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Abstract

After a brief review of the golden ratio in history and our previous exposition of the fine-structure constant and equations with the exponential function, the fine-structure constant is studied in the context of other research calculating the fine-structure constant from the golden ratio geometry of the hydrogen atom. This research is extended and the fine-structure constant is then calculated in powers of the golden ratio to an accuracy consistent with the most recent publications. The mathematical constants associated with the golden ratio are also involved in both the calculation of the fine-structure constant and the proton-electron mass ratio. These constants are included in symbolic geometry of historical relevance in the science of the ancients. International Journal of Mathematics and Physical Sciences Research, 5, 2, 89-100 (2018).
Michael A. Sherbon
Case Western Reserve University Alumnus
E-mail: michael.sherbon@case.edu
January 24, 2018
Abstract
After a brief review of the golden ratio in history and our previous exposition of the fine-
structure constant and equations with the exponential function, the fine-structure constant is
studied in the context of other research calculating the fine-structure constant from the golden
ratio geometry of the hydrogen atom. This research is extended and the fine-structure constant
is then calculated in powers of the golden ratio to an accuracy consistent with the most recent
publications. The mathematical constants associated with the golden ratio are also involved
in both the calculation of the fine-structure constant and the proton-electron mass ratio. These
constants are included in symbolic geometry of historical relevance in the science of the ancients.
Keywords fine-structure constant, dimensionless physical constants, history of mathematics,
golden ratio, sacred geometry, Fibonacci sequence, mathematical constants, fundamental physics.
As Willem Witteveen states in his book The Great Pyramid of Giza, “Every expression of the
golden mean, which includes: number, rectangle, triangle, spiral and frequency, is encoded in
the design of the Great Pyramid and illustrates the importance of the ratio in the universe in
which we belong. [1]. He also states, “The golden ratio, as well as the Great Pyramid as an
expression of it, is an important key to our universe containing the Earth and the Moon” [1] and
that “the ratio between the Earth and the Moon is in fact the basis for the mathematical concept
of ‘squaring the circle’ ...” [1]. Marja de Vries states, “The Golden Ratio defines the squaring
of a circle .... According to some, in ancient Egypt, this mathematical mystery was encoded
in the measurements of the Great Pyramid of Giza.” [2]. Continuing with her general theme of
1
universal laws and wholeness, de Vries says, “In short, the idea dawns that the one universal
principle ... embodiment of the Principle of Least Action ... indeed seems to be the Golden
Ratio Spiral.” [2]. Richard Heath has another description, “The Golden Mean was considered
a fundamental constant by the Egyptians and the fundamental division of the whole into two
parts.” [3]. Mario Livio says, “In fact, it is probably fair to say that the Golden Ratio has inspired
thinkers of all disciplines like no other number in the history of mathematics.” [4].
Alexey Stakhov explains that “one of the most important trends in the development of mod-
ern science ... is very simple: a return to the ‘harmonic ideas’ of Pythagoras and Plato (the
‘golden ratio’ and Platonic solids), embodied in Euclid’s ‘Elements’” [5], also see [6]. As stated
previously, “The golden ratio is an approximate harmonic of the Planck length in meters and
harmonics of fundamental units have a geometric basis in ancient metrology. [7, 8]. Further
consideration on the nature of the golden section is given by Scott Olsen [9]. Fundamental
modern applications are suggested by David Haight, “There is considerable evidence that the
golden proportion is the foundation for the unification of mathematics and physics. [10], also
see Gazalé [11]. In the description of Eckhart Schmitz, “Mathematics is a universal language
and it would be fitting to examine the Great Pyramid in this language to derive its meaning. It
has been suggested that the Great Pyramid is a repository of ancient knowledge.” [12].
Thousands of years ago the ancients had an advanced mathematical understand-
ing of universe that is revealed in many sources. There is a consistent link to knowl-
edge of the golden mean, but the way in which the ancients were able to formulate
and use this information speaks of a technical grasp of the subject that exceeds what
we know about it in the present day. –Alison Primrose [13].
David Haight says, “The golden proportion is the only one in which its (legato) addition and
(staccato) multiplication of itself are equivalent. It is both an arithmetic and a geometrical pro-
gression, two sides of the same coin (another “two that are one”), and is the basis of logarithms
and exponentials (logarithms transform multiplication into addition, and exponentials transform
addition into multiplication). [10].
In The Essence of the Cabalah, William Eisen describes the fundamental geometry of what he
described as the “Golden Apex of the Great Pyramid” [14]. Eisen’s description and interpretation
of Euler’s identity [15]-[17], exp(iπ) + 1=0 (“this most compact and mysterious formula” that
“Richard Feynman referred to as ‘the most remarkable formula in mathematics,’”[18]) in relation
to the Great Pyramid shows a drawing of four curves of exfrom x=0 to x=π, one curve on
each side and labelled the “Graphical Representation of the Exponential Function to the Base e.
ex=exp(x) =
n=0
xn
x!=lim
n1+x
nn.(1)
In addition to the exponential function [19, 20], Euler’s formula is exp(ix) = cos x+isinxand in
hyperbolic terms exp(x) = coshx+sinhx. Eisen was asking himself how the ancient Egyptians
could know so much about this and how it supports his effort to understand the role of imaginary
and complex numbers in the geometry of the Great Pyramid [14, 15]. The measures he found
in this model with exponential curves align with other traditional measures found in the Great
Pyramid design, including the golden ratio, also given as φ=exp(iπ/5) + exp(iπ/5).
