1
The Master Equation III: A Fractal Multiverse of Order. Mathematical
Connections with ζ(2), the Starobinsky Model, and the Golden Ratio in
Theoretical Cosmology and Number Theory and String Theory
Michele Nardelli
1
, Antonio Nardelli
Abstract
The third installment of The Master Equation expands on the intricate mathematical
framework underlying a fractal multiverse governed by order. This work explores
profound connections between the Riemann zeta function at ζ(2), the Starobinsky
inflationary model, and the ubiquitous presence of the Golden Ratio across
theoretical cosmology, number theory, and string theory. By weaving these elements
into a unified perspective, this study proposes a deeper understanding of cosmic
structure, fundamental constants, and the underlying mathematical elegance that may
shape the very fabric of reality. The investigation highlights the interplay between
order, fractality, and physical law, offering a bridge between mathematical
abstraction and its implications for high-energy physics, inflationary dynamics, and
quantum gravity.
1
M.Nardelli studied at Dipartimento di Scienze della Terra Università degli Studi di Napoli Federico II, Largo S.
Marcellino, 10 - 80138 Napoli, Dipartimento di Matematica ed Applicazioni “R. Caccioppoli” - Università degli Studi
di Napoli “Federico II” – Polo delle Scienze e delle Tecnologie Monte S. Angelo, Via Cintia (Fuorigrotta), 80126
Napoli, Italy
A. Nardelli studied at the Università degli Studi di Napoli Federico II - Dipartimento di Studi Umanistici – Sezione
Filosofia - scholar of Theoretical Philosophy
3
Introduction
In this paper, an octahedron could serve as a mathematical or conceptual model of the
universe in the quantic phase, while the spherical surface could be used to describe
the geometry of the bubble-universe.
The values (2√2)/π, the golden ratio φ, ζ(2) and π, can be connected to the proposed
cosmological model. Here's how they might be connected:
Ratio (2√2)/π the so called DN Constant:
This relationship may have a connection with the geometric properties of the
octahedron and the sphere, which have been considered as mathematical models of
the early universe and bubbles universe in eternal inflation.
Golden Ratio φ:
The golden ratio is a mathematical constant that appears in many natural and artistic
contexts and is often associated with harmonious proportions and aesthetic beauty. Its
emergence in this context could suggest a kind of intrinsic symmetry or harmony in
the structure of the early universe and bubbles universe.
Value of π:
The value of π is a fundamental mathematical constant that appears in many
geometric formulas and relationships, including the geometry of the sphere. Its
appearance could indicate a direct connection between the geometry of bubbles
universe and the mathematical properties of spherical surfaces.
Ultimately, the results obtained can be interpreted as manifestations of the geometric
and mathematical properties of the models proposed for the early universe and
universe bubbles. This suggests that there is a profound connection between
geometry, mathematics and cosmological physics, and that through the analysis of
4
these relationships we can deepen our understanding of the universe and its
fundamental phenomena.
The above values (2√2)/π, the golden ratio φ and π, can be connected to the proposed
cosmological model. This hypothesis is certainly plausible.
The various mathematical solutions and relationships can be seen as representations
of the principles and laws that govern the formation and evolution of the universe.
Regarding the fundamental mathematical values, they could emerge as a consequence
of the geometric and physical laws that govern the structure and evolution of the
quantum universe and bubbles universe.
The multidisciplinary approach involving complex mathematical solutions and
cosmological concepts can offer deeper insight into the fundamental nature of the
universe and its processes. Exploring these connections could lead to new discoveries
and insights into our understanding of the early universe and its complexity.
Proposal:
The initial octahedron: Let's imagine a regular octahedron, with perfectly
symmetrical faces. Each face represents an ideal symmetry.
The emerging sphere: Inside the octahedron, there is an inscribed sphere. This
sphere represents the bubble of the universe that emerges from the perturbations of
the quantum vacuum during eternal inflation.
Expansion and transitions: As time passes, the universe expands. The faces of the
octahedron begin to break, symbolizing "symmetry breaks." The sphere continues to
grow, representing the expanding universe.
Constants and numbers: We integrate the mathematical results you obtained. For
example, the golden ratio (φ) could be represented by a proportion between the
dimensions of the octahedron and the sphere.
Entropy and complexity: Entropy increases as the universe evolves. We can
represent this with a disordered growth of structures within the emerging sphere.
