Mathematical Constants of Natural Philosophy Michael A. Sherbon

e-mail: [email protected] July 21, 2010 Abstract Plato’s theory of everything is an introduction to a Pythagorean natural philosophy that includes Egyptian sources. The Pythagorean Table and Pythagorean harmonics from the ancient geometry of the Cosmological Circle are related to symbolic associations of basic mathematical constants with the five elements of Plato’s allegorical cosmology: Archimedes constant, Euler’s number, the polygon circumscribing limit, the golden ratio, and Aristotle’s quintessence. Quintessence is representative of the whole, or the one in four, extraneously considered a separate element or fifth force. This relationship with four fundamental interactions or forces also involves the correlation of constants with the five Platonic solids: tetrahedron, hexahedron, octahedron, icosahedron, and dodecahedron. The values of several fundamental physical constants are also calculated, and a basic equation is given for a unified physical theory in the geometric universe of Plato’s natural philosophy.

1 Introduction Plato described the geometric proportions of the Cosmological Circle in his allegorical study of the ideal City of Magnesia [1]-[8]. The Pythagorean geometry of the 3, 4, 5 right triangle and the “squared circle” form a twelve-sided theme of the Cosmological Circle. The heptagon, central to the Cosmological Circle, and its relation to the cycloid curve connect it to the foundation of calculus and the least action principle. Briefly, 3 + 4 + 5 = 12. The Pythagoreans considered 12 to be ideal for expressing musical proportions [9]-[18], 12:6 octave, 12:8 fifth, 12:9 fourth, and 12:12 unison. The geometry of the Cosmological Circle consists of 3, 4, 5 or 6, 8, 10 right triangles. With each side multiplied by 36, they form the 108, 144, 180 and 216, 288, 360 triangles. Then multiplied by 10 for the second set gives the large set dimensions for the diameter arranged in lengths

1

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2880+2160+2880 = 7920. 7 is the basic radius of the circle circumscribing the heptagon. The basic Earth square has a perimeter of 4 × 11 = 44 ' 14π, the circumference of the circle with the inscribed regular heptagon. John Michell emphasizes 6, 12, 37, and their multiples as most prominent in the canon of ancient geometry. 37 is most emphasized by William Eisen as special to the Egyptian initiates of antiquity [20]-[23]. Among several versions of the Cosmological Circle are the ground plan of the Great Pyramid, Stonehenge, Wolfgang Pauli’s vision of the World Clock [18], and Franklin Merrell-Wolff’s conceptual hyperbolic mandala [24]-[27]. Regarding Plato’s ideal City of Magnesia John Michell states that, “By Plato’s time, the very idea of a canon of music had been forgotten everywhere except in the academies of Egypt, but he himself had evidently studied and learned it, for the number code behind it is at the root of all his mathematical allegories and provided the scientific basis of his philosophy.” [5]. Plato alluded to the harmonic science of antiquity as the origin of the Pythagorean Table. John Michell interprets the Platonic dodecahedron as a solid form of the Cosmological Circle, with its twelve pentagonal sides. Aristotle is credited with the term quinta essentia for Plato’s fifth element, represented by the dodecahedron [28]. The involution and evolution of the regular dodecahedron contains all five regular Platonic solids [29]-[35]. Ratios found in tone-number form of the revised Pythagorean Table are harmonically related to fundamental physical constants [17]. The Pythagorean Table and versions of the Cosmological Circle encode the basic mathematical constants π, e, κ, φ, and Q. Expressing the archetypal quintessence in approximate numerical form: √ Q ' 377/370 ' (37 + 2/2)/37 ' 1.019. (1) 137 + 233 = 370. 2332 + 2882 = 137233 ' 3702 and 144 + 233 = 377, a Fibonacci prime and approximate harmonic of the characteristic impedance of free space; related to the equation defining the electromagnetic fine-structure constant, alpha [36]. √ √ √ C = Q2 +Q2 = 2Q ' 2 + 1/37 ' 1.44 (2) where C is a harmonic of the Pythagorean grid speed of light harmonic. Of relevance to the harmonic of light is the Pythagorean triangle relationship with quintessence. C

2

= Q2 +Q2 ' φf /φ ' 1/ ln φ

(3)

where φf is the reciprocal Fibonacci constant, see Eqs. (30, 32) and the discussion of Eq. (42); φ is the golden ratio. 144 is the fundamental tone-number harmonic found in William Conner’s revision of the classical Pythagorean Table [16]. The regular Platonic solids were the subject of the conclusion to Euclid’s Elements [37]. 37 is described as the “center of gravity” of Being by William Eisen [20]-[23]. Quintessence according to Carl Jung is the transcendent function of centering the “one in four” correlations of logical types such as Aristotle’s four causes, the four directions, and four elements [38]. This was also investigated by Arthur Young [39] and Peter Aleff [40], who rediscovered the symbolic associations of the four elements with four mathematical constants [41]. Also notice sin 37o ' 3/5 and right triangle 12, 35, 37.

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3 × 37 = 111 is also a most significant number for William Eisen, explained by the Egyptian sages. C 2 is the harmonic of William Conner’s “supernumber” representing the velocity v in v = f × λ, the fundamental tone 144 times its wavelength 144. “It appears as a constant throughout the base octave, and in all higher octave harmonics,” in the revised Pythagorean Table [16]. The main base octave tone-numbers include 144, 162, 180, 233, 240, 256, & 270. William Conner writes again, “... key dimensions of the Great Pyramid of Cheops often function as a magic invariance crystal through which the musical scale – the domain of Tone – Number – and the world of natural phenomena interpenetrate,” [16]. The hexagon, 3, 4, 5 right triangle, and approximate heptagon are hiding places for phi and thus the pentagon [23]. The square, hexagon, and heptagon are the main polygons of the Cosmological Circle. Another version of the Cosmological Circle is found in the conceptual mandala of mathematician and philosopher Franklin Merrill-Wolff featuring conjugate rectangular hyperbolas, fourfoldness, and circularity. The mandala symbolism addresses the foundational process of the physical universe and the question of how the subject-object relative manifold is made manifest. Similarly, the question concerning the Platonic origin and archetypal definition of “fundamental” physical measurement is raised in connection with the most fundamental physical units [42]-[44]. More simply, the dimensionless “harmonic” of quintessence is derived from the “squaring of the circle” found in the Cosmological Circle. William Conner also emphasizes, like Walter Russell, that the nature of matter is harmonic. In support of this he quotes the “advanced” source of Elizabeth Klarer from Beyond the Light Barrier, “Universal harmonics is the mathematics used by Akon’s civilization. He speaks of harmonic maths, in harmonic affinity with all substance, a resonance tuned to matter itself expressed in terms of light.” [45]. William Conner also repeats Bruce Cathie’s thesis “that the whole cosmological structure is built up from harmonic waveforms or angular velocities of light.” [16]. In ancient metrology fundamental units have a geometric basis of “earth measure” as an orientation toward the objective pole of physical existence in the relative manifold. In Plato’s Code, Malcolm Macleod’s geometry of angular momentum model [46], Q also has units of momentum. Q2 ' mP c/2π, thus signifying it as a primary fundamental physical constant. The dimensional equation describing quintessence suggests the circular motion attributed to the Aristotelian quintessence [19]. In Planck units, Planck momentum mP c = (~c 3 /G )1/2 where mP is the Planck mass, ~ is Planck’s constant h divided by 2π, c is the speed of light constant, and G is Newton’s universal gravitation constant. c 3 is an approximate harmonic of 1/37 and C 2 /77, see Eq. (18) discussion and reference [16]. The “harmonic” describes a dimensionless quantitative value which also has a qualitative meaning attributed to it in the Pythagorean/Platonic/Egyptian tradition [47]-[60]. Like the ideal forms of Plato, Carl Jung and Wolfgang Pauli explored together the inward nature of archetypes and synchronicity, “the acausal connecting principle,” [61]. Wilbert Smith addressed the foundations from a different point of view (“... from data obtained from beings more advanced than we are.”) with the quadrature concept, which presumably establishes the subject-object manifold as a “relationship of awareness to the understanding of reality.” [62]. This is explained √ as a geometric process of right-angled observation involving the rotation operator i = −1 [63]. Arthur Paul added to this