2
These exponential curves could be considered as having the esoteric properties of an alchem-
ical vessel as Steven Rosen explains that they “possess the feature of curving back into them-
selves.” [21]. Wolfgang Pauli’s i ring, part of his World Clock vision [22], was “used in his
description of microphysical spin” [21]. Rosen quotes Pauli, “The ring with the iis the unity
beyond particle and wave, and at the same time the operation that generates either of these.
[21]. Also see the following references for Pauli and the spin concept development [23]-[29].
The exponential growth function is the simplest possible differential equation,
the unique solution to dy/dx =y;y(0)1 and the most primitive (prime) derivative
in which state and rate, form and function, symmetry and dynamism, being and
becoming, structure and process, the integral and the derivative, the evolute and the
involute, the ’splice’ and the ’slice’ and the squaring of a root and the extraction of
a root are the same. –David Haight [30].
Dividing the sides of his mathematical model for the Great Pyramid by πlengths results in a
small square in the center called the Golden Apex, the geometry and symmetry thought to be
associated with the generation of the four fundamental forces of nature [14, 15]. Eisen provides
a description of the dimensions formed by the exponential function and the Golden Apex square,
A=eπ7π1'2/3π'0.1495.(2)
Ais the side length of the Golden Apex square. A'e/7 and A+1=eπ7π'R'1.152,
radius of the regular heptagon with side one. The heptagon is traditionally associated with
the geometry of ’squaring the circle’. The sin(2πA)'φ/2,where φis the golden ratio [31].
The tan(2πA)'1+A'K/2π,see Eq.(7) [32]. The polygon circumscribing constant is
K'2tan(3π/7)'φ2/2A,see Eq.(7) discussion and 2A'cosφ'p2/7π[7, 32, 33]. A
is also the reciprocal harmonic of Newton’s gravitational constant. Also, A'ln(φ)/φ'
3/e3'tan2(e1)and ln(A1)'π/e'p1+7'6/π'p1+φ2.The regular heptagon
radius, R=csc(π/7)/2'φ/2'cot2α1and 2πR'1/φα.RA '2α,where αis the
fine-structure constant, see the Eq.(7) discussion below [7, 32, 33].
RA 'pφ/e2'ln(π/7)'p7/π/K.(3)
R1'φsinα1and e2+φ2'RK.The cosh2(A)'π/e.A'αcosh(π/e).The p7/π'
coth2R'ARK [7, 32, 33]. The silver constant from the heptagon [34, 35] is approximated by
S'22R'tanφ'3.247.The S=2cos(π/7)'7φ/2πand 2πA'S/23'sin70.Also
relevant, 2A'S/11 '1/11,see the discussion of Eq.(9).
More approximations with the Great Pyramid’s Golden Apex A[7, 32, 33]:
A'11/7π'e/11 'πα '2παS.(4)
As Jean-Paul and Robert Bauval describe in the Secret Chamber Revisited how prime numbers
7 and 11 are significant keys to the Great Pyramid, 22/7'π[36]. Also noted by David Haight,
“When the Fibonacci number sequence is based on the number seven and its multiples, the
Fibonacci sequence self-reflexively reappears when differences are calculated between it and this
new number-seven-based Fibonacci sequence. The same thing happens with Lucas numbers.
3
[10]. With the fine-structure constant, 2πα is equal to the electron Compton wavelength divided
by the Bohr radius [21] and πα is the percentage of light absorbed by graphene [37]. The
α'R/4π'3A/2φ2and 2A'π/S.Also, 4/π'A/2A'pS/2 and see the Eq.(10)
discussion. Finally, e/φ'1+αφ2[7, 32, 33].
From David Haight, “prime numbers, the ‘atoms’ of mathematics, are necessarily related to the
atoms of nature because of the well-known Rydberg rule that follows the same pattern as Euler’s
harmonic zeta power series (derived from the self-derived exponential growth function). [30].
The Euler product formula for the Riemann zeta function [38, 39]:
ζ(2) =
n=1
1
n2=
pprime
1
1p2=π2
6'1.645.(5)
Approximations for ζ(2)'11A'71 and π2'K/2A'6e. The ln(A1)'Rζ(2)'
6/π,with the cube-sphere proportion. Quoting David F. Haight again, “There is geometry in the
humming of the primes, there is music in the spacing of the primes.” [30]. The sinhφ'ζ(2).
[32, 33]. The polygon circumscribing constant Kis the reciprocal of the Kepler–Bouwkamp
constant [40]-[42], related to “Pauli’s triangle” [22], with sides approximately proportional to
1, φ,qφ5 with the golden ratio φ= (1+5)/2 with 5=2cosh(lnφ).[31]. The prime
constant [32, 33], [43] is described as a binary expansion corresponding to an indicator function
for the set of prime numbers. Defining the prime constant Pfor p(k)as the k-th prime:
P=
k=1
2p(k)'ζ(2)αK'RA '0.4147.(6)
The prime constant, P'RA 'φ2/2π'p3/2K'21.Again, Kis the inverse Ke-
pler–Bouwkamp constant [40]-[42]. Introduced by Arnold Sommerfeld, the fine-structure con-
stant determines the strength of the electromagnetic interaction [44]-[49]. John S. Rigden states,
“The fine-structure constant derives its name from its origin. It first appeared in Sommerfeld’s
work to explain the fine details of the hydrogen spectrum.” [50].