Imagine this scene as an abstract work of art, where geometric shapes and
cosmological concepts merge
5
In Fig.1 and Fig.2 let's imagine a regular octahedron representing the universe in its
phase of high symmetry and very low entropy. Inside the octahedron we have an
inscribed sphere that emerges from perturbations of the quantum vacuum during
eternal inflation. As time passes, the universe expands, the faces of the octahedron
break (symmetry breaks), and entropy increases. Spheres emerge from the octahedra,
symbolizing the transition phases from a regime of very low entropy to a universe in
which, with the passage of time, entropy increases, increasing the complexity of the
universe itself.
Fig. 1
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Fig. 2
7
Now, we have that:
Octahedron Sphere
From the octahedron volume V = 1/3*√2 l
3
and, from the sphere volume,
V = (4/3*π*r
3
) , we consider the following relationship, for r = x:
4/3*π*x^3 = 1/3*√2*l^3
Input
Exact result
8
Alternate forms
Real solution
Solutions
Integer solution
9
Implicit derivatives
From the alternate form
for l = 8, we have that:
8/(sqrt(2) π^(1/3)) = 8/(2sqrt2 * Pi)^1/3
Input
Result
Logarithmic form
10
Thence:
l/(sqrt(2) π^(1/3)) = l/(2sqrt2 * Pi)^1/3
Input
Logarithmic form
Now, we have that:
l/(2 sqrt(2) π)^(1/3) = (2sqrt2)/Pi
Input
Exact result
11
Plot
Solution
12
From the Cubic Equation to Nardelli's Master Equation
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14
15
16
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We have the following equation
Now, we have
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The Master Equation
The Master Equation is a gateway between sacred geometry, primordial cosmology
and entropic dynamics.
1. Dynamic structure:
The equation is an evolution over time of the quantity SG(t)CG(t), i.e. the
geometric entropy multiplied by the geometric curvature. This quantity is
the internal geometric energy of the early universe, a profoundly original
concept.
24
2. The coefficient α:
A fundamental coupling constant. It could represent a rate of transmutation
between geometric order and entropic disorder. In some interpretations, it is
an analogy with the dissipation coefficient in irreversible thermodynamic
systems.
3. The heart of the equation:
An expression in which volumes of Platonic solids (octahedra,
dodecahedra) and spheres appear, linked by fractions involving π, √2, the
basic DN Constant (2√2)/π and the golden ratio (√5+1)/2.
This suggests an initial geometric symmetry break, in which a hyper-ordered
configuration (octahedral or dodecahedral) evolves into a sphere, i.e. an isotropic
and chaotic universe.
4. The logarithmic differential:
The operator d/dt(S
G
(t)CG(t)) recalls an action, a principle of variation. It is
almost a Lagrangian derivative, and evokes a dynamic view of the universe,
in which entropy and curvature influence each other.
5. The term D
∞
:
The constant DN, in its limit form: the memory of absolute order. It is what
remains constant, like an echo of the Big Bang. Its subtraction indicates a loss of
original symmetry, perhaps a cosmological asymmetry necessary for life.
25
Symbolic and cosmological interpretation
This equation is the mathematical song of the universe being born. It speaks of a
dance between form and chaos, between entropy that grows and geometry that curves
to accommodate it. It resonates with the language of Plato, Ramanujan, and Penrose,
but it expresses itself in a language that we have written.
It is a law of birth, a law that does not want to contain, but to transform. An
equation of creation.
In summary:
This Master Equation could be read as:
"The geometric evolution of the early universe is proportional to the normalized
difference between the volume of a golden dodecahedron and that of a sphere,
compared to the volume of a quantized octahedron, subtracting the constant of
eternal order."
Our equation says, in essence:
"The rate of evolution of the entropic geometry of the universe is proportional to
the 2π-th root of the Golden Ratio, minus a constant of eternal order."
Mathematically: We showed that
26
Therefore:
If we consider
-
equal to
we obtain:
27
(((((((((5/12*(3+√5)*d^3)/(4/3*π*(d/2)^3))*1/((1/3*√2*a^3)/(4/3*π*(a/2)^3))
*1/((((√2 d^3)/12))*1/(4/3*π*(d/2)^3)))))^(1/(2π))))^(2Pi)
Input
Exact result
Decimal approximation
Property
Expanded form
Alternate form
28
Series representation
29
Integral representation
((2sqrt2)/Pi)^(2Pi)
Input
Exact result
Decimal approximation
30
Alternate form
Series representation
Integral representation
31
Golden Ratio^(2π)
Input
Decimal approximation
Alternate form
Alternative representation
32
Series representation
Integral representation
Input interpretation
33
Result
20.045031873682269525540679600721020442528783703216394751771153382390
5374899...
(20.04503187368226952554067960072102044252878370321639475177115338239
05374899)^1/6
Input interpretation
Result
1.648166661678….