3

with, “The Possible Meaning of Imaginary Numbers” in Consciousness and Reality [64] with commentary by mathematicians Morris Kline and Charles Mus`es on the nature of Platonic mathematics [65]-[67]. Arthur Young related consciousness to the inherent nature of the i-deal quintessence with his third derivative proposal of “control” as the fifth element in the respective equations of physics [39]. Werner Heisenberg, one of the principal founders of quantum theory, held that mind and matter are structured by the same forces. “The particles of modern physics are representations of symmetry groups and to that extent they resemble the symmetrical bodies of Plato’s philosophy.” [68]. Heisenberg’s favorable view of Plato’s forms [69, 70] is extended in the symbolic association of elements with mathematical constants below.

2 Archimedes Constant A regular tetrahedron has 4 faces, 4 vertices, and 6 edges. The Platonic tetrahedron is associated with the fire element, Archimedes constant π [40], the strong force [71], and the tetrahedral symmetry group. π=

lim n→∞

Pn /d ∼ = 3.141 592 653 589,

(4)

a method of determining π attributed to the Greek Archimedes of Syracuse [72] from perimeter Pn of a regular polygon with n sides circumscribing a circle with diameter d. π = (2 ln i)/i = −i ln(−1),

(5)

various ways of simply expressing π in relation with the imaginary number i [73]. √ i = −1 = 1∠90o . (6) √ The equation for the rotation operator [73], i = −1 = x is the solution of x2 + 1 = 0. Q

' 12π/37.

(7)

122 + 352 = 372 , 23 + 37 = 60; reminders of the 6, 12, and 37 in the ancient canon. The inverse fine-structure constant resonates to the harmonic of 37 and the heptagon: α−1 ∼ = 37(3.7 + (337 − 20S )−1 )

(8)

with nearly the same value as Eq. (37). S = 2 + 2 cos (2π/7) ∼ = 3.246 979 603 717 ' 250/77 ' 12/3.7 and with the heptagon angle 2π/7. S , the silver constant [74], is the seventh Beraha constant. S ' 74/κφ2 ≈ 2φ, where κ is the polygon circle limit, φ is the p √ 3 golden ratio, and φ2 is the fifth Beraha constant. C 2 = 2Q2 ' S 1/φ . If x = 7 + 7 3 7 + ..., then S = 2 + (x + 2)/(x + 1). The silver constant is also involved in a calculation of the cosmological constant in the previous work [19], which has been suggested as a unifying link [75]. 87 + 250 = 360 − 23 = 337, see Eq. (37). 20S ' (77 + 377)/7, see Eq. (27) and notes to Eq. (39). 3.7 ' 12/S ' π ln S , 3.7 × 37 ' 137. 1752 + 2882 = 3372 , 5 × 35 = 175.

4

p Another Pythagorean harmonic √ sequence of interest: 10/α ∼ = 37.018 37 37 0. 18√is a harmonic of the heptagon chord ' S , see notes to Eqs. (33, 42), and [17] for the 10. α−1 ∼ = (35 + 144π − 7π 2 )S 4 /108π,

(9)

with the same value as Eq. (37), has the essential harmonics from the Cosmological Circle. 35 × 72 = 7 × 360 = 2520 [19], 1082 + 1442 = 1802 . S 4 /3 ' 37, 122 + 352 = 372 . Modern quantum chromodynamics describes the strong force, with the physical theory developed by many; Murray Gell-Mann and Gross-Politzer-Wilczek being among the most notable. The coupling constant for the strong force: αs ' 1/S φ2 ' 7π/187 ' 0.1176,

(10)

P DG 2009 [76] αs ∼ = 0.1176. √ Eq. (10) has the silver constant and golden ratio. αs ' A/ φ ' κ/74, where A is the edge of the “Golden Apex” in Eq. (14), and κ is the polygon circumscribing constant. 2 × 37 = 74. 7π ' 22, in the speed of light harmonic c ' (7π/(4 + φ/5))12 , see Eqs. (14, 33) and section 7. 3.74 ' 187 = 37 + 150 = 77 + 110, see Eq. (34) and Eq. (21) notes.

3 Euler’s Number The hexahedron has 6 faces, 8 vertices, and 12 edges. The cube is associated with the earth element, Euler’s number e, electromagnetic force, and octahedral symmetry group.   lim 1 n∼ e= 1+ (11) = 2.718 281 828 459. n→∞ n Euler’s number, named after Leonhard Euler, is also named after John Napier [77]. eiπ + 1 = 0.

(12)

Euler’s formula is a favorite equation of many. Wolfgang Pauli found the exponential function to be a useful archetype in regard to the unit circle in the complex plane [78]. eiπ = cos π + i sin π.

(13)

William Eisen explains the mystery of eiπ as found in the design of the Great Pyramid [22, 23]. A is the side length of the “Golden Apex” square in the capstone of the Great Pyramid from William Eisen’s construction. The edge length arises from a combination of π intervals with the geometry of four exponential curves at the base of the plan: A

= eπ − 7π − 1 ∼ = 0.149 544 057 650 7

(14)

where the curve of ex ends at x = π. Benjamin Pierce’s equation eπ = (−1)i , also from Euler’s formula [77], and noted by Paul Nahin in An Imaginary Tale; eπ = i−2i [73].

