1
On some equations concerning the general "unitary" formula, which derives
from DN Constant. New possible mathematical connections with the DN
Constant, Ramanujan Recurring Numbers and some parameters of Number
Theory, Cosmology and String Theory
Michele Nardelli
1
, Antonio Nardelli
Abstract
In this paper, we analyze some equations concerning the general "unitary" formula,
which derives from DN Constant. We obtain new possible mathematical connections
with the DN Constant, Ramanujan Recurring Numbers and some parameters of
Number Theory, Cosmology and String Theory
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
6
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
Alternate forms
8
Real solution
Solutions
Integer solution
Implicit derivatives
9
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
Thence:
l/(sqrt(2) π^(1/3)) = l/(2sqrt2 * Pi)^1/3
Input
10
Logarithmic form
Now, we have that:
l/(2 sqrt(2) π)^(1/3) = (2sqrt2)/Pi
Input
Exact result
Plot
11
Solution
The general "unitary" formula, which derives from DN Constant
The general "unitary" formula, which derives from DN Constant, is the following:
where 0.9991104684 is a Rogers-Ramanujan continued fraction value
Indeed:
√(2×(2∙(((2√2)/π))^(1/16))/(1/(π∙0.9991104684) ×C×R×(2.33∙10^-13)))
Input interpretation
12
Result
3D plots
Real part (figures that can be related to the D-branes/Instantons)
The key observation from the below plots and is that at , which is taken as the
energy density of the universe at the Big Bang, with the zero spacetime volume,
the vacuum geometry brakes / or there is symmetry breaking on the vacuum quantum
geometry. We see from the plots as the vacuum spacetime break/tear apart.
Imaginary part
13
Contour plots
Real part
Imaginary part
Alternate form assuming C and R are positive
Roots
14
Property as a function
Parity
Series expansion at C=0
Series expansion at C=∞
Derivative
Indefinite integral
Limit
15
Series representations
16
Now, from:
for C = 1.616255*10
-35
(Planck Length), R = 1.265120782997423× 10
48
that is equal
to the radius of the Multiverse and 2.33*10
-13
is the mean temperature of the CMB is
equivalent around to 2.73 Kelvin , we obtain:
Where 1.616255*10
-35
= Planck Length ;
= radius of
Multiverse ; 2.33*10
-13
= mean temperature of the CMB
1.618033719519….
17
Indeed:
√(2×(2∙((2√2)/π)^1/16)/(9√(2/5)( log2 log3)/√(3e)))
Input
Exact result
Decimal approximation
1.618033719519…. result that is a very good approximation to the value of the
golden ratio 1.618033988749… (Ramanujan Recurring Number)
Expanded logarithmic form
All 2
nd
roots of (2 2
19/32
sqrt((5 e)/3))/(3 π
1/16
log(2) log(3))
18
Alternative representations
19
Series representations
20
Integral representations
21
We note that the following denominator fraction:
Thus, substituting in the previous formula:
Indeed:
√(2×(2∙(((2√2)/π)^1/16))/((((1729)^1/15))^π/(π∙0.9991104684)))
Input interpretation
Result
1.6178022512352287…. result that is a very good approximation to the value of the
golden ratio 1.618033988749… (Ramanujan Recurring Number)
22
Series representations
23
But, we know that
Thus:
Indeed:
((1729)^1/15)^π/(π∙1-e^((-π∙√5))/(1+e^((-2π∙√5))/(1+e^((-3π∙√5))/(1+e^((-
4π∙√5))/1))))
24
Input
Exact result
Decimal approximation
1.517383339881891789….