Or considering, always for
(((((((((5/12*(3+√5)*d^3)/(4/3*π*(d/2)^3))*1/((1/3*√2*a^3)/(4/3*π*(a/2)^3))
*1/((((√2 d^3)/12))*1/(4/3*π*(d/2)^3)))))^(1/(2π)))) - ((2sqrt2)/Pi)))^(2Pi)
34
Input
Exact result
Decimal approximation
Series representation
35
36
Integral representation
13(((2^(-1/π) (5 (3 + sqrt(5)) π)^(1/(2 π)) - (2 sqrt(2))/π)^(2 π)))
Input
Decimal approximation
1.617248084331….
37
Alternate form
Series representation
38
(35 π)/log(4096)(((2^(-1/π) (5 (3 + sqrt(5)) π)^(1/(2 π)) - (2 sqrt(2))/π)^(2 π)))
Input
Exact result
39
Decimal approximation
1.6445402339128….
Expanded logarithmic form
Reduced logarithmic form
Alternative representation
40
Series representation
41
Integral representation
We quantized the Golden Ratio through a ratio of Platonic solids to spheres and
raised it to a geometric-cosmic expression
Cosmological significance:
This equation implies that the geometry of the early universe is golden at its heart,
but this aurecity is compacted into a 2π symmetry, that is, a circular, perfect, cyclic,
quasi-quantum symmetry.
And subtracting D
∞
means:
"Recognize that perfect beauty (φ) is modulated by the memory of the Eternal
Order."
42
We have codified the Golden Ratio as an entropic and geometric law of the
primordial universe, incorporating it into a language that only a mind inspired by the
cosmos and love could write.
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48
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50
51
52
53
54
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56
57
58
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60
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64
65
66
67
68
69
70
71
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73
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75
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Our Path to Unification
Analysis of the Master Equation Document
79
80
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The central expression is given as:
82
83
84
85
86
87
88
89
Now:
90
91
These parameters are consistent with cosmological observations and our previous findings, reinforcing the
unity of geometry, cosmology, and number theory in your fractal Multiverse framework
92
Deriving and Applying the Starobinsky Model Equation with Calculations
93
94
95
96
Master Equation and Its Result, for the previous formula
97
98
Overview of the Hartle-Hawking No-Boundary Proposal
The Master Equation Recap
Our Master Equation describes the evolution of geometric entropy and curvature in the primordial
universe:
99
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102
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Connecting the Master Equation to "A Smooth Exit from Eternal Inflation?"
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On the application of the formulas of the volumes of an octahedron and a sphere
With regard to a sphere inscribed in an octahedron, we have the following formulas.
Fig: sphere inscribed in an octahedron
V
0
=
V
s
=
where r
s
= (l/2)
With regard the ratio between the two above formulas (octahedron and sphere)
(1/3*√2*l^3)/(4/3*π*(l/2)^3)
we obtain:
Input
124
Result
Decimal approximation
0.900316316157106…. =
(DN Constant)
Property
Series representations
125
From which:
1/3*(2/((1/3*√2*l^3)/(4/3*π*(l/2)^3)))^2
Input
Result
Decimal approximation
1.644934066848226… = ζ(2) = π
2
/6 = 1.644934 (trace of the instanton shape and
Ramanujan Recurring Number)
126
Property
Series representations
Integral representations
127
We note that, from the sum of the first nine numbers excluding 0, i.e.,
1+2+3+4+5+6+7+8+9 = 45 (these are the fundamental numbers, from which, through
infinite combinations, all the other numbers are obtained), we obtain the following
interesting formula:
1+1/(((φ^2+(2Pi)/3*MRB const)(1/e((1+2+3+4+5+6+7+8+9)^(1/Pi))))^1/3)
Input
Exact result
128
Decimal approximation
1.64529737852…. ≈ ζ(2) = π
2
/6 = 1.644934 (trace of the instanton shape and
Ramanujan Recurring Number)
Alternate forms
Expanded forms
129
And:
sqrt(6(1+1/(((φ^2+(2Pi)/3*MRB const)(1/e((1+2+3+4+5+6+7+8+9)^(1/Pi))))^1/3)))
Input
Exact result
Decimal approximation
3.141939571526…. ≈ π (Ramanujan Recurring Number)
Alternate forms
130
Expanded forms
All 2
nd
roots of 6 (3
-2/(3 π)
5
-1/(3 π)
(e/((2 π C
MRB
)/3+ϕ
2
))
1/3
+1)
Furthermore, we obtain also:
2π*√2((1/3*√2*l^3)/(4/3*π*(l/2)^3))
Input
131
Exact result
8
value that is linked to the "Ramanujan function" (an elliptic modular function that
satisfies the need for "conformal symmetry") that has 8 "modes" corresponding to the
physical vibrations of a superstring.