5

The Golden Apex represented by A

−1

has several interesting relations, to begin with: √ (15) ' cosh(1/ A) ' S φ . A

Franklin Merrill-Wolff inspired the application of the hyperbolic cosine function [24]-[27]: cosh x = cos ix = (ex + e−x )/2. (16) √ √ Also, cosh S ' 35/e and A ' π/21 ' 5/(5 + 7 5)φ ' φ sin(π/5)/5, associated with icosahedron and dodecahedron proportions. A is a factor in another equation for Q: Q

' cosh2 A ' A2 (60 − 6j0,1 )

(17)

where j0,1 ∼ = 2.404 825 557 695 is the first root of the Bessel function of the first kind as calculated by WolframAlpha–Mathematica. Also, 23 + 37 = 3 × 4 × 5 = 60. “The Bessel functions of the first kind Jn (x) are defined as the solutions to the second order differential equation known as the Bessel differential equation x2 (d2 y)/(dx2 ) + x(dy)/(dx) + (x2 − n2 )y = 0 which are nonsingular at the origin. They are sometimes also called cylinder functions or cylindrical harmonics,” [79]. For the special case of n = 0: ˆ π −1 eiz cos θ dθ. (18) J0 (z ) = π 0

i−i = exp(π/2 + 2kπ) for integer k, providing another linkage between i and Q [19]. π C 2 ' 10 exp(−π/2). Q ∼ = (245 − 81j0,1 )e /2 /237 where 57 + 180 = 237, 162/2 = 81, 108 + 137 = 245, and 60 + 77 = 137. 77 is the side and radius of the hexagon in a Vesica Piscis forming Eisen’s construction of the “All-Seeing-Eye” [20] related to the “Wheel of Four Tunes” (the four elements), the “Wheel of Life,” the cycloid curve [18], and the “Golden Apex” of the Great Pyramid with its four faces. Another “advanced” source [80] affirms that an electromagnetic “standing columnar wave” was created by the Great Pyramid structure, which Nikola Tesla also found in his experimental electrical research [81, 82]. Aspects of this energy were also proposed in mathematical form by Paul Dirac [83], and reconsidered by others [84]-[88]. Related to this was Wolfgang Pauli’s obsession with the fine-structure constant and spectra, revealed by the Pauli-Jung letters [78] and interpretations of his World Clock vision [18]. The first root of the Bessel function j0,1 also gives an approximation for inverse alpha, or the inverse fine-structure constant: α−1 ' cosh(9 − 3j0,1 )/A2 ' 57j0,1 .

(19)

20 + 37 = 144 − 87 = 57, and see the discussion of Eqs. (24, 40). α−1 ' (77S − 15)/10δ0 , Q

' A cosh(76δ0 /5)

(20)

√ where the Hall-Montgomery constant δ0 = 1 + Li2 (1) + 2Li2 (− e) ∼ = 0.171 500 493 141 5 P k /k 2 is the dilogarithm, and ζ(s) is the [90], Li2 (1) = ζ(2) = π 2 /6, where Li2 (z ) = ∞ z k=1 Riemann zeta function; both found in the perturbative form of determining the finestructure constant [91]. cos2 Q ' 2π/7eζ(3), j0,1 ' 2ζ(3), α−1 ' S (ζ(3) + 41), and see

6

the discussion of Eqs. (39-40). 12 + 27 + 37 = 76 ≈ 8.72 ≈ κ2 , κ is the polygon circle limit in Eq. (23), and κ2 is an effective harmonic of the apex angle of the Great Pyramid. C

2

∼ = cosh(587δ0 /74)

(21)

where δ0 is the Hall-Montgomery constant again, C 2 = 2Q2 ' S 1/φ . 37 + 550 = 587, and 2 × 37 = 74. Q−1 ∼ = cos((13e − 14)/110), see Eq. (34). 13 + 14 = 27, 27 + 110 = 137. Faraday and Maxwell are most notable for the development of electromagnetic theory, Freeman Dyson and Feynman-Schwinger-Tomonaga, for quantum electrodynamics. The fine-structure constant determines the strength of the electromagnetic interaction and is related to the least action principle, which is a generalization of many facets of modern physical theory [18]. α = e2 /~c, in cgs units. ~ = h/2π = h-bar, the reduced Planck’s constant, c is the speed of light constant, and e is the elementary charge. James Gilson’s mathematical polygon theory of the fine-structure constant [92] can be applied to the regular polygons of the Cosmological Circle and their related Pythagorean geometry: α−1 ∼ = 137θ/ tan(θ) cos(π/137)

(22)

with the same value as Eq. (37). θ = π/3930 ' A/187 radians. 60+77 = 370−233 = 137. 137 + 256 = 105 + 288 = 60 + 333 =23 + 370 = 110 + 283 = 483 − 90 = 393 ' 53 π ' 35 φ, and see the discussions of Eqs. (33, 34, 42). 3930 is the reciprocal of the primary harmonic from Bruce Cathie’s global electromagnetic grid [93] (from yet another source thought to be more “advanced” and rediscovered by Cathie, further explained by William Conner as a significant harmonic associated with numerous Pythagorean harmonics). This is the same grid connected with the Cosmological Circle geometry and grid speed of light harmonic 144, also related to the Great Pyramid of Cheops where the effective quartz crystal resonant frequency is the second octave harmonic of 393 × 103 [16].

4 Polygon Circle Limit A regular octahedron has 8 faces, 6 vertices, and 12 edges. The Platonic octahedron is associated with the air element, the weak force, and the octahedral symmetry group. Peter Aleff rediscovered the ancient symbolic associations of the four elements with four basic mathematical constants including κ, the polygon circumscribing constant [40, 94]. κ=

∞ Y

sec(π/n) ∼ = 8.700 036 625 208.

(23)

n=3

An archetypal approximation of the polygon circumscribing constant: κ ' 74/S φ2

(24)

again with the silver constant and the golden ratio appearing together. S φ2 ' eπ. √ 2×37 = 74. A circle of radius C has a circumference of 2π C ' 1+ π+2π ' (35κ−60)/27. Stonehenge diameter of 87 ' 10κ ' (1 + π 3 )e. 57 + 87 = 144, 37 + 137 = 2 × 87 = 174, and 10/κ ' 1.15, regular heptagon radius with a side equal to 1, see reference [19].

7

Various quintessence approximations involving the polygon circumscribing constant: Q

' ln(κ/π) ' eπ /(14 + κ).