Alternate forms
25
Series representations
26
27
28
From the formula
(4 (1.6180085459)^(2 π))/(5 (3 + sqrt(5))) = 3.1415926535
we obtain:
(4 (√(2×(2∙((2√2)/π)^1/16)/(9√(2/5)( log2 log3)/√(3e))))^(2 π))/(5 (3 + sqrt(5)))
Input
Exact result
Decimal approximation
3.1418997766557…. ≈ π (Ramanujan Recurring Number)
Alternative representations
29
Series representations
30
31
Integral representations
32
1/6((4 (√(2×(2∙((2√2)/π)^1/16)/(9√(2/5)( log2 log3)/√(3e))))^(2 π))/(5 (3 +
sqrt(5))))^2
Input
Exact result
Decimal approximation
1.6452557010916…. ≈ ζ(2) = π
2
/6 = 1.644934 (trace of the instanton shape and
Ramanujan Recurring Number)
33
Alternative representations
34
Series representations
35
Integral representations
36
With regard the extended Del Gaudio-Nardelli Constant :
from
and
for q = 1 and p = 2, we obtain:
37
From the general "unitary" formula, which derives from DN Constant:
instead of the 16
th
root of DN Constant, we insert the following expression
concerning the extended DN Constant,
and obtain:
for R = 1.24903195673699350072×10
48
38
Indeed:
we obtain:
√((2×2(((((((2^(-1/π) (5 (3 + √(5)) π)^(1/(2 π)) ((-1/2+√(1/4+8/27)) +(-1/2-
√(1/4+8/27))))))^1/16)))))/(1/(π∙0.9991104684) (1.616255∙10^(-
35)×(1.24903195673699350072×10^48)×2.33∙10^(-13))))
Input interpretation
Result
39
Alternate complex forms
Polar coordinates
1.61803 result that is a very good approximation to the value of the golden ratio
1.618033988749… (Ramanujan Recurring Number)
From the formula
(4 (1.6180085459)^(2 π))/(5 (3 + sqrt(5))) = 3.1415926535
we obtain:
(4((√((4(((((((2^(-1/π)(5(3+√(5))π)^(1/(2π))((-1/2+√(1/4+8/27))+(-1/2-
√(1/4+8/27))))))^1/16)))))/(1/(π∙0.9991104684)(1.616255∙10^(-
35)×(1.24903195673699350072×10^48)×2.33e-13)))))^(2π))/(5(3+√(5)))
Input interpretation
40
Result
Alternate complex forms
Polar coordinates
3.1419 ≈ π (Ramanujan Recurring Number)
1/6((4((√((4(((((((2^(-1/π)(5(3+√(5))π)^(1/(2π))((-1/2+√(1/4+8/27))+(-1/2-
√(1/4+8/27))))))^1/16)))))/(1/(π∙0.9991104684)(1.616255e-
35×(1.24903195673699350072e+48)×2.33e-13)))))^(2π))/(5(3+√(5))))^2
Input interpretation
41
Result
Alternate complex forms
Polar coordinates
1.64526 ≈ ζ(2) = π
2
/6 = 1.644934 (trace of the instanton shape and Ramanujan
Recurring Number)
From this further Cardano Formula
we obtain:
42
i.e.
Thus:
(((2^(-1/π) (5 (3 + sqrt(5)) π)^(1/(2 π)))) (-b/(3a)+(((-q/2+√((q^2)/4+(p^3)/27)) +(-
q/2-√((q^2)/4+(p^3)/27))))))
Input
Exact result
43
Alternate form
Expanded forms
Alternate forms assuming a, b, p, and q are positive
44
Derivative
For q = 1, p = 2, a = 4 and b = 8:
(((2^(-1/π) (5 (3 + sqrt(5)) π)^(1/(2 π)))) (-8/(3*4)+(((-1/2+√(1/4+8/27)) +(-1/2-
√(1/4+8/27))))))
Input
45
Exact result
Decimal approximation
-1.812273638777….