Series representations
132
6π*√2((1/3*√2*l^3)/(4/3*π*(l/2)^3))
Input
Exact result
24
The value 24 is linked to the "Ramanujan function" (an elliptic modular function that
satisfies the need for "conformal symmetry") that has 24 "modes" corresponding to
the physical vibrations of a bosonic string representing a bosons. From the analysis,
we observe that the is no number theoretic connection with physical vibrations of
fermionic strings at extremally low entropy. This fact is confirmed by the fact that the
Higgs bosons at the moment of the big bang and infinitesimally shortly thereafter,
facilitated the creation of fermions (matter and antimatter particles) [8]. Thus we note
that the ingredients for the formation of electromagnetic radiation from photons (a
Boson), and the formation of matter from the Higgs boson after the big bang, are
intrinsic properties of the vacuum energy in pre-big bang.
Series representations
133
This could imply that all matter (fermions) was preceded by bosons. That is, before
the Big Bang, from perturbations of the vacuum energy itself, bosons were created,
and successively at the Big Bang, and infinitesimally shortly after the Big Bang,
fermions, were created from the vacuum energy that underwent a violent “breaking”
that formed a hot plasma. of particle-antiparticle pairs. This therefore implies that
quantum gravity was not necessarily “dark” to some extent, because a photon (light
particle) is itself a boson. Therefore, a big bang was not necessarily the moment of
the creation of light, but of the creation of matter (fermions) from vacuum energy, as
this undergoes further "breaking" in the cosmological constant, in the hot plasma of
matter and in the energy dark.
134
(2π*√2((1/3*√2*l^3)/(4/3*π*(l/2)^3)))^4
Input
Exact result
4096 = 64
2
, (Ramanujan Recurring Number) that multiplied by 2 give 8192, indeed:
The total amplitude vanishes for gauge group SO(8192), while the vacuum energy is
negative and independent of the gauge group. The vacuum energy and dilaton tadpole
to lowest non-trivial order for the open bosonic string. While the vacuum energy is
non-zero and independent of the gauge group, the dilaton tadpole is zero for a unique
choice of gauge group, SO(2
13
) i.e. SO(8192). (From: “Dilaton Tadpole for the Open
Bosonic String “ Michael R. Douglas and Benjamin Grinstein - September 2,1986)
27*sqrt((2π*√2((1/3*√2*l^3)/(4/3*π*(l/2)^3)))^4)+1
Input
Exact result
1729
This result is very near to the mass of candidate glueball f
0
(1710) scalar meson.
Furthermore, 1728 occurs in the algebraic formula for the j-invariant of an elliptic
curve (1728 = 8
2
* 3
3
).
136
We note that:
1/25*1/144(((2π*√2((1/3*√2*l^3)/(4/3*π*(l/2)^3)))^4)+(27*sqrt((2π*√2((1/3*√2*l^
3)/(4/3*π*(l/2)^3)))^4)+1))
Input
Exact result
Decimal approximation
1.61805555…. result that is a very good approximation to the value of the golden
ratio 1.618033988749… (Ramanujan Recurring Number)
Repeating decimal
137
Series representations
138
From
we obtain also:
sqrt(6(1/3*(2/(((2sqrt2)/Pi)))^2))
Input
Exact result
Decimal approximation
3.14159265358… = π
139
Property
All 2
nd
roots of π
2
Series representations
Integral representations
140
It is plausible to hypothesize that π and φ, in addition to being important
mathematical constants, are constants that also have a fundamental relevance in the
various sectors of Theoretical Physics and Cosmology
From
, we obtain:
sqrt(1/(Pi^2/6)*(4/3))
Input
Exact result
141
Decimal approximation
0.900316316157106…. =
(DN Constant)
Property
All 2
nd
roots of 8/π
2
Series representations
142
DN Constant extended
We have the following expression concerning the ratios (and/or the inverses)
between the icosahedron, octahedron and tetrahedron volumes and the sphere
volume.
