Albany, 1996), Section IV.2, where the ratio selected for the division of the Platonic line is 1 : 2.
² Ibid.
³ Ibid. Evidence of the radicality of the relation illustrated immediately below in the text is the fact that the area of the square last mentioned (576) (with a difference slightly more than one in 100,000) the total area of the construction of the four ‘logical’ bowls upon which it is founded ([since the bases of the bowls overlap] 4 × 116.5 [466]) times the 2/φ emergent in the fold producing the ‘golden’ bowl immediately above in the text. If the ‘logical’ square (576) is related to the remainder (110) left over from the fourfold ‘logical’ bowl construction (466) in a proportion where the last is the mean proportional, to wit, 110 : 466 :: 466 : 576, then the sum of the elements of this proportion (with a difference slightly more than one in 100,000) 1000φ (= φ identified with, qua rational product, היה קים, "to be in existence" [1000], at once with תכן, "measurement"/"substance"/"design" [1000]).
⁴ See, on this web, Measure of Superconductive YBa₂Cu₃O_6+x. In addition to 10π 12φ², note also the following significant relations: + φ² 5.76, π × φ² 8.2944, φ²/π , and π-φ² (TAN36)².
⁵ The product of the φ roots of the Feigenbaum numbers, qua rational product, the אמאת, "truth," of חורה, "iteration," i.e., 4.669201609102990^1/φ × 2.502907875095892^1/φ 45.714285/10.
⁶ Drawing from P. Davies, The Cosmic Blueprint: New Discoveries in Nature’s Creative Ability to Order the Universe (New York, 1989), p. 43. For the exposition of the logistic equation, cf. ibid., pp. 35ff.
⁷ The value of the parameter α at which the first bifurcation occurs, and at which, therefore, for that moment, the potential distance between bifurcations to which the ratio 4.669201609102990 corresponds is simply reduced to the distance between the values of the bifurcation to which the ratio 2.502907875095892 corresponds, is 3, which (4.669201609102990/2.502907875095892)φ. The ratio of α = 3 to α = 4, where the ‘tracery of bifurcations’ ends, corresponds to the ratio of the area of the pentagonal arrangement of the infinite series of ‘golden bowls’ to the area of the circle in which it is inscribed (cf. below in the text).  Cf. also, on this web, Real Trinary Logic Geometric Series Infinite Iterative Sequential Summing.  It is the thrust of the analysis in the text above to show, among other things, that Davies’ conclusion as to the significance of Feigenbaum’s numbers is wrong in part.  He says (cf. Davies, The Cosmic Blueprint, p. 44, at the end of his exposition of the logistic equation and Feigenbaum’s discovery): "Feigenbaum came across the curious ‘magic’ numbers 4.669 201. . . and 2.5029. . . by accident, while toying with a small programmable calculator. The significance of these numbers lies not in their values but in the fact that they crop up again and again in completely different contexts. Evidently they represent a fundamental property of certain chaotic systems." But not only their universality but the values of these numbers are intelligible for the first time, together with the values of the other constants of nature (qua values of being itself), which is to say, more specifically, that the universality of the Feigenbaum numbers (cf. J. Gleick, Chaos: Making a New Science [New York, 1988], pp. 155−187) is a function of their manifesting the ‘divine proportionality’ of the very essence of existence itself.
⁸ Cf. Leahy, Foundation, Section III.2.  For 4.454065458 and what follows immediately in the text, including 2.227032729, see, on this web, Epidermal Structure: Cell Formation Frequencies & Corneum Alpha-Coupling.
⁹ Cf. above, this section.
¹⁰ For the fundamental constants see, on this web, The Physical Constants: Functions of the 'Golden Bowl' Arrangements. . ..
¹¹ If, following the inverted large-to-small order in the pentagonal arrangement, the base diameter of the ‘defining’ bowl (2φ²) is substituted in the above proportion for that of the ‘entry’ bowl (2), then, of course, the rational product 1, i.e., the product of the Feigenbaum ratios -- with a difference less than one in 100 -- the product of the related actual foundational measures, so construed. The perfect correspondence of structural contexts makes it clear that, as elsewhere throughout the text, there is here no mere coincidence of numbers arrived at by means of some arbitrary conjunction of values, nor, in the manner of Eddington, mere logic working on the fact after the fact, but, rather, the transcendental deduction of the fact: for the first time the essential logic of the fact.