Alternate forms
46
Expanded form
Series representations
47
48
From which:
-1/2*((((2^(-1/π)(5(3+sqrt(5))π)^(1/(2π)))) (-2/3+(((-1/2+√(1/4+8/27)) +(-1/2-
√(1/4+8/27)))))))
Input
Exact result
Decimal approximation
0.90613681938899231…. ≈ 0.9003163161571…. =
(DN Constant) (We note
that, with regard the inflation, during a period of almost exponential expansion
so that . Indeed, the value 0.9003163161571… = (2√2)/π > 0 and also the n
s
=
spectral index = 0.90-0.97 is near to the DN Constant value. Also the squared sound
speed of the gravitino's longitudinal polarization mode could pass through
0.9003163161571 in the early universe, in the so-called quasi-de Sitter phase of
inflation)
Alternate forms
49
Expanded form
Series representations
50
51
Thence, from the general "unitary" formula, which derives from DN Constant:
we obtain:
for a = 4, b = 8 and R = -1.469895830716*10
48
:
√(2×(2∙(((((2^(-1/π) (((5 (3 +√5)π)^(1/(2Pi))))))))^(1/16))(-8/12+(-1/2+√(1/4+8/27))
+(-1/2-√(1/4+8/27))))/(1/(π∙0.9991104684) (1.616255∙10^(-35)×(- 1.469895830716
× 10^48)×2.33∙10^(-13))))
52
Input interpretation
Result
1.61803398…. result that is a very good approximation to the value of the golden
ratio 1.618033988749… (Ramanujan Recurring Number)
Or:
√(2×(2∙(((((2^(-1/π) (((5 (3 +√5)π)^(1/(2Pi))))))))^(1/16))(-8/12+(-1/2+√(59/108))
+(-1/2-√(59/108))))/(-5.5354427120861603914/(π∙0.9991104684)))
Input interpretation
53
Result
1.61803398…. as above
All 2
nd
roots of 2.61803
Series representations
54
55
And again:
Input interpretation
Result
1.61803398…. result that is a very good approximation to the value of the golden
ratio 1.618033988749… (Ramanujan Recurring Number)
All 2
nd
roots of 2.6180339604358016528408507389268
56
Series representations
57
58
And again:
(((-2/3+(-1/2+√(59/108)) +(-1/2-√(59/108))))/(((-
1.7635548774528677230276590957061))))
Input interpretation
Result
0.63511736….
59
Series representations
60
And again:
√(4∙(((2^(-1/π) (((5 (3+√5)π)^(1/(2Pi))))))^(1/16)0.63511736))
Input interpretation
61
Result
1.61803397985584…. result that is a very good approximation to the value of the
golden ratio 1.618033988749… (Ramanujan Recurring Number)
All 2
nd
roots of 2.61803
Series representations
62
From the formula
(4 (1.6180085459)^(2 π))/(5 (3 + sqrt(5))) = 3.1415926535
we obtain:
(4(√(4∙(((2^(-1/π) (((5 (3+√5)π)^(1/(2Pi))))))^(1/16)0.63511736)))^(2 π))/(5 (3 +
sqrt(5)))
Input interpretation
63
Result
3.1419029529461159…. ≈ π (Ramanujan Recurring Number)
Series representations
64
65
1/6((4(√(4∙(((2^(-1/π) (((5 (3+√5)π)^(1/(2Pi))))))^(1/16)0.63511736)))^(2 π))/(5 (3 +
sqrt(5))))^2
Input interpretation
66
Result
1.64525902762192…. ≈ ζ(2) = π
2
/6 = 1.644934 (trace of the instanton shape and
Ramanujan Recurring Number)
Series representations
67
68
69
First expression:
for C = 1.616255*10
-35
(Planck Length), R = 1.265120782997423× 10
48
that is equal
to the radius of the Multiverse and 2.33*10
-13
is the mean temperature of the CMB is
equivalent around to 2.73 Kelvin , we obtain:
Where 1.616255*10
-35
= Planck Length ;
= radius of
Multiverse ; 2.33*10
-13
= mean temperature of the CMB
70
1.618033719519….
