Copyright © 2017 Michael A. Sherbon. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. International Journal of Physical Research, 5 (2) (2017) 46-48 International Journal of Physical Research Website: www.sciencepubco.com/index.php/IJPR doi: 10.14419/ijpr.v5i2.8084 Research paper Fundamental physics and the fine-structure constant Michael A. Sherbon * Case Western Reserve University Alumnus, United States *Corresponding author E-mail: michael.sherbon@case.edu Abstract From the exponential function of Euler's equation to the geometry of a fundamental form, a calculation of the fine-structure constant and its relationship to the proton-electron mass ratio is given. Equations are found for the fundamental constants of the four forces of nature: electromagnetism, the weak force, the strong force and the force of gravitation. Symmetry principles are then associated with traditional physical measures. Keywords: Euler's Equation; Fine-Structure Constant; Forces of Nature; Fundamental Constants; Symmetry Principles. 1. Introduction Leonhard Euler was one of the greatest mathematicians of the eighteenth century. As a mathematician Lokenath Debnath states in The Legacy of Leonhard Euler, "It is remarkable that Euler discovered two elegant and most beautiful formulas in mathematics." [1]. eiπ + 1 = 0 and e2πi − 1 = 0. (1) William Eisen's description and interpretation of Euler's equation eiπ + 1 = 0 in relation to the Great Pyramid design shows four curves of ex from x = 0 to x = π, one curve on each side. Dividing the sides by π lengths results in a small square in the center called the Golden Apex A, the geometry and symmetry thought to generate the four fundamental forces of nature [2]. Eisen then asks the obvious question about the exponential function and Golden Apex interpretation, "Just how could the builders of the Great Pyramid have been so knowledgeable of the mathematics of the universe ...?" A = eπ − 7π − 1 ≃ √ (2) ⁄ 3π ≃ 0.1495. (2) A is the side of the Golden Apex square. √ (A) ≃ e ⁄ 7 and A + 1 = eπ − 7π ≃ R ≃ 1.152, radius of the regular heptagon with side one. The sin (2πA) ≃ φ ⁄ 2, where φ is the golden ratio, 1 ⁄ 2φ ≃ φ√(A). The tan (2πA) ≃ 1 + √(A) ≃ K ⁄ 2π, see Eq. (7) [3]. The polygon circumscribing constant is K ≃ 2tan(3π ⁄ 7) ≃ φ2 ⁄ 2A, see Eq. (7) and discussion [3]. A is also the reciprocal harmonic of the gravitational constant. Also, A ≃ tan2e − 1 and A − 1 ≃ cosh2(√(7 ⁄ e)) ≃ φ√(2πe). ln (A − 1) ≃ π ⁄ √(e) ≃ 6 ⁄ π, with the cube-sphere proportion. The regular heptagon radius, R = csc(π ⁄ 7) ⁄ 2 ≃ φ ⁄ √(2) ≃ cot2α − 1 and 2πR ≃ 1 ⁄ φ√(α). Also, RA ≃ 2√(α), where alpha α is the fine-structure constant, see the Eq. (7) discussion. RA ≃ √(φ) ⁄ e2 ≃ ln(π ⁄ √(7)) ≃ √(7 ⁄ π) ⁄ K. (3) R − 1 ≃ √(φ)sinα − 1 with eπ − πe ≃ sinα − 1 [3]. The cosh2(√(A)) ≃ eA − Ae ≃ π ⁄ e. The silver constant from the heptagon is S ≃ √ (π ⁄ 2A) ≃ 2√(2)R ≃ 3.247. √(S) = 2cos(π ⁄ 7) ≃ 7φ ⁄ 2π, 2πA ≃ S ⁄ 2√(3) and S ⁄ √(π) ≃ √(11 ⁄ S) ≃ √(1 ⁄ 2A) ≃ ln(2π). Again, Golden Apex A: A ≃ √(11) ⁄ 7π ≃ √(e) ⁄ 11 ≃ √(πα) ≃ 2παS. (4) With the fine-structure constant, 2πα is equal to the electron Compton wavelength divided by the Bohr radius and πα is the percentage of light absorbed by graphene [4]. 4 ⁄ π ≃ √(A) ⁄ 2A ≃ √(S ⁄ 2), with Eq. (8) discussion. K + 2R ≃ 11 and √(e) ⁄ φ ≃ 1 + αφ2 [3]. 