(25)

p 7/2. Q ' (κφ/π)2 ' e3 ' 20. φπ/e ' 5φ2 /7 ' 1 + κ/10 ' 1.87 ' 144/77 ' √ √ √ √ √ 60/π(κ + 10) ' κ/eπ. −(κ2 − π 2 − e2 − φ2 )/( κ − π − e − φ) ' (κ2 − κ − κ)/2 ' 10κ/e ' 1 + π 3 ' 32 and (27 + 37)/2 = 32 [40]. 32 is the archetypal number essential to Wolfgang Pauli’s visionary World Clock geometry [78]. 1.87 × Q ' 1.91 ' 6/π, the cube/sphere ratio favored by the Egyptians. 187 is a harmonic of 374 and 37κ ' φ12 . The parameter of the electroweak force relating the force carrier masses of the W and Z bosons is the Weinberg angle θW , the weak mixing angle of the Glashow-Weinberg-Salam electroweak theory [71]. θW ≈ 30o is the angle of the twelve-sided regular dodecagon. sin2 θW ' S /14 ' tan2 (π/7) ' 0.2319,

(26)

P DG 2009 [76] sin2 θW ∼ = 0.2319. Additional expressions the polygon circumscribing constant and light harmonic: √ include 2 −4 2 sin θW ' C ' 2 77/κ ' 37/7κφ2 , see Eqs. (41-42) for κφ2 . Q12 ' tan (2π/7) [19]. The inverse fine-structure constant α−1 ' A2 cosh(7 + 21/κ), see Eqs. (17, 19), and α−1 ∼ = (370/2.7) − (105κ + 50)−1

(27)

with the same value as Eq. (37). 3 × 35 = 50 + 55 = 15 + 90 = 162 − 57 = 105, see Eq. (19). 162 is the phi harmonic. 92 + 122 = 152 , see Eq. (42) notes. 1372 − 882 = 1112 − 362 = 1052 . 2 × 44 = 11 + 77 = 137 − 72 = 88. 20S ' 105/φ, and see Eq. (8).

5 Golden Ratio A regular icosahedron has 20 faces, 12 vertices, and 30 edges. The Platonic icosahedron is associated with the water element, the golden ratio, gravitational force, and icosahedral symmetry group. φ is the golden ratio [95]-[97]. For the Fibonacci numbers F (n): φ=

lim F (n + 1) ∼ = 1.618 033 988 749. n→∞ F (n)

(28)

φ is a solution to the quadratic equation x2 − x − 1 = 0, with the conjugate root −1/φ. √ φ = (1 + 5)/2 ' 60/37. (29) √ √ 23+37 = 60, Also, φ = 1+1/φ = φ2 −1 and φ = (a+b)/a = a/b. Q ' 35/27 φ ' e/φ. C

2

∼ = κ(6φf + 3)/97.

(30)

P ∼ Again, C 2 = Q2 + Q2 ' S 1/φ , φf is the reciprocal Fibonacci constant, φf = ∞ n=1 1/F (n) = 3.359 885 666 243, [98]. The ratio of successive terms of the sum approaches the reciprocal

8

of the golden ratio. Recall from the √introduction above C 2 ' φf /φ, also see section 7. √ 3 φ2 ' 97/37. 37 + 60 = 97, 137/97 ' 2, and 97/77 ' 2 is the Delian constant [37]. Q

' cosh(1/φπ).

(31)

Additional expressions include cosh φ ' φ2 ' 10 − e2 , e ' φ2 × Q 2 , and cosh φf ' 14.4. α ' A2 / cosh(6/φf )

(32)

2 where φf is the reciprocal Fibonacci constant again. C 2 ' 2 cosh(φ−1 f ) and φf ' 7φ. φ 2 −φ Also of interest, 87 ' 37 , and the Golden Apex again from Eq. (15) A ' S .

α−φ ∼ = 3π223 /35 ∼ = 2867.2867 + 28672−1 .

(33)

For the geometry of 3/5 and 22/7 see [19]. 122 + 352 = 372 and 37 + 2830 = 2867, 3700 − 870 = 2830, harmonic of the fundamental energy reciprocal in William Conner’s harmonic theory, approximately proportional to 90π. 47 × 61 = (37 × 77) + 18 = 2867, see the discussion of Eq. (42) and [15, 16]. Relating quintessence, the golden ratio, and Pythagorean harmonics from the Cosmological Circle with the fine-structure constant: Q

2 ∼ = 1 + α (φ + 9α/110φ) ∼ = 1.019 113 431 9,

(34)

with the value for α that is determined from Eq. (37). 33 + 77 = 37 + 73 = 110 = 10 × 11 = 2 × 55 = 187 − 77. See the discussion of Eqs. (22, 36, 38, 41) and reference [19] for a connection with the classical Foundation Stone from ancient Greek geometry. The heptagonal 7 and silver constant S are found in a “harmonic” determination [19] of Sir Isaac Newton’s universal gravitation constant of the equation for gravitational force: F = GM m/r2 . G is also the harmonic of the inner diameter of the All-Seeing-Eye [20]. G

∼ = 6.674 26 × 10−11 , = (7 + S /70)−12 ∼

(35)

N IST 2006 [100] G = 6.674 28 (67) × 10−11 m3 kg −1 s−2 . From Eq. (3) and commentary G ' ~c3 /π 2 C 4 , showing the fundamental constants of quantum theory and Albert Einstein’s theory of relativity from angular momentum and the harmonics of light; C 2 ' mP c/π. Also, A−1 ' S φ is the harmonic of G, see Eq. (15).

6 Aristotle’s Quintessence The fifth Platonic solid is a regular dodecahedron, having 12 faces, 20 vertices, and 30 edges. The dodecahedron is associated with the elemental aether, the unified force of quintessence, and the icosahedral symmetry group. The regular dodecahedron and its “embroidered figures” represented the whole universe in Plato’s natural philosophy. The dodecahedron can be considered a three dimensional version of the Cosmological Circle, with its twelve pentagonal faces. The faces of a regular dodecahedron are pentagons, and the diagonal of a regular pentagon with side equal to one is φ, the golden ratio.

9

√ The inner radius r of the dodecahedron is r = (250 + 110 5)1/2 /20 ' 1.1135 ' 7/2π. C 2 ' 7/er 2 , Q2 ' π/er, and see reference [19] for other ways of expressing quintessence. Another geometric relationship, recalling Euler’s formula and Eisen’s explanation, shows a connection between the golden ratio, the dodecahedron, and Aristotle’s quintessence: √ eiQ ' φ2 /5 + 5i/φ2 . (36) √ cos−1 (φ2 /5) ' 58o , sin−1 ( 5/φ2 ) ' 59o which is nearly the regular hexagon angle of 60o and the sum of both is the regular dodecahedron dihedral angle ≈ 117o . The dihedral angle is the internal face angle where the two adjacent faces of the polyhedron meet. Fine-structure constant with the hyperbolic cosine function of Pythagorean harmonics: α−1 ∼ = cosh((337 ln 2 + 266)/89) ∼ = 137.035 999 16,

(37)