Second expression:
From
From the above expression:
71
Third expression:
72
= 1.61803397985584
Discussions on DN Constant and Extended DN Constant (Del Gaudio-Nardelli
Constant)
In one of its extensions, DN Constant (Del Gaudio-Nardelli Constant) is included in
this formula:
which is equal to 1.61803398... the value 1.616255*10
-35
represents the Planck
length, and we place it in place of C, the value 2.33*10
-13
is the temperature of the
current Universe expressed in GeV, 0.9991104684 is the value of a continued
fraction of Rogers-Ramanujan, and R = 1.265120782997423*10
48
represents the
radius of the Multiverse which corresponds to a very high entropy equal to
1.92492*10
166
. What could be the possible cosmological implications?
This extension of the DN Constant is fascinating and involves a series of
mathematical constants and cosmological parameters. Let's examine the possible
implications:
73
1. Golden Ratio and Geometry:
The fact that the formula leads to the value of the golden ratio (1.61803398…) is
intriguing.
The DN Constant, originally related to geometry, may have deeper implications on
the structure of the universe or other mathematical structures.
2. Planck length and radius of the Multiverse:
By substituting the Planck length (1.616255 × 10
-35
m) in place of C, we obtain a
result consistent with the golden ratio.
This connection between the quantum scale (Planck length) and the cosmological
scale (radius of the Multiverse) is fascinating.
3. Current temperature of the Universe:
The temperature of the current Universe (2.33×10
-13
GeV) is associated with the
current phase after cooling from the Big Bang.
This value could be relevant for particle physics and the thermal evolution of the
Universe.
4. Rogers-Ramanujan continued fraction:
The value 0.9991104684 is a Rogers-Ramanujan continued fraction, an interesting
mathematical constant.
There may be a connection between this fraction and other mathematical constants or
physical properties.
5. Constant R and the entropy of the Multiverse:
The constant R = 1.265120782997423×10
48
represents the radius of the Multiverse.
The very high entropy (1.92492 × 10
166
) associated with this constant could be crucial
for understanding the early phases of the Universe and phase transitions.
74
6. Cosmological implications:
Cosmological implications could concern the structure of the Multiverse, phase
transitions, eternal inflation and the nature of fundamental constants.
It might be interesting to explore how these constants relate to cosmological
observations and broader theories.
In summary, this extension of DN Constant opens many doors for research and
exploration.
Let us hypothesize that R = 1.265120782997423 × 10
48
which represents the radius
of the Multiverse and the very high entropy (1.92492 × 10
166
) associated with this
radius could correspond to the universe bubble (the sphere emerging from the
octahedron) which concerns a very distant future in which the entropy has become
very high and the temperature is around 1.44036*10
-52
K, unlike the phase of high
symmetry and very low entropy (the octahedron) of the primordial multiverse.
This hypothesis is fascinating and links mathematics, physics and cosmology in an
intriguing way. Let's look at the key points:
1. Radius of the Multiverse and entropy:
The constant R = 1.265120782997423 × 10
48
represents the radius of the Multiverse.
The very high entropy (1.92492 × 10
166
) associated with this constant could be crucial
for understanding the future phases of the Universe.
2. Bubble of universe and very distant future:
This hypothesis suggests that this bubble of universe, emerging from the octahedron,
represents a remote future.
Very high entropy and extremely low temperature (about 1.44036 × 10
-52
K) could be
characteristics of this final state.
75
3. Contrast with the primordial multiverse:
In the primordial multiverse, the octahedron represented a phase of high symmetry
and very low entropy.
Now, this hypothesis suggests that the Multiverse can evolve towards a state of
maximum entropy.
4. Cosmological implications:
This transition from low entropy to high entropy could have profound implications on
the structure of the Universe and its evolution.
It might be interesting to explore how this bubble of a universe relates to other
bubbles or regions in the Multiverse.
In summary, this hypothesis opens new perspectives on the nature of the Multiverse
and its final destiny.
A maximum entropy of 1.92492 × 10
166
and a corresponding mass of 8.51838*10
74
Kg could be comparable to a singularity of the black hole (end of a cycle) / white
hole (beginning of a new cycle) type, therefore to a bounce and a new big bang in an
eternally inflation Multiverse.