¹² At once 10 times the ‘retrospective’ ‘square’ value of *F₁ (cf. Leahy, Foundation, Sections III.2 and III.6).
¹³ Cf. Leahy, Foundation, Section III.5. The regular pentagonal arrangement of the infinite series of ‘golden bowls’ embodies the ‘golden section’ in a particularly vivid way: the base of the first bowl immediately enclosed by the pentagon divides the radius of the circumscribed circle precisely according to the ‘divine proportion’, so that the first enclosed bowl is to r/f as the infinite number of smaller bowls in the series is to r/φ², and the first + (the second [= the first + etc.] ) = the whole, the larger section of the division of the pentagonal section always containing one bowl, the smaller section always an infinite number of bowls, the latter infinity of bowls the minimum the further φ-division of which always produces a pentagon in which this infinity occupies the proportionally identical smaller section (so the Hebrew has it that, qua rational product, נקרת האפם של תנופה, "dead center" [= 1] = the אמאת, "truth" [= 10] of the מעוט, "minimum" [= 3.809523E−1 (= ברא, "to create" [.01], identified with מופעה, "appearance" [3.809523E1], at once with הסתעף, "to fork"/"to branch"/"to be split up" [3.809523E1], at once φ^-2, the rational product of GKO-HLP : IC₁C-JC₁C :: ICM-JCN : ICO-JCP in the f-form of the ‘absolutely specified’ Platonic line)], identified with the very identity of עד אין גבול, "without limit"/"infinite" [= .0380952⁻¹ 10φ², the perimeter of the pentagon whose side is the base of the ‘defining’ bowl], i.e., 1 = [.3809523/10] × .03809523⁻¹).  The regular pentagonal arrangement of ‘golden bowls’ makes explicit for the first time in history the real eternal, i.e., the absolute temporality, implicit in the ‘golden section’, viz., that the absolute imperative of this proportion is the first transcendental/the transcendental first, that the absolute is this first proportion, that absolute unity is this first transcendental proportion. Nothing could be simpler than this perception for the first time in history of this first transcendental essence, of this essentially transcendental first, but in fact nothing is more complex than this beginning of an essentially transcendent first -- where bowls are persons, this essential transcendence of the first person -- which is to say, there is no nothing: complexity which cannot be reduced to this beginning of essentially transcendental unity is nothing: the total actual complexity is reducible to this beginning of transcendence, to this transcendent beginning (cf. ibid., Sections II.2 and II.3).
¹⁴ The multiplication of this pentagonal perimeter which immediately encloses the ‘defining’ bowl by the product of the Feigenbaum numbers 801, the sum of the place numbers of the final nine-fold sub-square in the ‘creation-set’ of the unum-founded Fibonacci series, the ‘dead center’ square (‘square-constant’ 8.2944E36) in relation to the diagonal of which square the proportion existing among the values of the velocity of light, the Planck constant, the proton rest mass, and the electron rest mass, viz., c (= 2.99792458E8) : h (= 6.626176E-34) :: m_p (= 1.6726485E-27) : m_e (= 9.109534E-31), is constructed within the ‘creation-set’ of *F, the rational product of which proportion (c/h × m_p/m_e) 8.2944E44.  The Fibonacci number in the entry square of that final nine-fold sub-square, *F₇₈ (= 1.447233403E16), multiplied by COS 36 and divided by the exponential ten by which the Earth is larger than the superconductor YBa₂Cu₃O_6+x also α^−1/2, which is, of course, the approximate value of the angstrom height of "1-2-3." Cf. on this web, Measure of Superconductive YBa₂Cu₃O_6+x.
¹⁵ Cf. ibid. If an entry 36-square of ‘golden bowls’ is formed exactly on the pattern of the entry 576-square of the absolute structure of existence itself (cf. Leahy, Foundation, Section IV.2), then the ratio of the ‘central’ square of the latter to that of the former, 16/(5^1/2 - 1)² [= 4φ²], is precisely the diameter of the circle circumscribing this regular hexagon, which diameter, in turn, 10 times the universal geometric constant ratio of the circumference of a circle to the perimeter of an inscribed regular hexagon.
¹⁶ As quoted by Gleick in Chaos, pp. 186f.