N IST 2002 [99] α−1 = 137.035 999 11 (46). The alternating sum of the reciprocals of the integers is equal to ln 2, and ln 2 = cosh−1 (5/4) = 2 tanh−1 (1/3). Also, C ' 1/ ln 2. 87 + 250 = 283 + 54 = 360 − 23 = 337, see Eq. (8) and the discussion of Eq. (33). 12 + 77 = 233 − 144 = 89. 89 is a Fibonacci prime and 52 + 82 = 89. The decimal fraction expansion of 1/89 is the Fibonacci series [95]. 393 − 127 = 377 − 111 = 73 − 77 = 288 − 22 = 33 + 233 = 266, see Eq. (22) notes. 266 is also the harmonic width of the All-Seeing-Eye [20]. 1752 + 2882 = 3372 and 337 ln 2 ' 233. Also, 27 + 33 = 23 + 37 = 60 [17, 18], and 37 ' 23φ ' 10π ln S ' 120/S . The proton-electron mass ratio: √ √ mp /me ∼ (38) = 5π 187/α 257 ∼ = 1836.152 673, N IST 2006 [100] mp /me = 1836.152 672 47 (80). The harmonics involved are 37 + 150 = 77 + 110 = 324 − 137 = 187 and 77 + 180 = 70 + 187 = 740 − 483 = 257, see Eq. (42) discussion. 187 + 257 = 12 × 37 = 444. Also, mp /me ' (37κ − 7)/δ0 where δ0 is the Hall-Montgomery constant of Eq. (20). For other harmonic derivations involving the fine-structure constant see the references [18, 19]. The neutron-electron mass ratio: √ mn /me ∼ (39) = (80 − 505)/25δ0 α ∼ = 1838.683 660, N IST 2006 [100] mn /me = 1838.683 660 5 (11). √ 5 2 2 ∼ 370 + p 270/2 = 505, 25 + 55 = 89 √ − 39 = 80. mn /me = 6π + 6 − (20ζ(3))/7. Also, mn /me ' 40S − 87 ' 3/( φ − ζ(3)) and the last part of the equation was found by David Eagleman [101]. ζ(3) is the Riemann zeta function ζ(s) for s = 3 called Ap´ery’s constant, ζ(3) ∼ = 1.202 056 903 159, found in the perturbative determination of the fine-structure constant from quantum electrodynamic theory and the experimentally measured value of the electron’s gyromagnetic ratio. See the discussion of Eqs. (20, 40) and for supplementary connections with the Riemann zeta function see reference [19].

10

The electron magnetic moment anomaly, the electron g factor, is one of the most precisely measured values in modern physics, as a hyperbolic function of harmonics: ge /2 ∼ = cosh((1554 − 505π)/675) ∼ = 1.001 159 652 180 87,

(40)

N IST 2006 [100] ge /2 = 1.001 159 652 181 11 (74). Eq. (40) essentially has the factors of 5, 27, and 37. 25 × 27 = 675 = (2 × 270) + 270/2 and 2 × 777 = 1440 + 114 = 1554 = 37 × 42 = 372 + 370/2. α−1 ' 114ζ(3). Also, 370/S ' 2 × 57 = 114, see Eq. (19) and the discussion of Eq. (20). 370 + 270/2 = 505. Q

∼ = πe2 /κφ2 − (φ/11)4 /10 ∼ = 1.019 113 431 9.

(41)

A circle of diameter Q has a circumference of π Q ' 10 − 11/φ. The second part of Eq. (41) could also be written entirely as powers of phi, φ2 + φ−2 = 3, φ4 + φ−4 = 7, 3 + 7 = 10, φ5 − φ−5 = 11; obviously highlighting the role of the golden ratio. Another interesting harmonic proportion is from the basic Vesica Piscis construction forming the Cosmological Circle, 370/37 = 10 [8]. Remember 11 is the side of the basic square of the Cosmological Circle. 10/(φ/11)4 ' (77φ − 9)/φA3 ' 252 + 1442 , Pythagorean harmonics from the archetypal geometry of the All-Seeing Eye and Cosmological Circle, [17].

7 Geometric Universe Charles Mus`es found an interesting equation revealing a relationship between e, φ, and π in 1980 [102], which includes the e/7 ratio; see the discussion of Eqs. (21, 25) and beginning of section 6. From previous work [17] connections can be found with Arthur Young’s seven stage toroidal process of the hypersphere [39, 67], and remember [103] with renewed significance the Great Pyramid of Cheops as one of the seven wonders of the ancient world [104]. Wonder also relates to the nature of imaginary numbers and their interaction with mathematical constants of archetypal geometry. Quintessence is equated with mathematical constants π, e, A, κ, φ. Q

∼ = (πe2 − A)/κφ2

(42)

where A ∼ = A/(S + 137). A is the Golden Apex of Eq. (14), S is the silver constant with the archetypal 137 [61], and see the discussion of Eqs. (8, 22). A ' A3 /π ' φ/483π, see Eq. (16) discussion. 87 + 396 = 21 × 23 = 483 ' harmonic of C −2 , the original height of the Great Pyramid, and the circumference of the circle with radius 77 in the All-Seeing √ Eye construction. From the Golden Apex, 7π ' 483. 233 + 250 = 570 − 87 = 483. Equation (42) has symbolic associations [40] with all four forces [105, 106], the “one in four” ethereal quintessence of alchemy, or “life force” [107]-[110] of the “whole universe” [111, 112], the five Platonic solids, the rotation operator i of Eq. (6) represented by the Golden Apex, and all the basic elements for a unified physical theory [113]-[120]. The rotation operator is considered to be at the “root” of unification schemes [121]-[129], as it generates hypercomplex operations from the quadrature concept of consciousness [62].

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In p a previous work [17], (3/α)12 was found as an approximate harmonic of 2.330 618 and φf φ ' 2.33 with 1/φ ' 0.618. The multiple of 3 and the power of 12, or 1/12, applied to fundamental physical constants revealed sequences containing Pythagorean harmonics of the Cosmological Circle, see Eq. (35). Doing the same with quintessence, (3Q)12 ' 666 999, which also has a significant esoteric meaning [20] giving an additional metaphysical emphasis to the description of quintessence. Also, (3/C )1/12 ' 12/φ2f , and c1/12 ' φπ. φ = −2 sin 666 = −2 cos 144, 18 × 37 = 666, and 27 × 37 = 999. In another form, 3/(3Q)1/3 ' 2.0 66 99 77, see the discussion of Eq. (8). 77 × 87 = 66 99. Again we note that the 77 geometry [20] represents the “All-Seeing Eye” and the “Great Architect of the Universe.” Nikola Tesla seems to have understood this mystery [130], as he is frequently quoted, “If you only knew the magnificence of the 3, 6, and 9; then you would have a key to the universe.” See [131] for additional explanatory suggestions. Harmonic radius of the Cosmological Circle, 9 × 44 = 11 × 36 = 396, relates to the mathematics of yet another “advanced” source [132]. From ancient geometry, 666 − 99 = 9 × 63 = 567 is the archetypal solar mass harmonic [16]. With the 162 phi harmonic, 162+567 = 729, the signature of the classical Foundation Stone [19]. 729 + 63 = 792 = 2 × 396. In addition, 729−396 = 483−150 = 9×37 = 333, and 567/333 is proportional to the archetypal earth mass harmonic. This gives an abbreviated summary of the main archetypal harmonics in the geometric universe [133] beginning from the natural philosophy of Plato [47]-[60].