Arnold Sommerfeld generalized Bohr’s model to include elliptical orbits in three
dimensions. He treated the problem relativistically (using Einstein’s formula for the
increase of mass with velocity) .... According to historian Max Jammer, this success
of Sommerfeld’s fine-structure formula “served also as an indirect confirmation of
Einstein’s relativistic formula for the velocity dependence of inertia mass. –Stephen
G. Brush [51].
Charles P. Enz writes, “For Pauli the central problem of electrodynamics was the field concept
and the existence of an elementary charge which is expressible by the fine-structure constant ...
1/137. This fundamental pure number had greatly fascinated Pauli .... For Pauli the explanation
of the number 137 was the test of a successful field theory, a test which no theory has passed up
4
to now.” [52]. And again, Pauli is quoted by Varlaki, Nadai and Bokor concerning his evaluation
and regard for the fine-structure constant, “The theoretical interpretation of its numerical value
is one of the most important unsolved problems of atomic physics.” [53].
Michael Eckert, “Even among physicists of the twenty-first century, the ‘Bohr-Sommerfeld-
Atom’ and the ‘Sommerfeld fine-structure constant,’ remain current concepts. [49]. David
Haight states, “Since the ideal divergence angle in nature is the golden proportion of 137.5
degrees, could this unique proportion be the reason why 137 is the ‘prime’ prime number or
paradigm prime in nature, beginning with hydrogen?” [10]. Harald Fritzsch recalls that “Richard
Feynman, the theory wizard of Caltech in Pasadena, once suggested that every one of his theory
colleagues should write on the blackboard in his office: 137 –how shamefully little we under-
stand!” [54]. Alpha,the electromagnetic coupling constant is α=e2/¯
hc in cgs units [32, 33].
Calculation of the inverse fine-structure constant as an approximate derivation from prime num-
ber theory:
α1'157 337P/7'137.035999168.(7)
This equation gives the same approximate value for the inverse fine-structure constant as deter-
mined in Eqs.(10-12), having three prime numbers and the prime constant [55]. Also, α1'
7tan1(φ)'3/A2.The ln(A1)'1/Pφ'p1+7'p1+φ2and 117 +157 '2×
137,see discussion of Eq.(17). 337157 =180 and the ratio 528/337 'π/2 [7], see discussion
of Eq.(10). The fine-structure constant is described by David Haight as the “basis or boundary
condition of the ‘pre-established harmony’ at the prime ‘atomic’ level of mathematics, physics,
and cosmology, for the reason that the essences of things are numbers, especially primes.” [30].
Other approximations with the Golden Apex of the Great Pyramid: α'Ae/φK'R/4π
and 2πφR'1/α.Also, ln(A1)'K P SPand 2A'2πα 'φ2/K'p2/7π.The
heptagon is a feature of the Cosmological Circle, a geometric template for the Great Pyramid and
many ancient architectural designs; related to the cycloid curve and the history of the least action
principle [8, 22]. From the Golden Apex Aand the silver constant S,2A'tanhS1'tan2(1/2)
, see Eq.(4).
The Wilbraham-Gibbs constant is Gwand the sinc function, sincx=sin(x)/x[56]:
Gw=Zπ
0
sincx dx 'esinα1'K/7π.(8)
The Wilbraham-Gibbs constant Gw'φlnπ'φ2/2'1.852.The Wilbraham-Gibbs constant
is related to the overshoot of Fourier sums in the Gibbs phenomena [56]. Additional approxima-
tions: Gw'sec(1)'exp(φ1)'pφ/πA,see the discussion of Eq.(18).
The inverse Kepler-Bouwkamp constant is the polygon circumscribing constant K[40]-[42]:
K=π
2
n=1
sinc2π
2n+1=
n=3
secπ
n.(9)
The polygon circumscribing constant, K'3/2RA '2 tan(3π/7)'8.700 and KA 'pe/φ'
p5/3'R+A.Half the face apex angle of the Great Pyramid plus half the apex angle is ap-
proximately 70and sinα1'2cos70[7]. The cscα1'Rφ'p7/S'85/2πand
528/504 '7A'π/3,see discussion of Eq.(10). First level sum of Teleois proportions is
85,foundational in Great Pyramid design [7] and 85 '360/φ3,see Eq.(11) discussion. Also,
5
85/11 'R/Aand 528/85 '2π,see Eq.(10) discussion below. The latest value for the inverse
fine-structure constant by Aoyama et al, α1'137.035999157(41),from experimental results
and quantum electrodynamics [57]. The inferred value [58] determined from quantum electro-
dynamic theory and experiment [59] with the least standard uncertainty in CODATA results is
α1'137.035999160(33)[58].
sinα1'504/85K=7!/(713 +137)K.(10)
Again, this equation gives α1'137.035999168 [60]. p13/7'2sinα1and from the har-
monic radii of the Cosmological Circle, 108 +396 =504 [7]. 504/396 =14/11 '4/π.Ad-
ditionally, sinα1'ζ(2)P'1π1'e/πφ'2π/85.504/144 =7/2 and φsin α1'
3/e'φ/R'Ae2.From Eq.(7), sin2α1'157/337.Mamombe demonstrates the link be-
tween 137 and 117 in his analysis of the Fibonacci sequence and describes “its significance to
the theory of the golden section and its relationship with the fine-structure constant,” [61], also
see the discussion of Eq.(17) and [62]. Also, 144/85 'e/φ'1/4A. 528/504 '7A,and the
Great Pyramid Key is 528 'ln(7/A)/α[33]. Pyramid base angle θB'tan1(4/π)'51.85.