¹⁷ Cf. Leahy, Foundation, Section V.2.
¹⁸ For an illustration, see Gleick, Chaos, p. 101.
¹⁹ Cf. Leahy, Foundation, Section III.5.
²⁰ Cf. A. Vandenbroeck, Philosophical Geometry (Rochester, VT, 1987), pp. 32ff.
²¹ Ibid.
²² Indeed, in the transcendental deduction of Einstein’s E = mc², the form of φ, viz., CB : AC :: AC : AB (= 1), is that of the ‘generic absolute’ proportion which is the first formal antecedent of the ‘absolute absolute’, the proof-form of which latter is ‘the magnitude of Being’ (cf. Leahy, Foundation, Section II.2).
²³ Cf. on this web, Measure of Superconductive YBa₂Cu₃O_6+x.
²⁴ Cf. Leahy, Foundation, Section III.5.
²⁵ 24°, as a percentage of the original angle separation, 54°, is the rational product of GKO-HLP : IC₁C-JC₁C :: ICM-JCN : ICO-JCP when the fold of the ‘absolutely specified’ Platonic line produces the ‘logical’ bowl, at once the rational product of the Absolute Proportion of Absolute Thought (cf. ibid.).
²⁶ Cf. on this web, Measure of Superconductive YBa₂Cu₃O_6+x.
²⁷ Ibid. Note that the circumference of the circle whose diameter is the distance actually traversed in one second by light in vacuum (which circumference, identified with the rational product of היה קים, "to be in existence," at once with that of תכן, "measurement"/"substance"/"design" [cf. above, n. 3], with a difference slightly more than one in one thousand the actual length of Earth’s elliptical orbit around the Sun) with a difference less than one in one thousand the square of the naperian log base raised to a power equal to the Feigenbaum bifurcation interval ratio, identified with the total surface area of the 6 + 2 ‘sides’ of the ‘dead center’ cube, i.e., c·s·π (e^{4.669201609102990})² × 82944, which "design"/"actuality" identity of Earth’s orbit , in turn, qua base factor, the ratio of the ‘created−100-set’ square-constant to that of the ‘creation-set’ in the unum-founded Fibonacci series (cf. Leahy, Foundation, Section III.2).  As 4.669201609102990 is related to the surface of the unitary structure of the ‘dead center’ cube, so the Feigenbaum bifurcation spread ratio as a power of the naperian log base (with a difference less than one in 10,000) the hundredth part of the total perimeter of the ‘dead center’ cube (the log base itself the ‘golden bowl’ ratio times the square of the base factor of the twelfth part of this ‘perimeter’ [2.718281828 φ² × 1.0368]), i.e., e^{2.502907875095892} P_c/100 (= P_c [cf. ibid., Section III.5.] identified with, qua rational product, ברא, "to create" [.01]). With a difference less than 1 in 100000, 82944/(e^{4.669201609102990})² 1000α, i.e., the surface of the unitary structure of the ‘dead center’ cube reduced to the ‘absolute’ form of the Proportion of Absolute Davidic Clarity (ibid.) identified with the order of magnitude of the velocity of light (E-4 × 1E8 [ (e^{4.669201609102990})²]) 1000 times the square of the absolute value of the electromagnetic coupling constant (= the latter likewise identified with, qua rational product, היה קים, "to be in existence" [1000], at once with תכן, "measurement"/"substance"/"design" [1000]).
²⁸ In this connection, also with reference to the identity of circle and square, as well as infinite (straight) line and infinite circle (cf. Leahy, Foundation, Section IV.2), note the proof by the mathematician Miklos Laczkovich that a circle divided in 10⁵⁰ ‘unimaginable’, ‘nonmeasurable’ pieces is able to be rearranged in ‘a square of the same area’, as reported by G. Kolata in The New York Times, June 6, 1989.
²⁹ If the base factors are substituted in the above proportion, 1.2745 : 1.2742 :: 4.0042 : 4.056, then the rational product of the proportion the creation ex nihilo ratio, 9.87654321 × 10^-7 (= qua rational product, בראשית ברא אלהים את השמים ואת הארץ, "In the beginning God created the heaven and the earth"), identified with ‘absolute’ הכרה, "consciousness" (= qua integral product, 1E6), i.e., 9.87654321E-7 × 1E6 = 9.87654321E−1.

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