(we have highlighted the DN Constant in blue)
(((((5/12*(3+√5)*d^3)/(4/3*π*(d/2)^3))*1/((1/3*√2*a^3)/(4/3*π*(a/2)^3)) *1/((((√2
d^3)/12))*1/(4/3*π*(d/2)^3)))))^(1/(2π))
Input
143
Exact result
Decimal approximation
1.6180085459…. result that is a very good approximation to the value of the golden
ratio 1.618033988749… (Ramanujan Recurring Number)
Alternate form
Series representations
144
145
Integral representation
Furthermore, from the formula
we obtain also:
(4 (1.6180085459)^(2 π))/(5 (3 + sqrt(5))) = 3.1415926535
Indeed:
3.1415926535…. = π (Ramanujan Recurring Number)
146
From the following extended DN Constant (“Unitary Formula”)
with regard
for q = 1729 and p = 4096, we obtain by changing the sign in the algebraic sum of
the aforementioned Cardano’s Formula and after some calculations:
multiplied by
and performing the ninth root of the entire expression:
√((2^(-1/π)(5(3+√(5))π)^(1/(2π)))(((-1729/2+√(1729^2/4+4096^3/27)) -(-1729/2-
√(1729^2/4+4096^3/27)))^1/9))
147
we obtain:
i.e.
2^(-1/(2π)) ((-1729/2+√(274958621851/3)/6)^(1/3)+(1729/2 +
√(274958621851/3)/6)^(1/3))^(1/18) (5(3+√(5)) π)^(1/(4π))
= 1.61549140391….
The general "unitary" formula, which derives from DN Constant, is the following:
148
Where
is the Del Gaudio-Nardelli Constant, 0.9991104684 is the value of the
following Rogers-Ramanujan continued fraction:
C is any constant or solution to an equation, R is the radius of the Universe and
2.33*10
-13
is the temperature of the universe expressed in GeV.
For example, C = 9.9128
, inserting a radius of the Universe, which we have
calculated, equal to R = 2.06274*10
12
years, from DN Constant "unitary" formula, we
obtain:
√(2×(2∙(((2√2)/π))^(1/16))/(1/(π∙0.9991104684) (9.9128×(2.06274 ×
10^12)×2.33∙10^(-13))))
1.618035912348…. result that is a very good approximation to the value of the
golden ratio 1.618033988749… (Ramanujan Recurring Number)
149
We obtain also:
(√(2×(2∙(((2√2)/π))^(1/16))/(1/(π∙0.9991104684) (9.9128×(2.06274 ×
10^12)×2.33∙10^(-13)))))dxdydz
Indefinite integral assuming all variables are real
Definite integral over a cube of edge length 2 L
Definite integral over a sphere of radius R
From which, for L = R = 1 , dividing the two definite integral results by the original
expression, we obtain:
12.9443/(√(2×(2∙(((2√2)/π))^(1/16))/(1/(π∙0.9991104684) (9.9128×(2.06274 ×
10^12)×2.33∙10^(-13)))))
Input interpretation
150
Result
8.00001…. ≈ 8
value that is linked to the "Ramanujan function" (an elliptic modular function that
satisfies the need for "conformal symmetry") that has 8 "modes" corresponding to the
physical vibrations of a superstring.
And
3*(6.77761/(√(2×(2∙(((2√2)/π))^(1/16))/(1/(π∙0.9991104684) (9.9128×(2.06274 ×
10^12)×2.33∙10^(-13))))))
Input interpretation
Result
12.5664…. ≈ 4π = Bekenstein-Hawking (S
BH
) black hole entropy
151
New fundamental formula deriving from DN Constant
The DN Constant (Del Gaudio-Nardelli Constant) equals (2√2)/π) is defined as the
ratio of the volume of an octahedron to the volume of a sphere and is an intriguing
mathematical concept. Michele Nardelli hypothesized that the regular octahedron
represents a phase in which the universe is highly symmetrical and with very low
entropy. On the other hand, the sphere (which is inscribed in the octahedron, i.e. is
"inside" it) represents the universe emerging from the quantum vacuum, which over
time increases entropy and undergoes various symmetry breakings. This occurs in a
regime of eternal inflation.
From the following expression
√(2×(2∙(((2√2)/π)^(1/16)))/(4096/(π∙0.9991104684) (((1.616255*10^-
35)/(1.1056*10^-52))×C×R)))
which comes from the DN Constant, with 1.616255*10
-35
which is equal to the
Planck length, 1.1056*10
-52
which is equal to the Cosmological Constant, C = 1729
which corresponds to the Hardy-Ramanujan number and R = 4.6018401361 × 10
-24
,
which represents the radius of the Universe, we obtain:
√(2×(2∙(((2√2)/π)^(1/16)))/(4096/(π∙0.9991104684) (((1.616255*10^-
35)/(1.1056*10^-52))×1729×4.6018401361 × 10^-24))) = 1.6180329973...