8 Conclusion The unifying principle [134] of Franklin Merrell-Wolff’s hyperbolic mandala and the Platonic Cosmological Circle reveals the harmonic nature of the space-time continuum, the ethereal quintessence, and fabric of the mind-matter field; the relative subject-object manifold. Quintessence is representative of the whole, and this relationship with the four fundamental interactions or forces also involves the correlation of constants with the five Platonic solids. The Golden Apex of the Great Pyramid represents the quintessential rotation operator, the imaginary number, responsible for generating the tempic field and its directional nature [39] by the quadrature conception [62]. Sir Roger Penrose writes “... the more deeply we probe Nature’s secrets the more profoundly we are driven into Plato’s world of mathematical ideals as we seek our understanding.” [135]. Finally, Sir James Jeans [136] in the spirit of Plato [137], “From the intrinsic evidence of his creation, the Great Architect of the Universe now begins to appear as a pure mathematician.” Acknowledgments Special thanks to Case Western Reserve University, Franklin Merrell-Wolff Fellowship, Social Science Research Network, Philosophical Research Society, and WolframAlpha.

References [1] Schneider, M. A Beginner’s Guide to Constructing the Universe: Mathematical Archetypes of Nature, Art, and Science, New York, NY: Harper-Collins, 1995.

12

[2] Schneider, M. Constructing the Cosmological Circle, San Anselmo: CA, 2006. [3] Michell, J. Ancient Metrology, Bristol: Pentacle Books, 1981. [4] Michell, J. How the World Is Made, New York: Inner Traditions, 2009. [5] Michell, J. The Dimensions of Paradise, New York: Inner Traditions, p.9, 2008. [6] Brumbaugh, R.S. Plato’s Mathematical Imagination: The Mathematical Passages in the Dialogues and Their Interpretation, Indiana: Indiana University Press, 1954. [7] McClain, E.G. The Pythagorean Plato, Stony Brook, NY: Nicolas Hays, 1978. [8] Gaunt, B. Beginnings: The Sacred Design, Kempton: Adventures Unlimited, 2000. [9] Levin, F.R. tr. The Manual of Harmonics of Nicomachus the Pythagorean, Grand Rapids, MI: Phanes Press, 1993. [10] Barker, A. The Science of Harmonics in Classical Greece, Cambridge, UK: Cambridge University Press, 2007. [11] Graham P. “Philosophy and Science of Music in Ancient Greece: Predecessors of Pythagoras and their Contribution,” Nexus Network Journal, 6, 1 (2004). [12] Wortmann, E. The Law of the Cosmos: The Divine Harmony According to Plato’s Republic-Timaeus, Godwin, A. tr., Godwin, J. ed. German ed. 1965, Idyllwild, CA: Sacred Science Translation Society, 2008. [13] Stakhov, A.P. The Mathematics of Harmony, Singapore: World Scientific, 2009. [14] Babbitt, E.D. Harmonic Laws of the Universe, New York: Kessinger, 2006. [15] Conner, W.B. Harmonic Mathematics, Chula Vista, CA: Tesla Book Co. 1982. [16] Conner, W.B. Psycho-Mathematics, Chula Vista, CA: Tesla Book Co. 1983. [17] Sherbon, M.A. “Pythagorean Geometry and Fundamental Constants,” SSRN Classics: Journal of Philosophical & Scientific Texts (2007) abstract=1032181. [18] Sherbon, M.A. “Constants of Nature from the Dynamics of Time,” SSRN Classics: Journal of Philosophical & Scientific Texts (2008) abstract=1296854. [19] Sherbon, M.A. “Classical Quintessence and the Cosmological Constant,” SSRN Classics: Journal of Philosophical & Scientific Texts (2009) abstract=1433068. [20] Eisen, W. The English Cabalah, Volume 1: The Mysteries of Pi, Camarillo, CA: DeVorss & Company, 1980. [21] Eisen, W. The English Cabalah, Volume 2: The Mysteries of Phi, Camarillo, CA: DeVorss & Company, 1982.

13

[22] Eisen, W. The Essence of the Cabalah, Camarillo, CA: DeVorss & Company, 1984. [23] Eisen, W. Universal Language of Cabalah, Camarillo, CA: Devorss & Company, 1989. [24] Leonard, R. “On the Mandala: Genesis and Symbolism of Wolff’s Mandala,” Phoenix, AZ: Phoenix Philosophical Press, 1995. [25] Leonard, R. The Transcendental Philosophy of Franklin Merrell-Wolff, Albany, NY: State University of New York Press, 1999. [26] Merrell-Wolff, F. Mathematics, Philosophy, and Yoga, 1966 Lecture Series, Phoenix, AZ: Phoenix Philosophical Press, 1995. [27] Merrell-Wolff, F. Transformations in Consciousness: The Metaphysics and Epistemology, Albany, NY: State University of New York Press, 1995. [28] Krauss, L. Quintessence, New York, NY: Basic Books, 2000. [29] Plummer, L.G. Mathematics of the Cosmic Mind : A Study in Mathematical Symbolism, Wheaton, IL: Theosophical Publishing House, 1970. [30] Cromwell, P.R. Polyhedra, New York: Cambridge University Press, 1999. [31] Brown, K. “Platonic Solids and Plato’s Theory of Everything,” mathpages.com/home/kmath096.htm. [32] Gray, R.W. “Short Overview of Polyhedra Systems,” Presentation at the University of Arizona (2007) rwgrayprojects.com/Lynn/Presentation20070926/p001.html. [33] MacLean, K.J.M. A Geometric Analysis of the Platonic Solids and Other SemiRegular Polyhedra, Ann Arbor, MI: Loving Healing Press, 2007. [34] Atiyah, M. & Sutcliffe, P. “Polyhedra in Physics, Chemistry and Geometry,” Milan Journal of Mathematics, 71, 1, 33-58 (2003) arXiv:math-ph/0303071v1. [35] Williams, R. The Geometrical Foundation of Natural Structure: A Source Book of Design, Mineola, NY: Dover Publications, 1979. [36] Mac Gregor, M.H. The Power of Alpha, Singapore: World Scientific, 2007. [37] Heath, T.L. A History of Greek Mathematics, New York: Dover Publications, 1981. [38] Black, J.A. The Four Elements in Plato’s Timaeus, Lewiston: Mellen Press, 2000. [39] Young, A. Mathematics, Physics, and Reality, Portland: Briggs Associates, 1990. [40] Aleff, H.P. “Genesis Equations” & “Outer Limit Circle,” recoveredscience.com. [41] Storm, L. The Enigma of Numbers, Pari, Italy: Pari Publishing, 2008.