The ln(4/π)'Aφand π/4'cscα1sinα1'sinθB.792/5280 =54/360 =3/20 'Aand
φ=2sin54[7].
Helmut Warm found a significant ratio between the Venus/Earth synodic period and the Mars
orbital period to be 17/20 =13/20 '1/ln(2φ)'p2π/K.This had the strongest cycle reso-
nance found among the planets [63]. Also, 17/20 =0.85,the fraction contained in the base angle
of the Great Pyramid. The pyramid apex angle θA,sin α1'θB/θA'2 tan(π/7)'pS/7'
2π/85 [33]. Apex angle of the regular heptagon triangle is 3π/7 and an approximation to the
apex angle of the Great Pyramid. 528/396 =4/3 and 396/85 'pS/A.
While twentieth-century physicists were not able to identify any convincing math-
ematical constants underlying the fine structure, partly because such thinking has
normally not been encouraged, a revolutionary suggestion was recently made by
the Czech physicist Raji Heyrovska, who deduced that the fine structure constant ...
really is defined by the [golden] ratio .... –John Calleman [64].
Raji Heyrovska says, “On noticing the closeness of the fine structure constant ... to the ratio
of the angles, 360/φ2... the author suggested that the small difference ... could be due to the
Sommerfeld’s relativity correction factor. [65]. “It was also pointed out that the ratio 360/φ2
... which is a Golden section of 360, differs from the inverse fine structure constant by ... 2/φ3
... probably due to the difference in the g-factors for the electron and proton ... arising from the
magnetic momenta of the two particles,” with the result of α1'(360/φ2)(2/φ3)[66].
α1'360
φ22
φ3+A2
Kφ4A3
K2φ5+A4
K3φ7.(11)
This extension of Heyrovska’s equation also gives α1'137.035999 168 [67, 68]. Raji Hey-
rovska found it a “surprise to find for the first time that the Bohr radius is divided into two unique
sections at the point of electrical neutrality, which is the Golden point. The Golden ratio, which
manifests itself in many spontaneous creations of Nature, was thus found to originate right in
the core of atoms. [66]. In degrees, the modern golden angle [69], θG=360/φ2and related
6
θg'26.57'tan1(1/2)(twice the Cabibbo angle) is the ancient Golden Angle of Resurrection
noted by Robert Temple [7, 70]. Along with the Golden Angle of Resurrection, Robert Temple
states the Pythagorean Comma, P
cwas one of the greatest secrets of the ancient Egyptians [59].
Highly complex numbers like the Comma of Pythagoras, Pi and Phi (sometimes
called the Golden Proportion) ... lie deep in the structure of the physical universe,
and were seen by the Egyptians as the principles controlling creation, the principles
by which matter is precipitated from the cosmic mind. –Jonathan Black [71].
Aubrey Meyer finds that the “Pythagorean Comma and the Fibonacci series converge on the
Golden Section. [72]. For the same approximate value of α1'137.035999 168,the three
terms on the right of Eq.(11) can be reduced to 2(SA)2/3K2φ44A3/φ6and the last two
terms can be reduced to one term with the Pythagorean Comma, A3/P
cK2φ5.P
c'p2φ/π'
1+α/2π'2ζ(2)/Sand P
c= (3/2)12/27'1.014 [7].
The fourth approximation of the inverse fine-structure constant with this particular value:
α1'1/tanh1τ1'137.035999168,(12)
where τis the root of x4−137x3−6x2+98x+73=0, “which represents a particular quartic plane
curve, different combinations of the coefficients of the general curve give rise to the lemniscate
of Bernoulli. [32]. Number 7 also appears as 2 ×72=98 and 73=343 =6+337. The root
τ'137.0384316101 [73]. The other root is approximately 1.523 'πSA 'πφ.
In another representation of Eq.(12), cothα'τ. Other relations include τ/cosh φ'7π/P'
117/5 (see the discussion of Eq.(18) below) and τ/cosh2φ'7Gw+7'7L2
r7,also from
the discussion of Eq.(18). The cosh φ'1/2Aφ'P
c/A'KA(P
c+1)φ2.“Gauss’s and
Euler’s study of the arc length of Bernoulli’s lemniscate, a polar curve having the general form
of a toric section, led to later work on elliptic functions.” [32]. Also, “the lemniscate, inverse
curve of the hyperbola with respect to its center, has the lemniscate constant Lwhich functions
like πdoes for the circle. [32, 74]. The lemniscate constant L'coshφφ2and sinα1'
6L/3π,where L/πis Gauss’s constant and 6/3 is the diameter of the inscribing sphere of
the octahedron [32].