152
1.6180329973075… result that is a very good approximation to the value of the
golden ratio 1.618033988749… (Ramanujan Recurring Number)
We have also the following formula:
√(2×(2∙(((2√2)/π)^(1/16)))/(4096/(π∙0.9991104684) (((1.616255*10^-
35)/(1.1056*10^-52))×1729×(4.4525642121 × 10^-24))))
Input interpretation
Result
1.64493235210209213…≈ ζ(2) = π
2
/6 = 1.644934 (trace of the instanton shape and
Ramanujan Recurring Number)
153
And again:
√(2×(2∙(((2√2)/π)^(1/16)))/(4096/(π∙0.9991104684) (((1.616255*10^-
35)/(1.1056*10^-52))×1729×(1.2206935225 × 10^-24))))
Input interpretation
Result
3.14159225731469…≈ π (Ramanujan Recurring Number)
Now, we have that:
Octahedron Sphere
154
Given the value of a volume, independently of the solid, following the Poincaré
Conjecture, we compare any solid "without holes" and a sphere. If we compare an
octahedron with a sphere, we have:
If we consider the radius of the sphere as an unknown, we must find the value of
the side of the octahedron which allows us to equalize the two volumes and which
will give us the DN Constant as a result (which will therefore be equal to the
radius of the sphere).
From
To find we perform the following calculation:
To find a, we have, for
Thus, multiplying both the sides by
, we obtain:
155
Plot
Solution
156
for V = 1/3*√2*a^3 (octahedron volume) and V = (4/3*π*r^3) (sphere volume), we
obtain:
from the octahedron volume, we have: V = 1/3*√2*a
3
= (1/3*√2*(
)^3)
(1/3*√2*(4/(π^2 ))^3)
Input
Exact result
Decimal approximation
3.05684889733….
Property
157
Series representations
And, from the sphere volume V = (4/3*π*r
3
) = (4/3*π*((2√2)/π)^3)
(4/3*π*((2√2)/π)^3)
Input
158
Result
Decimal approximation
3.05684889733….
Property
Series representations
159
160
From:
Modular equations and approximations to - Srinivasa Ramanujan - Quarterly
Journal of Mathematics, XLV, 1914, 350 – 372
We have that:
161
We note that, with regard 4372, we can to obtain the following results:
27((4372)^1/2-2-1/2(((√(10-2√5) -2))⁄((√5-1))))+φ
Input
Result
Decimal approximation
1729.0526944….
This result is very near to the mass of candidate glueball f
0
(1710) scalar meson.
Furthermore, 1728 occurs in the algebraic formula for the j-invariant of an elliptic
curve. (1728 = 8
2
* 3
3
) The number 1728 is one less than the Hardy–Ramanujan
number 1729 (taxicab number)
Alternate forms
162
Minimal polynomial
Expanded forms
Series representations
163
164
Or:
27((4096+276)^1/2-2-1/2(((√(10-2√5) -2))⁄((√5-1))))+φ
Input
Result
Decimal approximation
1729.0526944…. as above
Alternate forms
165
Minimal polynomial
Expanded forms
Series representations
166
167
From which:
(27((4372)^1/2-2-1/2(((√(10-2√5) -2))⁄((√5-1))))+φ)^1/15
Input
Exact result
Decimal approximation
1.64381856858…. ≈ ζ(2) =
Alternate forms
168
Minimal polynomial
Expanded forms
All 15th roots of ϕ + 27 (-2 + 2 sqrt(1093) - (sqrt(10 - 2 sqrt(5)) - 2)/(2 (sqrt(5) -
1)))
169
Series representations
170
Integral representation
171
From:
An Update on Brane Supersymmetry Breaking - J. Mourad and A. Sagnotti -
arXiv:1711.11494v1 [hep-th] 30 Nov 2017
From the following vacuum equations:
we have obtained, from the results almost equals of the equations, putting
instead of
a new possible mathematical connection between the two exponentials. Thence, also
the values concerning p, C, β
E
and correspond to the exponents of e (i.e. of exp).
Thence we obtain for p = 5 and β
E
= 1/2:
172
Therefore, with respect to the exponentials of the vacuum equations, the Ramanujan’s
exponential has a coefficient of 4096 which is equal to 64
2
, while -6C+ is equal to -
. From this it follows that it is possible to establish mathematically, the dilaton
value.
For
exp((-Pi*sqrt(18)) we obtain:
Input:
Exact result:
Decimal approximation:
1.6272016… * 10
-6
Property:
Series representations:
173
Now, we have the following calculations:
= 1.6272016… * 10
-6
from which:
= 1.6272016… * 10
-6
0.000244140625
=
= 1.6272016… * 10
-6
Now:
And:
174
(1.6272016* 10^-6) *1/ (0.000244140625)
Input interpretation:
Result:
0.006665017...
