THE GOLDEN BOWL STRUCTURE:

THE PLATONIC LINE, FIBONACCI, AND FEIGENBAUM

The DifferenceThe 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

quaintegral product, and itself a function of φ-geometry,[137.082039324994 - (8.2944φ

^{2})^{-1}]^{-1}=0.00729735 314. . .

that is,

[20φ

^{4}- (8.2944φ^{2})^{-}^{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 = φ^{2}, and *EC** ^{2}*/(

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 *KC*_{1}*L*
where with *KCL/MNC* it forms the decisive interior base of the triangles *IC*_{1}*C-JC*_{1}*C*,
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*
: *IC*_{1}*C*-*JC*_{1}*C* :: *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 existence^{2} -- 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, φ^{2} 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* : *IC*_{1}*C*-*JC*_{1}*C* :: *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, YBa_{2}Cu_{3}O_{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

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 φ^{2}, 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,
φ^{2} +
φ^{2}, minus the
critical value of *a* at which chaotic motion begins ([φ^{2} +
φ^{2}] - ~3.6) identified with the conjunction of
the Feigenbaum constants ([(φ^{2} +
φ^{2}) - ~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 = 10φ^{2}, the diameter of the (undrawn) circle in which
it is inscribed = 8.908130915 (**F*_{10}/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 constants^{10} 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,
φ^{2},

which, differently expressed, is to say that the product of the
Feigenbaum numbers reduced to φ^{2} 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
φ^{2}*r*, 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
φ^{2}*r *
.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 8.2944^{2}, 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, *R _{}c*, divided by the exponential
ten by which the Earth is larger than YBa

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 sponge^{18} 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 YBa_{2}Cu_{3}O_{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 [(φ^{2} +
φ^{2}) - 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,

^{D}E ^{:}* ^{C}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 light^{27}) 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 YBa**** _{2}Cu_{3}O_{6+x}**.
In addition to 10π 12φ