With the Aand Kconstants shown above, approximate values for the proton-electron mass ratio
including the golden ratio and fine-structure constant:
mp/me'Aφ2/4α2'4K/αφ2.(13)
Other approximations of proton-electron mass ratio with the golden ratio and Golden Apex [7]:
mp/me'4(φ+3)/α'2/Aα.(14)
Also, φsin70'2πφ A'ln(mp/me)/ln(α1)[3, 7] and ln(αmp/me)'φ2cos A.Malcolm H.
Mac Gregor has stated, “The bridge between the electron and the other elementary particles is
7
provided by the fine structure constant .... An expanded form of the constant leads to equations
that define the transformation of electromagnetic energy into electron mass/energy ....” [75]. The
expζ(2)'4πcot2(1)and replacing expζ(2)in mp/me'exp ζ(2)/αAwith 73/(102 191P)
gives a more accurate value, 1836.152673 485 [76]. This is approximate to the value, mp/me'
1836.152673346(81),that has been derived from the most recent experimental measure of the
proton mass by Heiße, F. et al. [77]. Additionally, 73 +191 =102+162 =528/2=264,related
to Great Pyramid geometry in several ways [7, 33].
The half-wavelength associated with a frequency resonant to quartz crystal is 191 meters [33]
and the height of the Great Pyramid without the capstone is about 137 meters: 191/137 '
7/5'P/2A'K/2π'5/φ'528/382.From the discussion of Eq.(7) 528 337 =191,
191/73 'φ2and 102/85 =6/5'π/φ2.The length of the horizontal passage in the Great
Pyramid is 102 feet [78]. 117/102 'Rand 117/73 'φ'191/117.The electron/pion mass
ratio, me/mπ'α/2 and mπ/mp'A,where mπis the pion mass [32].
The omega constant [79] is defined as Lambert W(1)[80]-[82], the attractive fixed point of
ex.W(1) = =exp()'sinh1(4A)'0.5671 and 4A'φ/e.
=exp()'Aln(mp/me)/2.(15)
W(x), the Lambert W-function, is an analog of the golden ratio for exponentials and is expressed
as exp[W(1)] = W(1). Also, '2Aln(A1)'α/A'1/πand ln(α1)'K.From the
zeta function, ζ(2)/'1+ln(A1).
In the Foundation Stone of classical harmonics the alpha harmonic is equal to the sum of the
golden ratio harmonic and the omega constant harmonic, relating the ancient Greek Pythagorean
form of metaphor for the fine-structure constant to golden ratio geometry and architecture [32].
The reciprocal Lucas constant is Lr'1.9628 [81, 82] with a derivation from the golden ratio:
Lr=
n=1
1
Ln
=
n=1
1
(φ)n+φn.(16)
Approximately, Lr'1/pA3'2K/3'φcosh(1)'pe/φ/λ,where λis the Laplace
limit [85] of Kepler’s equation described previously with the fine-structure constant [32]. Also,
cothλ1'φsinα1,see the discussion of Eq.(10) and Eq.(18).
Lr'P/2A'S/4P'p2/φ/.(17)
The cosh1Lr'pe/φ'KA and ln(mp/me)'Lr/A. The reciprocal Lucas constant Lr'
tan117'3φ2/4,the square of the radius of the circumscribing sphere of the dodecahe-
dron. The golden ratio φ'A117 and sinh1(117)'φ/2A.Also, 137 '117 sinh(1)and
117 '2K/A'10/α.The approximate dihedral angle of the dodecahedron 117'180
tan1(2)'Lrtan1(φ1)/T,see Eq.(18) for Tau T. Also, the Pythagorean relation 1172+
1372'1082+1442=1802and 144/117 '2/φ[22]. With the Great Pyramid Key 528 again,
528/137 'L2
rand 137/21 'L2
r/4A; related to the hydrogen emission resonance of the King’s
Chamber [1].
8
Regarding the golden ratio again, Boeyens and Thackeray [86] are quoted by Mamombe [61],
“We suggest that there is a strong case that the so-called, ‘Golden Ratio’ (1.61803 ...) can be
related not only to aspects of mathematics but also to physics, chemistry, biology and the topol-
ogy of space-time.” [86]. Xu and Zhong state, “... we would like to draw attention to a general
theory dealing with the noncommutativity and the fine structure of spacetime which comes to
similar conclusions and sweeping generalizations about the important role which the golden
mean must play in quantum and high energy physics.” [87]. In the Fibonacci inspired sequences
studied by Mamombe the number 117 marks a transition point, which is also a harmonic of the
square root of the inverse fine-structure constant [48] and he describes a “geometrical basis for
the fine-structure constant in the golden section. [61].
The Egyptian Royal Cubit is a traditionally known measure basis for the Royal Cubit and the
numerical harmonic of the meter is also found in the Great Pyramid geometry, π/6'φ2/5'π
φ2'0.5236 meters, which is approximately equal to a Royal Cubit [33]. Lr'(T+7)/φ'
2Tφlnπ,coshT'lnπand Tau Tis described as:
T=α+π/6'2A/'ln(Lr)/pφ.(18)
T'7/5'2πα'0.5309.From the Great Pyramid, with a height of 280 Royal Cubits,
280/117 '7π/Lr'2.4 is the golden angle in radians or 360/φ2'137.5.117'θG(1A)
with θGin degrees. The modern golden angle, θG'φ/T'ln(A1) + 1/2'2.4 radians, see
the discussion of Eq.(11). Also, T'3
A2Aπ'Lr/2Gw'Lr(1A)/π'φ2/ln(α1)'
Lrpφ/7π'cos58.The base of the face angle on the Great Pyramid is approximately 58.