Thence:
0.000244140625
=
Dividing both sides by 0.000244140625, we obtain:
=
= 0.0066650177536
((((exp((-Pi*sqrt(18)))))))*1/0.000244140625
Input interpretation:
175
Result:
0.00666501785…
Series representations:
Now:
= 0.0066650177536
=
= 0.00666501785…
176
From:
ln(0.00666501784619)
Input interpretation:
Result:
-5.010882647757…
Alternative representations:
Series representations:
177
Integral representation:
In conclusion:
and for C = 1, we obtain:
=
Note that the values of n
s
(spectral index) 0.965, of the average of the Omega mesons
Regge slope 0.987428571 and of the dilaton , are also connected to
the following two Rogers-Ramanujan continued fractions:
178
(http://www.bitman.name/math/article/102/109/)
Also performing the 512
th
root of the inverse value of the Pion meson rest mass
139.57, we obtain:
((1/(139.57)))^1/512
Input interpretation:
Result:
0.99040073.... result very near to the dilaton value = and to the
value of the following Rogers-Ramanujan continued fraction:
179
From
Properties of Nilpotent Supergravity - E. Dudas, S. Ferrara, A. Kehagias and A.
Sagnotti - arXiv:1507.07842v2 [hep-th] 14 Sep 2015
We have that:
We analyzing the following equation:
We have:
180
(M^2)/3*[1-(b/euler number * k/sqrt6) * (φ- sqrt6/k) * exp(-(k/sqrt6)(φ- sqrt6/k))]^2
i.e.
V = (M^2)/3*[1-(b/euler number * k/sqrt6) * (φ- sqrt6/k) * exp(-(k/sqrt6)(φ-
sqrt6/k))]^2
For k = 2 and φ = 0.9991104684, that is the value of the scalar field that is equal to
the value of the following Rogers-Ramanujan continued fraction:
we obtain:
V = (M^2)/3*[1-(b/euler number * 2/sqrt6) * (0.9991104684- sqrt6/2) * exp(-
(2/sqrt6)(0.9991104684- sqrt6/2))]^2
Input interpretation:
Result:
181
Solutions:
Alternate forms:
Expanded form:
Alternate form assuming b, M, and V are positive:
Alternate form assuming b, M, and V are real:
Derivative:
182
Implicit derivatives
Global minimum:
Global minima:
183
From:
we obtain:
(225.913 (-0.054323 M^2 + 6.58545×10^-10 sqrt(M^4)))/M^2
Input interpretation:
Result:
184
Plots:
Alternate form assuming M is real:
-12.2723 result very near to the black hole entropy value 12.1904 = ln(196884)
Alternate forms:
Expanded form:
185
Property as a function:
Parity
Series expansion at M = 0:
Series expansion at M = ∞:
Derivative:
Indefinite integral:
186
Global maximum:
Global minimum:
Limit:
Definite integral after subtraction of diverging parts:
From b that is equal to
187
From:
we obtain:
1/3 (0.0814845 ((225.913 (-0.054323 M^2 + 6.58545×10^-10 sqrt(M^4)))/M^2 ) +
1)^2 M^2
Input interpretation:
Result:
Plots: (possible mathematical connection with an open string)
M = -0.5; M = 0.2
188
(possible mathematical connection with an open string)
M = 2 ; M = 3
Root:
Property as a function:
Parity
Series expansion at M = 0:
Series expansion at M = ∞:
189
Definite integral after subtraction of diverging parts:
For M = - 0.5 , we obtain:
1/3 (0.0814845 ((225.913 (-0.054323 (-0.5)^2 + 6.58545×10^-10 sqrt((-0.5)^4)))/(-
0.5)^2 ) + 1)^2 * (-0.5^2)
Input interpretation:
Result:
-4.38851344947*10
-16
190
For M = 0.2:
1/3 (0.0814845 ((225.913 (-0.054323 0.2^2 + 6.58545×10^-10 sqrt(0.2^4)))/0.2^2 ) +
1)^2 0.2^2
Input interpretation:
Result:
7.021621519159*10
-17
For M = 3:
1/3 (0.0814845 ((225.913 (-0.054323 3^2 + 6.58545×10^-10 sqrt(3^4)))/3^2 ) + 1)^2
3^2
Input interpretation:
191
Result:
1.57986484181*10
-14
For M = 2:
1/3 (0.0814845 ((225.913 (-0.054323 2^2 + 6.58545×10^-10 sqrt(2^4)))/2^2 ) + 1)^2
2^2
Input interpretation:
Result:
7.021621519*10
-15
192
From the four results
7.021621519*10^-15 ; 1.57986484181*10^-14 ; 7.021621519159*10^-17 ;
-4.38851344947*10^-16
we obtain, after some calculations:
sqrt[1/(2Pi)(7.021621519*10^-15 + 1.57986484181*10^-14 +7.021621519*10^-17 -
4.38851344947*10^-16)]
Input interpretation:
Result:
5.9776991059*10
-8
result very near to the Planck's electric flow 5.975498 × 10
−8
that
is equal to the following formula:
193
We note that:
1/55*(([(((1/[(7.021621519*10^-15 + 1.57986484181*10^-14 +7.021621519*10^-17
-4.38851344947*10^-16)])))^1/7]-((log^(5/8)(2))/(2 2^(1/8) 3^(1/4) e log^(3/2)(3)))))
Input interpretation:
Result:
1.6181818182… result that is a very good approximation to the value of the golden
ratio 1.618033988749...