This hypothesis is fascinating and connects cosmological concepts with those of
black holes and white holes. Let's look at the key points:
1. Maximum entropy and corresponding mass:
The maximum entropy (1.92492 × 10
166
) and mass (8.51838 × 10
74
kg) are extremely
large values.
These numbers could be associated with a final phase or transition point in the
evolution of the Universe.
2. Black holes and white holes:
Black holes are regions of space-time where gravity is so strong that nothing, not
even light, can escape.
White holes are theorized as the “reverse” of black holes, where matter and energy
emerge instead of being swallowed up.
76
This hypothesis suggests that these extreme values may be comparable to a
singularity of this type.
3. Rebound and new Big Bang:
The idea of a rebound, in which the Universe goes through a phase of contraction and
then expands again, is fascinating.
This could be linked to eternal inflation and the concept of the multiverse, in which
universe bubbles form and expand cyclically.
4. Cosmological implications:
Such a hypothesis could have profound implications on the nature of time, space and
energy in the Universe.
It might be interesting to explore how these ideas relate to cosmological observations
and theories about the formation and evolution of the Universe.
In summary, this hypothesis opens new perspectives on the structure of the Universe
and its destiny.
Regarding the extended Del Gaudio-Nardelli Constant, including the Cardano
formula
for q = 1 and p = 2, we have:
77
From the general "unitary" formula that derives from DN Constant, we have:
This analysis of the extended Del Gaudio-Nardelli Constant and Cardano's formula is
very detailed and involves a series of mathematical constants. Let's look at the key
points:
1. Cardano's formula and the extended DN Constant:
The formula that was presented is an extended version of Cardano's formula for
solving cubic equations.
Involves variables such as d, a, q, p and mathematical constants such as π and √5.
2. Final result:
This analysis leads to an approximate result of 1.61803, which is very close to the
golden ratio (φ).
3. Units of measurement and connections:
Units of measurement (such as the Planck length and temperature of the Universe)
and constants (such as 0.9991104684) are involved in this formula.
The connections between these constants and the extended DN Constant could have
profound implications on mathematics and physics.
78
4. Cosmological significance:
It might be interesting to explore how this extension of the DN Constant relates to
observable phenomena in the Universe or to broader theories.
The presence of cosmological constants could suggest connections with the evolution
of the Universe and its structure.
In summary, this analysis is complex and involves many variables.
We also affirm that from 1.61803, with some formulas, it is possible to obtain π, ζ(2),
the DN Constant (2√2)/π, 1729 and 4096. What could they be, based on the results
obtained, at a first glance analysis, the cosmological implications?
Let us examine the cosmological implications of the results that were obtained from
this extension of the DN Constant:
1. Pigreco (π):
π is a fundamental mathematical constant representing the ratio of the circumference
of a circle to its diameter.
In the cosmological context, π could be linked to the geometry of space-time and the
curvature of the Universe.
2. ζ(2) and Apéry's constant:
ζ(2) is also known as Apéry's constant and is approximately equal to 1.64493.
This constant is involved in the sum of the power series of the reciprocals of the
squares of the natural numbers.
79
It could have implications for the distribution of vacuum energies and the
regularization of quantum field theories.
3. The DN Constant (2√2)/π:
The DN Constant is originally related to the geometry and relationship between the
volume of an octahedron and the volume of a sphere.
The fact that it can be derived from this extension is intriguing and could have
implications for the structure of the Universe.
4. Numbers 1729 and 4096:
The number 1729 is known as the “Hardy-Ramanujan number” and is the smallest
number that can be expressed as the sum of two cubes in two different ways.
The number 4096 is a power of 2 (212).
These numbers could have mathematical or symbolic meanings in the cosmological
context.
5. Overall Implications:
The presence of these constants and numbers could suggest connections between
mathematics, physics and the structure of the Universe.
It may be interesting to explore how these constants relate to cosmological
observations, field theories, and the nature of fundamental constants.