Manu Seyfzadeh explains that “The pyramid’s core design is based on a Kepler Triangle whose
hypotenuse (356 cubits) is in the golden ratio (1.618. . .) to its short leg (220 cubits) incidentally
producing a side-slope angle of 51500.” [88]. T/φ'117/356.
The ratio of angles, 117/51.85'φ π;φπ is another frequently encountered measure in
the Great Pyramid. 356/70 'φ π,from another Kepler Triangle. Seyfzadeh says, “Two classic
interrelated geometric relationships can be approximated from an observable measure of Earth’s
curvature and a sidereal lunar year.” [88]. With another significant number from the Great
Pyramid measures 432,432/117 '6/φ'Lr/T.Also, R/T'2S/3'2ζ(4)and RT '1/ζ(2).
70/117 '4Aand KA 'T6.
The Royal Cubit again, π/6=ζ(2)/π'T/P
c,where P
cis the Pythagorean Comma from the
discussion of Eq.(11). Also, 6/π'PKT 'p1+7. From tetrahedral geomery, cos 19.5'
sin(6/π),in radians. From Great Pyramid geometry, sin1φ2'69.4+90137and the
tan69.4'φ ζ (2)'2/T'tan(φ2θg),see the discussion of Eq.(11). The ln(Lr/T)'φ2/2.
T1'4πA'ζ(2)lnπ'φ+φ1'L2
rLr,T'1/Lr(12A)'2π/Kand 3
T'φ/2.
T/'2πAand T'2A'KA 1.
The Royal Cubit in inches, 144/7'Rc[89]. Alpha, α'A/Rc,related to the symbolism in
historical references concerning the fine-structure constant [90]. Multiplying, α××T'
1/7πRc.Adding, α++T'φ/R,see the discussion of Eq.(10). Also, A++T'
2p140/360 =14/3'φtan(πφ)'φpφ/e'2/2'Rc/2λH,with Rcand λH(21cm
hydrogen emission wavelength) in meters and T'P
cRc.From the hieroglyph for gold, 14/9'
tan(1)and A+'9/4π'pφ/π[7, 22]: “The vertex angle of the nonagon, 140,when
considered as the central angle of a triangle in a circle forms the Egyptian hieroglyph for neb or
9
gold related to quintessence, or the ‘unified field’ of physics. [22].
These suggestive references to the quintessential aether are interpretations of Herschel’s Alpha-
Omega-Taurus Star Gate [91], the Ark of Osiris from Coppens [92] and Hardy’s Pyramid Energy
[93]. Inverse Golden Apex, A1as a harmonic of the Earth frequency utilized by Tesla of 6.67
Hz [94], divided by Tesla’s magnifying transmitter frequency of 11.78 Hz: 6.67/11.78 '.
With Parr’s 51.5 Hz pyramid orb frequency, the ratio is 51.5/11.78 '2π(6/π)'6T'K/2
[95, 96].
Euler’s equation and the exponential function applied to the geometry of the Golden Apex of the
Great Pyramid were part of four different calculations of the inverse fine-structure constant with
the same approximate value. The fine-structure constant and the Golden Apex were related to
the proton-electron mass ratio, golden ratio and other fundamental constants; finally speculating
briefly on the metaphorical physics and mathematical metaphysics of the Great Pyramid [97, 98].
These results illustrate clearly the highly interrelated nature of the fundamen-
tal constants. Dependent as our knowledge of them is upon many different fields of
physics, we have here a good example of the importance of making occasional anal-
yses of the consistency situations of sufficiently inclusive scope to serve as valuable
guides to further research. The present example emphasizes especially that a better
knowledge of the Sommerfeld constant, α, would be of great value to physics at the
present time. –Jesse DuMond [99].
David Haight quotes “Physicist John Wheeler, who coined the term ‘geometrodynamics’ put it
this way, ‘Physics is really geometry .... Some profound connection exists between the funda-
mental constants of microphysics and the geometry of the cosmos.’” [10].
This unification of mathematics through Phi should not come as a complete sur-
prise to us since Phi is related to all three means that are essential to mathematics-
-the arithmetic, the geometric and the harmonic. (These three means are the result
of the calculus of differences, just as the harmonic intervals in music are the result
of the calculus of variations.) –David Haight [10].
Eckhart Schmitz emphasizes, “The Great Pyramid of Giza may clearly be regarded as a reposi-
tory of ancient knowledge. [12]. In conclusion, Witteveen is quoted again on Pythagoras, “All
is number, everything is movement, everything is music in the harmony of the spheres. [1].
This ‘harmony of the spheres’ is exemplified by the Golden Apex [7] in a relation which also
‘squares the circle’ [1], pA/2'3/11.As Robin Heath explains, “This ratio of 3:11 is exactly
the ratio between the Moon’s diameter ... and the diameter of the mean Earth (the first major
treatment of this was in City of Revelation by John Michell [100], who was responsible for its
rediscovery). [3]. With reference to the revelation of this harmony Mary Anne Atwood states,
And here the external and internal worlds were seen to blend together in confluent harmony,
proving and establishing each other, and leaving reason nothing more to doubt of, or the senses
to desire, but a fulfillment under the universal law. [101].