2. The nature of the fine-structure constant Introduced by Arnold Sommerfeld, the fine-structure constant determines the strength of the electromagnetic interaction. Alpha, the fine-structure constant is α = e2 ⁄ ħc in cgs units [3]. The finestructure constant related to the Golden Apex of the Great Pyramid: 2A ≃ 2√(πα) ≃ 4me ⁄ mpα ≃ φ2 ⁄ K. (5) Also, 2A ≃ tanhS − 1 ≃ tan2(1 ⁄ 2) ≃ √(K) ⁄ π2 and √(2A) ≃ √(π) ⁄ S. When substituting the fine-structure constant value and approximate value for the proton-electron mass ratio αmp ⁄ 4me ≃ φ + √(3) and ln(mp ⁄ me) ⁄ ln(α − 1) ≃ 2πφA, see the discussion of Eq. (14) [3]. The Wilbraham-Gibbs constant is Gw and the sinc function is the sincx = sinx ⁄ x [5]: Gw = ∫ sinc xdx π 0 ≃ e sinα − 1 ≃ K ⁄ √(7π). (6) The Wilbraham-Gibbs constant Gw ≃ φlnπ ≃ φ2 ⁄ √(2) ≃ 1.852. The Wilbraham-Gibbs constant is related to the overshoot of Fourier sums in the Gibbs phenomena [5] and other approximations: Gw ≃ sec (1) ≃ exp(φ − 1) ≃ 5 ⁄ 7√(A), see the discussion of Eq. (17). The inverse Kepler-Bouwkamp constant is the polygon circumscribing constant K [3]: International Journal of Physical Research 47 K = (π/2) ∏ sinc((2π)/(2n + 1))∞n=1 = ∏ sec((π)/(n)). ∞ n=3 (7) Also, the polygon circumscribing constant K ≃ 2π(1 + √(A)) ≃ √(7 ⁄ 4πα) ≃ √(11)φ2 and K ≃ φ2 ⁄ 2A ≃ 5A ⁄ √(α) ≃ 4π√(SA) ≃ 3 ⁄ 2RA ≃ 2tan(3π ⁄ 7) ≃ 8.7. KA ≃ √(e ⁄ φ) ≃ R + A. Half the face apex angle of the Great Pyramid plus half the apex angle is approximately 70○ and sinα − 1 ≃ 2cos70○ [3]. The cscα − 1 ≃ R√(φ) ≃ √(7 ⁄ S) ≃ √(85) ⁄ 2π ≃ ln(372 ⁄ 85), see below. 528 ⁄ 504 ≃ 7A, see discussion of Eq. (17) [3]. First level sum of Teleois proportions is 85, foundational in Great Pyramid design [3]. 85 ⁄ 11 ≃ R ⁄ A and 528 ⁄ 85 ≃ 2π, see Eq. (17). The latest value for the inverse fine-structure constant by Aoyama et al, α − 1 ≃ 137.035 999 157 (41), from the most recent experimental results and quantum electrodynamics [6]. Eq. (8) derives the approximate value for α − 1 ≃ 137.035 999 168 [7]: sinα − 1 ≃ 504 ⁄ 85K = 7! ⁄ (713 + 137)K. (8) The sum of the eight main resonant nodes on the Turenne Rule is 372 ≃ e ⁄ α, part of a spectrum analysis that is also related to crystallographic groups [8]. K ≃ φln(372 − 137), ln372 ≃ 1 + ln137 ≃ √(π) ⁄ 2A and ln(137 × 372) ≃ φ ⁄ A. 1372 + 3722 ≃ 3962 and from the harmonic radii of the Cosmological Circle, 108 + 396 = 504 [3]. 504 ⁄ 396 = 14 ⁄ 11 ≃ 4 ⁄ π. Pyramid base angle θB≃ tan − 1(4 ⁄ π). The ln(4 ⁄ π) ≃ Aφ and π ⁄ 4 ≃ cscα − 1 − sinα − 1 ≃ sinθB. The pyramid apex angle θA, sinα − 1 ≃ θB ⁄ θA≃ √(2)tan(π ⁄ 7) ≃ √(S ⁄ 7)≃2π ⁄ √(85) [3]. 528 ⁄ 396 = 4 ⁄ 3 and 396 ⁄ 85 ≃ √(S ⁄ A). The quartz crystal harmonic is Qc = 786432 and 1 ⁄ √(α) ≃ ln(AQc). Base octave harmonic of Qc, 192 ≃ 7πK and φ ⁄ π ≃ 192 ⁄ 372 [7]. 3. The four fundamental forces of nature The heptagon is a feature of the Cosmological Circle, geometric template for many ancient architectural designs; related to the cycloid curve and history of the least action principle [7]. Golden Apex A, silver constant S and fine-structure constant, see Eq. (4). αE ≃ exp(− 2 ⁄ Ae) ≃ A ⁄ 2πS ≃ 7.29 × 10 − 3. (9) The electromagnetic coupling constant is αE = α = e2 ⁄ ħc in cgs units [7]. Together with Eq. (12) the ratio αW ⁄ αE ≃ Aπ3 ≃ √(π)φ2. √(α) ≃ Ae ⁄ φ√(K) ≃ √(R) ⁄ 4π and 2πφR ≃ 1 ⁄ √(α). G = ħc ⁄ mP2 ≃ 6.67191(99) ×10 − 11m3kg − 1s − 2 is the gravitational constant with Planck mass mP [9, 10]. Gravitational coupling is αG = Gme2 ⁄ ħc = (me ⁄ mP)2, with Golden Apex A: αG ≃ exp(− K√(π) ⁄ A) ≃ 1.752 × 10 − 45. (10) Also, − ln(αG) ≃ 2R ⁄ A2 ≃ exp(Aπ3) ≃ 2πRd ⁄ √(α), where dodecahedron circumradius Rd = φ√(3) ⁄ 2. Aπ3 ≃ KsinθC, where θC ≃ 32○, half the Great Pyramid face apex angle. An approximation with the Golden Apex