14

[42] Barrow, J.D. The Constants of Nature: From Alpha to Omega–the Numbers that Encode the Deepest Secrets of the Universe, New York, NY: Pantheon Books, 2002. [43] Fritzsch, H. The Fundamental Constants, Singapore: World Scientific, 2009. [44] Rees, M. Just Six Numbers: The Deep Forces That Shape the Universe, New York, NY: Basic Books, 2000. [45] Klarer, E. Beyond the Light Barrier, New York: Aquarian Book, 1991. [46] Macleod, M. Plato’s Code (Original title: The Chess Board Universe, Minsk: Magic Books, 2003) 2007. [47] Vlastos, G. Plato’s Universe, Las Vegas: Parmenides Publishing, 2005. [48] Hall, M.P. Lectures on Ancient Philosophy, New York: Tarcher, 2005. [49] Kennedy, J.B. “Plato’s Forms, Pythagorean Mathematics, and Stichometry,” Apeiron: A Journal for Ancient Philosophy and Science, 43, 1, 1-32 (2010). [50] Grant, E. A History of Natural Philosophy, NY: Cambridge University Press, 2007. [51] Johansen, T.K. Plato’s Natural Philosophy, NY: Cambridge University Press, 2004. [52] Gregory, A. Plato’s Philosophy of Science, London: Duckworth Publishers, 2001. [53] Smith, W. The Wisdom of Ancient Cosmology, Oakton, VA: Foundation for Traditional Studies, 2009. [54] McEvoy, J. “Plato and the Wisdom of Egypt,” Irish Philosophical Journal, 1, Autumn, 1-24 (1984). [55] Cleary, J.J. “Plato’s Mathematical Cosmology,” in Aristotle and Mathematics, Leiden: Brill Academic Publishers, 1995. [56] Cornford, F.M. tr. Plato’s Cosmology: The Timaeus of Plato, Indianapolis, IN: Hackett Publishing Company, 1997. [57] Taylor, A.E. A Commentary on Plato’s Timaeus, Oxford: Clarendon Press, 1928. [58] Miller, I.E. The Significance of the Mathematical Element in the Philosophy of Plato, Chicago: The University of Chicago Press, 1904. [59] Heiberg, J.L. Mathematics and Physical Science in Classical Antiquity, London, UK: Oxford University Press, 1922. [60] Critchlow, K. “The Platonic Tradition on the Nature of Proportion”, in Bamford, C. Homage to Pythagoras: Rediscovering Sacred Science, Lindisfarne Press, 1994. [61] Miller, A.I. Deciphering the Cosmic Number: The Strange Friendship of Wolfgang Pauli and Carl Jung, New York: W.W. Norton & Company, 2009.

15

[62] Smith, W.B. The New Science, Canada: Fenn-Graphic Publishing, 1964. [63] Perkins, J.S. A Geometry of Space-Consciousness, Wheaton, IL: Theosophical Publishing House, 1995. [64] Paul, A. “The Possible Meaning of Imaginary Numbers,” in Mus`es, C. & Young, A.M. eds. Consciousness and Reality, New York: Avon Books, 1974. [65] Charon, J.E. The Real and the Imaginary, New York: Paragon House, 1987. [66] Penrose, R. Shadows of the Mind, New York: Oxford University Press, 1996. [67] Mus`es, C. & Young, A.M. eds. Consciousness and Reality: The Human Pivot Point, New York: Avon Books, 1974. [68] Heisenberg, W. “The Nature of Elementary Particles,” Physics Today, 29, 3, 32-39 (1976) quoted in Cassidy, D.C. Beyond Uncertainty, New York, NY: Bellevue Literary Press, p.407, 2009. [69] Lloyd, D.R. “Symmetry and Beauty in Plato,” Symmetry 2, 2, 455-465 (2010). [70] Hargittai, I. & Hargittai, M. Symmetry: A Unifying Concept, London: Shelter Publications, 1994. [71] Wilczek, F. The Lightness of Being: Mass, Ether, and the Unification of Forces, New York: Basic Books, 2008. [72] Beckmann, P. A History of Pi, New York, NY: St. Martin’s Griffin, 1974. √ [73] Nahin, P.J. An Imaginary Tale: The Story of −1 , Princeton, NJ: Princeton University Press, pp.53, 67, 1998. [74] Weisstein, E.W. “Silver Constant,” From MathWorld–A Wolfram Web Resource, mathworld.wolfram.com/SilverConstant.html. [75] Sivaram, C. “The Cosmological Constant as a Unifying Link,” Saha, S.K. & Rastogi, V.K. eds. 21st Century Astrophysics, pp.16-25, 2005. [76] Amsler, C. et al. (Particle Data Group), Physics Letters, B667, 1 (2008) and (2009) update, pdg.lbl.gov/2009/reviews/rpp2009-rev-phys-constants.pdf. [77] Maor, E. e: The Story of a Number, Princeton, NJ: Princeton University Press, 1994. [78] Meier, C.A. ed. Atom and Archetype: The Pauli/Jung Letters 1932-1958, Princeton, NJ: Princeton University Press, 2001. [79] Weisstein, E.W. “Bessel Function of the First Kind,” From MathWorld–A Wolfram Web Resource, mathworld.wolfram.com/BesselFunctionoftheFirstKind.html. [80] Hardy, D., Hardy, M., Killick, M., & Killick, K. Pyramid Energy, Allegan, MI: Delta-K Products, 1987.