10
Special thanks to Case Western Reserve University, PhilPapers, MathWorld and WolframAlpha.
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... Raji Heyrovska found that the Bohr radius was divided by the golden ratio into two different sections giving α −1 (360/φ 2 ) − (2/φ 3 ) [29], with a difference from experiment possibly due to the g-factors of the electron and proton [30]. Her equation was then extended with the Golden Apex A and the polygon circumscribing constant K [31]. Related to the Golden Apex geometry [27], 667 − 178 − 49 = 440, 4 + 49 + 49 + 178 = 280, 667 + 136 − 137 = 666 and 4 + 49 + 667 = 280 + 440 = 720 [24]. ...
... In 1939 Alfred Landé, who found the Landé g-factor of the electron, stated that the sinh(2π) was significant to an understanding of the fine-structure constant [32,33]. From the Golden Apex view, α −1 λ sinh(2π)/KA sinh(2π)/λ √ K and α sinh(2π) tan −1 (4/π)/ tan −1 (1/2), base angle of the physical structure defining the Golden Apex A divided by the ancient 'golden angle' [24,31]. Another Landé inspired approximation of α sinh(2π) is a root of a cubic equation. ...
... The Golden Apex A π/(3 × 7) 2παS. [28,31]. The conceptual structure defining the Golden Apex has a significant relationship to advanced algebras, modular forms and fundamental physics [27]. ...
Preprint
Full-text available
Research into ancient physical structures, some having been known as the seven wonders of the ancient world, inspired new developments in the early history of mathematics. At the other end of this spectrum of inquiry the research is concerned with the minimum of observations from physical data as exemplified by Eddington's Principle. Current discussions of the interplay between physics and mathematics revive some of this early history of mathematics and offer insight into the fine-structure constant. Arthur Eddington's work leads to a new calculation of the inverse fine-structure constant giving the same approximate value as ancient geometry combined with the golden ratio structure of the hydrogen atom. The hyperbolic function suggested by Alfred Landé leads to another result, involving the Laplace limit of Kepler's equation, with the same approximate value and related to the aforementioned results. The accuracy of these results are consistent with the standard reference. Relationships between the four fundamental coupling constants are also found. Published version: Sherbon, M.A. “Physical Mathematics and The Fine-Structure Constant,” Journal of Advances in Physics, 14, 3, 5758-64 (2018) doi:10.24297/jap.v14i3.7760.
... Raji Heyrovska found that the Bohr radius was divided by the golden ratio into two different sections giving ≃ (360 ⁄ ) − (2 ⁄ ) [29], with a difference from experiment possibly due to the g-factors of the electron and proton [30]. Her equation was then extended with the Golden Apex A and the polygon circumscribing constant K [31]. Related to Golden Apex geometry [27] In 1939 Alfred Landé, who found the Landé g-factor of the electron, stated that the sinh(2π) was significant to an understanding of the fine-structure constant [32,33]. ...
... Related to Golden Apex geometry [27] In 1939 Alfred Landé, who found the Landé g-factor of the electron, stated that the sinh(2π) was significant to an understanding of the fine-structure constant [32,33]. From the Golden Apex view, ≃ λsinh(2π) ⁄ KA ≃ sinh(2π) ⁄ λ√K and αsinh(2π) ≃ (4 ⁄ π) ⁄ (1 ⁄ 2), base angle of the physical structure defining the Golden Apex A divided by the ancient 'golden angle' [24,31]. Another Landé inspired approximation of αsinh(2π) is a root of a cubic equation. ...
... Related to the squaring of the circle are more approximations involving the four fundamental coupling constants. The proportion significant to 'squaring the circle' in the classical tradition was found by John Michell and presented in his study of what he named the Cosmological Circle: 3 ⁄ 11 ≃ √φ − 1 ≃ √φα ⁄ ≃ ≃ √φ ⁄ ln(ln ) [31]. S is the silver constant from the regular heptagon [34]. ...
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Research into ancient physical structures, some having been known as the seven wonders of the ancient world, inspired new developments in the early history of mathematics. At the other end of this spectrum of inquiry the research is concerned with the minimum of observations from physical data as exemplified by Eddington's Principle. Current discussions of the interplay between physics and mathematics revive some of this early history of mathematics and offer insight into the fine-structure constant. Arthur Eddington's work leads to a new calculation of the inverse fine-structure constant giving the same approximate value as ancient geometry combined with the golden ratio structure of the hydrogen atom. The hyperbolic function suggested by Alfred Landé leads to another result, involving the Laplace limit of Kepler's equation, with the same approximate value and related to the aforementioned results. The accuracy of these results are consistent with the standard reference. Relationships between the four fundamental coupling constants are also found.
... In living systems, fractals must satisfy the state of quantum coherence, which maximizes global cohesion as well as local autonomy and enables energy from any local level to spread to the global and conversely concentrate energy to any domain from the entire system. Fractals with fractal dimension in the golden ratio might be the most effective in giving autonomy to the greatest number of cycles because of the golden ratio's link to the fine-structure constant (Miller, 2010;Sherbon, 2018). Global cohesion is also ensured because cycles in a fractal hierarchy are quantum coherent; energy can be shared between global and local. ...