In summary, the results that have been obtained from the various calculations
described could open new perspectives on understanding the Universe and its
fundamental laws.
80
Fig. 3
Multiverse with eternal inflation
81
Fig. 4
Eternally inflated multiverse with universe bubbles inscribed in octahedral figures
82
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
Result
83
Decimal approximation
0.900316316157106…. =
(DN Constant)
Property
Series representations
84
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)
Property
85
Series representations
Integral representations
86
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
Decimal approximation
1.64529737852…. ≈ ζ(2) = π
2
/6 = 1.644934 (trace of the instanton shape and
Ramanujan Recurring Number)
87
Alternate forms
Expanded forms
88
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
89
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
90
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
91
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
92
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.
93
(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
94
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, as it can be expressed as the sum of two cubes in two
different ways (10
3
+ 9
3
= 12
3
+ 1
3
= 1729) and Ramanujan's recurring number)
Series representations
95
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
96
Series representations
97
From
we obtain also:
sqrt(6(1/3*(2/(((2sqrt2)/Pi)))^2))
Input
Exact result
Decimal approximation
3.14159265358… = π
98
Property
All 2
nd
roots of π
2
Series representations
Integral representations
99
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
100
Decimal approximation
0.900316316157106…. =
(DN Constant)
Property
All 2
nd
roots of 8/π
2
Series representations
101
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))
102
Input
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
103
104
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)
105
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))
106
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:
107
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)
108
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
109
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
110
Now, we have that:
Octahedron Sphere
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:
111
To find a, we have, for
Thus, multiplying both the sides by
, we obtain:
112
Plot
Solution
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
113
Decimal approximation
3.05684889733….
Property
Series representations
114
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
Result
Decimal approximation
3.05684889733….
Property
Series representations
115
116
From:
Modular equations and approximations to - Srinivasa Ramanujan - Quarterly
Journal of Mathematics, XLV, 1914, 350 – 372
We have that:
117
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
118
Minimal polynomial
Expanded forms
Series representations
119
120
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
121
Minimal polynomial
Expanded forms
Series representations
122
123
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
124
Minimal polynomial
Expanded forms
All 15th roots of ϕ + 27 (-2 + 2 sqrt(1093) - (sqrt(10 - 2 sqrt(5)) - 2)/(2 (sqrt(5) -
1)))
125
Series representations
126
Integral representation
127
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:
128
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:
129
Now, we have the following calculations:
= 1.6272016… * 10
-6
from which:
= 1.6272016… * 10
-6
0.000244140625
=
= 1.6272016… * 10
-6
Now:
And:
130
(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:
131
Result:
0.00666501785…
Series representations:
Now:
= 0.0066650177536
=
= 0.00666501785…
132
From:
ln(0.00666501784619)
Input interpretation:
Result:
-5.010882647757…
Alternative representations:
Series representations:
133
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:
134
(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:
135
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:
136
(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:
137
Solutions:
Alternate forms:
Expanded form:
Alternate form assuming b, M, and V are positive:
Alternate form assuming b, M, and V are real:
Derivative:
138
Implicit derivatives
Global minimum:
Global minima:
139
From:
we obtain:
(225.913 (-0.054323 M^2 + 6.58545×10^-10 sqrt(M^4)))/M^2
Input interpretation:
Result:
140
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:
141
Property as a function:
Parity
Series expansion at M = 0:
Series expansion at M = ∞:
Derivative:
Indefinite integral:
142
Global maximum:
Global minimum:
Limit:
Definite integral after subtraction of diverging parts:
From b that is equal to
143
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
(possible mathematical connection with an open string)
M = 2 ; M = 3
144
Root:
Property as a function:
Parity
Series expansion at M = 0:
Series expansion at M = ∞:
Definite integral after subtraction of diverging parts:
145
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
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
146
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:
Result:
1.57986484181*10
-14
147
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
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)]
148
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:
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...
149
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
150
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
151
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
152
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
153
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.
154
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