16

[81] Brennan, H. The Secret History of Ancient Egypt, New York: Berkley, 2001. [82] Barrett, T.W. & Grimes, D.M. eds. Advanced Electromagnetism: Foundations, Theory, and Applications, Singapore: World Scientific, 1995. [83] Dirac, P. “Is There an Ether?” Nature, 168, 906-907 (1951). [84] Decaen, C.A. “Aristotle’s Aether and Contemporary Science,” The Thomist 68, 375-429 (2004). [85] Saunders, S. & Brown, H.R. The Philosophy of Vacuum, Oxford: Clarendon, 1991. [86] Brown, K. “A Quintessence of So Subtle A Nature,” in Reflections on Relativity, mathpages.com/rr/rrtoc.htm. [87] Kostro, L. Einstein and the Ether, Montreal: Apeiron, 2000. [88] Solomon, D. “Dirac’s Hole Theory and the Pauli Principle,” Advanced Studies in Theoretical Physics, 3, 9-12, 323-332 (2009). [89] King, M.B. Quest for Zero Point Energy, Kempton: Adventures Unlimited, 2002. [90] Finch, S. R. “Hall-Montgomery Constant,” Mathematical Constants, Cambridge, UK: Cambridge University Press, pp.205-207, 2003. [91] Gabrielse, G. “Determining the Fine Structure Constant,” in Roberts, B.L. & Marciano, W.J. eds. Lepton Dipole Moments, Singapore: World Scientific, 2009. [92] Gilson, J.G. “Calculating the Fine Structure Constant,” Physics Essays, 9, 2 June, 342-353 (1996). [93] Cathie, B.L. The Energy Grid, Kempton: Adventures Unlimited, 1997. [94] Pickover, C.A. Keys to Infinity, New York: John Wiley, pp.147-151, 1995. [95] Livio, M. The Golden Ratio: The Story of Phi, New York: Broadway Books, 2002. [96] Dunlap, R.A. The Golden Ratio and Fibonacci Numbers, Singapore: World Scientific, 1997. [97] Huntley, H.E. The Divine Proportion: A Study in Mathematical Beauty, New York: Dover Publications, 1970. [98] Weisstein, E.W. “Reciprocal Fibonacci Constant,” From MathWorld–A Wolfram Web Resource, mathworld.wolfram.com/ReciprocalFibonacciConstant.html. [99] Mohr, P.J. & Taylor, B.N. “CODATA Recommended Values of the Fundamental Physical Constants: 2002,” Reviews of Modern Physics, 77, 1-107 (2005). [100] Mohr, P.J., Taylor, B.N. & Newell, D.B.“CODATA Recommended Values of the Fundamental Physical Constants: 2006,” Reviews of Modern Physics, 80, 633 (2008).

17

[101] Wikipedia, The Free Encyclopedia, “Mathematical Constant: Ap´ery’s Constant,” en.wikipedia.org/wiki/Mathematical constant. [102] Mus`es, C. “The e, Phi, Pi Connection,” Abstracts of the American Mathematical Society, 1, 384, (1980). [103] Needleman, J. Consciousness and Tradition, New York: Crossroad, 1982. [104] Verheyen, H.F. “The Icosahedral Design of the Great Pyramid,” in Hargittai, I. Fivefold Symmetry, Singapore: World Scientific, 1992. [105] Hesse, M.B. Forces and Fields, New York: Dover Publications, 2005. [106] Weinberg, S. “The Forces of Nature,” Bulletin of the American Academy of Arts and Sciences, 29, 4, 13-29 (1976). [107] McTaggart, L. The Field: The Quest for the Secret Force of the Universe, New York, NY: Harper-Collins Publishers, 2008. [108] Coats, C. Living Energies: An Exposition of Concepts Related to the Theories of Viktor Schauberger, Chicago: Gateway, 2002. [109] Murchie, G. The Seven Mysteries of Life, Boston: Houghton Mifflin, 1978. [110] Laszlo, E. Science and the Akashic Field, New York: Inner Traditions, 2007. [111] Mohr, R. & Sattler, B. One Book, The Whole Universe: Plato’s Timaeus Today, Las Vegas: Parmenides Publishing, 2010. [112] Shorey, P. The Unity of Plato’s Thought, Chicago: University of Chicago Press, 1903. [113] G¨ onner, H.F.M. “On the History of Unified Field Theories,” Living Reviews in Relativity, 7, 2 (2004). [114] Parker, B. Einstein’s Dream: The Search for a Unified Theory of the Universe, New York, NY: Basic Books, 2001. [115] Davies, P. Superforce: The Search for a Grand Unified Theory of Nature, New York, NY: Simon & Schuster, 1984. [116] Gross, D. “Einstein and the Search for Unification,” Special Section: The Legacy of Albert Einstein, Current Science, 89, 12, 2035-2040 (2005). [117] Oerter, R. The Theory of Almost Everything, New York: Plume, 2006. [118] Barrow, J.D. New Theories of Everything, London: Oxford University Press, 2007. [119] Morrison, M. Unifying Scientific Theories: Physical Concepts and Mathematical Structures, Cambridge: Cambridge University Press, 2007. [120] Holton, G. The Scientific Imagination, Cambridge: Harvard University Press, 1998.

18

[121] Greiter, M. & Schuricht, D. “Imaginary in All Directions,” European Journal of Physics, 24, 397, (2003) arXiv:math-ph/0309061v1. [122] Kuipers, J. Quaternions and Rotation Sequences: A Primer with Applications ..., Princeton, NJ: Princeton University Press, 2002. [123] Ruthenberg, K. “Quaternionic Structure of 3-dimensional Natural Geometry,” Journal of Natural Geometry, 16, 125-140 (1999). [124] MacFarlane, A. “Hyperbolic Quaternions,” Proceedings of the Royal Society at Edinburgh, 1899-1900 Session, 169–181 (1900). [125] K¨ oplinger, J. “Dirac Equation on Hyperbolic Octonions,” Applied Mathematics and Computation, 182, 443-446 (2006). [126] Gogberashvili, M. “Rotations in the Space of Split Octonions,” Advances in Mathematical Physics, 2009, 483079 (2009) arXiv:0808.2496v1. [127] Furey, C. “Unified Theory of Ideals,” (2010) arXiv:1002.1497v2. [128] Rowlands, P. Zero to Infinity, Singapore: World Scientific, 2007. [129] Sirag, S-P. “Consciousness: a Hyperspace View,” Appendix in Mishlove, J. The Roots of Consciousness, 2nd ed. Tulsa, OK: Marlowe & Company, 1997. [130] McClain, J. “The Rule of Nines: Resonant Geometry and the Zero Point,” (1994) rexresearch.com/mra/1mra.htm#rule9. [131] Stiskin, M.N. The Looking-Glass God, New York: Random House, 1978. [132] Nelson, S.C. “The Seal of Solomon–A Mathematical Key to the Great Pyramid,” (2002) markorodin.com/content/view/19/42/. [133] Huggett, S.A. et al. The Geometric Universe: Science, Geometry, and the Work of Roger Penrose, London: Oxford University Press, 1998. [134] Scoular, S. First Philosophy, Boca Raton, FL: Universal Publishers, 2007. [135] Penrose, R. The Road to Reality, New York: Alfred A. Knopf, p.1029, 2006. [136] Jeans, J.H. The Mysterious Universe, New York: Cambridge University Press, 1930, quoted in Pickover, C. Archimedes to Hawking: Laws of Science and the Great Minds Behind Them, New York: Oxford University Press, p.263, 2008. [137] Uˇzdavinys, A. The Golden Chain: An Anthology of Pythagorean and Platonic Philosophy, Bloomington, IN: World Wisdom Books, 2004.

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