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In recent work in cognitive science, it has been proposed that synchronicity is a self-organizing dynamic in the evolution of nonlinear neural systems. However, capturing the real-time dynamics of synchronicity has been a formidable challenge. This paper provides an overview of a recent model that applies a joint dynamical-statistical approach to predict synchronicity. This model, termed the Harmonic Model, uses time-series data based on the Fibonacci sequence to forecast phase-space trajectories over the lifespan. The multiscale features limit the predictability of synchronicity and can be quantified in terms of conditional probabilities. Fitting model predictions to experimental data enables underlying synchronicity dynamics to be inferred, giving a new quantitative approach to the study of synchronicity. The model will be useful for empirical assessment of synchronicity, and for new therapeutic strategies to be developed on it.
... An approximation from previous work [3], the inverse fine-structure constant −1 is a root of: (33), from experimental measurement and quantum electrodynamics [14]. This calculation of the inverse fine-structure constant gives the same approximate value as ancient geometry combined with the extension of Raji Heyrovska's work on the golden ratio structure of the hydrogen atom [15]. 697 − 137 = 280 + 280 and 365 + 365 − 10 = 280 + 440 = 720. ...
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The golden ratio is found to be related to the fine-structure constant, which determines the strength of the electromagnetic interaction. The golden ratio and classical harmonic propor- tions with quartic equations give an approximate value for the inverse fine-structure constant the same as that discovered previously in the geometry of the hydrogen atom. With the former golden ratio results, relationships are also shown between the four fundamental forces of nature: electromagnetism, the weak force, the strong force and the force of gravitation. (PDF) Golden Ratio Geometry and the Fine-Structure Constant. Available from: https://www.researchgate.net/publication/336937435_Golden_Ratio_Geometry_and_the_Fine-Structure_Constant [accessed Nov 01 2019].
... The latest determination from the Gabrielse group [13] is α −1 137.035 999 150 (33), from experimental measurement and quantum electrodynamics [14]. This calculation of the inverse fine-structure constant gives the same approximate value as ancient geometry combined with the extension of Raji Heyrovska's work on the golden ratio structure of the hydrogen atom [15]. 697 − 137 = 280 + 280. ...
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Abstract: The golden ratio is found to be related to the fine-structure constant, which determines the strength of the electromagnetic interaction. The golden ratio and classical harmonic propor- tions with quartic equations give an approximate value for the inverse fine-structure constant the same as that discovered previously in the geometry of the hydrogen atom. With the former golden ratio results, relationships are also shown between the four fundamental forces of nature: electromagnetism, the weak force, the strong force and the force of gravitation. (PDF) Golden Ratio Geometry and the Fine-Structure Constant. Available from: https://www.researchgate.net/publication/336663995_Golden_Ratio_Geometry_and_the_Fine-Structure_Constant [accessed Oct 25 2019].
... [17]. This calculation of the inverse fine-structure constant gives the same approximate value as ancient geometry combined with the extension of Raji Heyrovska's work on the golden ratio structure of the hydrogen atom [18]. ...
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The fine-structure constant, which determines the strength of the electromagnetic interaction, is briefly reviewed beginning with its introduction by Arnold Sommerfeld and also includes the interest of Wolfgang Pauli, Paul Dirac, Richard Feynman and others. Sommerfeld was very much a Pythagorean and sometimes compared to Johannes Kepler. The archetypal Pythagorean triangle has long been known as a hiding place for the golden ratio. More recently, the quartic polynomial has also been found as a hiding place for the golden ratio. The Kepler triangle, with its golden ratio proportions, is also a Pythagorean triangle. Combining classical harmonic proportions derived from Kepler’s triangle with quartic equations determine an approximate value for the fine-structure constant that is the same as that found in our previous work with the golden ratio geometry of the hydrogen atom. These results make further progress toward an understanding of the golden ratio as the basis for the fine-structure constant.
... [17]. This calculation of the inverse fine-structure constant gives the same approximate value as ancient geometry combined with the extension of Raji Heyrovska's work on the golden ratio structure of the hydrogen atom [18]. The inverse fine-structure constant is a root of: ...
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The fine-structure constant, which determines the strength of the electromagnetic interaction, is briefly reviewed beginning with its introduction by Arnold Sommerfeld and also includes the interest of Wolfgang Pauli, Paul Dirac, Richard Feynman and others. Sommerfeld was very much a Pythagorean and sometimes compared to Johannes Kepler. The archetypal Pythagorean triangle has long been known as a hiding place for the golden ratio. More recently, the quartic polynomial has also been found as a hiding place for the golden ratio. The Kepler triangle, with its golden ratio proportions, is also a Pythagorean triangle. Combining classical harmonic proportions derived from Kepler's triangle with quartic equations determine an approximate value for the fine-structure constant that is the same as that found in our previous work with the golden ratio geometry of the hydrogen atom. These results make further progress toward an understanding of the golden ratio as the basis for the fine-structure constant. . . . . . . Journal of Advances in Physics https://doi.org/10.24297/jap.v16i1.8402
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