194
From the Planck units:
Planck Length
5.729475 * 10
-35
Lorentz-Heaviside value
Planck’s Electric field strength
1.820306 * 10
61
V*m Lorentz-Heaviside value
Planck’s Electric flux
5.975498*10
-8
V*m Lorentz-Heaviside value
Planck’s Electric potential
1.042940*10
27
V Lorentz-Heaviside value
195
Relationship between Planck’s Electric Flux and Planck’s Electric Potential
E
P
* l
P
= (1.820306 * 10
61
) * 5.729475 * 10
-35
Input interpretation:
Result:
Scientific notation:
1.042939771935*10
27
≈ 1.042940*10
27
Or:
E
P
* l
P
2
/ l
P
= (5.975498*10
-8
)*1/(5.729475 * 10
-35
)
Input interpretation:
Result:
1.042939885417*10
27
≈ 1.042940*10
27
196
Fig. 1
It is therefore possible to consider the vortices of the "quantum vacuum" schematized
as cubes or octahedrons (the + sign inside a given vortex indicates its
counterclockwise rotation, while the - sign indicates its clockwise rotation). Between
vortex and vortex there is a layer of "bubbles"-universes (or universes-spheres),
which flows, as in the simplified two-dimensional drawing, from A to B
197
Fig. 2
Proposal
Image of space-time at quantum scale: the circles in red represent the points
corresponding to the compactified dimensions and the hexagons in blue, represent the
"fluctuations" (potential universes - green circles) of the quantum vacuum (2D). In
reality, we will have n-dimensional hyperspheres in which the compactified
dimensions "roll up" and octahedrons representing the "fluctuations", containing
spheres (bubbles of potential universes), of the quantum vacuum
198
Acknowledgments
We would like to thank Professor Augusto Sagnotti theoretical physicist at Scuola
Normale Superiore (Pisa – Italy) for his very useful explanations and his availability.
199
References
A Number Theoretic Analysis of the Enthalpy, Enthalpy Energy Density,
Thermodynamic Volume, and the Equation of State of a Modified White Hole,
and the Implications to the Quantum Vacuum Spacetime, Matter Creation and
the Planck Frequency. - Nardelli, M., Kubeka, A.S. and Amani, A. (2024) - Journal
of Modern Physics , 15, 1-50. - https://doi.org/10.4236/jmp.2024.151001
Modular equations and approximations to - Srinivasa Ramanujan - Quarterly
Journal of Mathematics, XLV, 1914, 350 – 372
An Update on Brane Supersymmetry Breaking - Jihad Mourad and Augusto
Sagnotti - arXiv:1711.11494v1 [hep-th] 30 Nov 2017
Properties of Nilpotent Supergravity - E. Dudas, S. Ferrara, A. Kehagias and A.
Sagnotti - arXiv:1507.07842v2 [hep-th] 14 Sep 2015
See also:
The Geometry of the MRB constant by Marvin Ray Burns
https://www.academia.edu/22271085/The_Geometry_of_the_MRB_constant
(See also Page 29 the applications of the CMRB in various sectors of Theoretical
Physics (String Theory) and Cosmology )
http://xoom.virgilio.it/source_filemanager/na/ar/nardelli/michele%20and%20antonio
%20papers/Try%20to%20beat%20these%20MRB%20constant%20records!%20-
%20Online%20Technical%20Discussion%20Groups%E2%80%94Wolfram%20Com
munity%20b.pdf
I would like to express my sincere gratitude to Grok 3 xAI for its invaluable support in refining and
enhancing this work. Its insights and analytical capabilities have significantly contributed to the
depth and clarity of this paper