A for the strong force coupling constant [11]: αS ≃ exp (A − √(2)φ) ≃ A ⁄ √(φ) ≃ 1.177 × 10 − 1. (11) The tetrahedral angle θt ≃ 109.5○, csc2(θt) = 9 ⁄ 8, the whole tone. αS ≃ ln(9 ⁄ 8) ≃ √(α)tan(2πA) ≃ √(2 ⁄ A) ⁄ π3 ≃ K√(α) ⁄ 2π, √(2)φ − A ≃ e ⁄ √(φ) and αS − 1 ≃ 7π√(A) ≃ π2 ⁄ R [7]. Approximation with the Golden Apex for the coupling constant of the weak force [12]: αW ≃ exp(− 1 ⁄ 2A) ≃ 2A ⁄ K ≃ 3.4 × 10 − 2. (12) Also, αW ⁄ αE ≃ Aπ3 ≃ √(π)φ2. Coupling constant αW = gw2 ⁄ 4π ≃ A ⁄ √(2)π, where gw is the coupling constant mediating the weak interaction. The Fermi coupling constant GF determines the strength of Fermi's interaction that explains the beta decay caused by the weak nuclear force. GF = gw2 ⁄ 4√(2)mW2 = αWπ ⁄ √(2)mW2 [13], mW is the mass of the W boson. The weak interaction is mediated by the exchange of W and Z gauge bosons. Weinberg angle θW, sin2θW ≃ √(α)φ2 ≃ 3A ⁄ 2 [12], [14], Eq. (17); cosθW = mW ⁄ mZ ≃ √(2) ⁄ φ. K2 ≃ cos(2θW) ⁄ α ≃ A√(6 ⁄ π). The pyramid base angle θB, θW ⁄ θB ≃ √(2A) ≃ √(π) ⁄ S. 4. Symmetry principles and physical measures Steven Weinberg is often quoted, "... there are symmetry principles that dictate the very existence of all the known forces of nature." [15]. Saul-Paul Sirag says, "By far the deepest theoretical advance afforded by the group-theory approach is the set of ADE Coxeter graphs. It is plausible to think of the ADE graphs as the ultimate Platonic archetypes." [16]. the discovery of symmetry principles and their application [17], [18] has advanced to the monster group and the modular j-function (referred to as "monstrous moonshine") [19], the umbral moonshines and string theory; along with vertex operator algebras in related physical approaches with conformal field theory [20]. For a half-period ratio of τ, the modular function j(τ) with q = exp(2πiτ) has the Fourier expansion [21]: J (τ) = q − 1 + 744 + 196884q + 21493760q2 + .... (13) The ln(21493760 ⁄ 196884) ≃ √(7π) ≃ √(2) ⁄ 2A. The ln196884 ≃ √(2 ⁄ A) ⁄ 2A ≃ KRd and Rd = φ√(3) ⁄ 2 is the circumradius of the regular dodecahedron with the side equal to one. From Grumiller et al, the chiral half of the monster group conformal field theory originally proposed by Ed Witten has a partition function given by the j-function. "The number 196884 is interpreted as one Virasoro descendant of the vacuum plus 196883 primary fields corresponding to flat space cosmology horizon microstates." The quantum correction in the respective low-energy entropy is then proportional to the ln196883 [22]. G.f. is a Fourier series which is the convolution square root of j (τ), see Eq. (13) [23]. G.f. = q − 1 + 372q + 29250q3 − 134120q5 + .... (14) The ln (29250 ⁄ 372) ≃ φ2 ⁄ 4A ≃ K ⁄ 2 and the ln(2 × 372) ≃ A − 1. The ln372 ≃ φ√ (2 ⁄ A) ≃ √(π) ⁄ 2A ≃ 2Kcos70○ and sin70○ ≃ 372 ⁄ 396 ≃ 2πA, with the hieroglyphic geometry for gold. An application in a special case of umbral moonshine is the Mathieu moonshine work of Eguchi, Ooguri and Tachikawa [24] and followed by others, includes the q-series e(q) whose coefficients are then " ... twice the dimension of some irreducible representation of the Mathieu group M24." Pierre-Philippe Dechant continues, "Modularity is therefore very topical, also in other areas and a Clifford perspective on holomorphic and modular functions could have profound consequences." [25]. A normalized q-series e (q): e (q) = 90q + 462q2 + 1540q3 + 4554q4 + 11592q5 + .... (15) The ln(462 ⁄ 90) ≃ √(7) ⁄ φ ≃ √(A)φ3 with the ln90 ≃ √(3)φ2 ≃ 2 ⁄ 3A ≃ U√(K), see Eq. (16). Also correlated, 90 ⁄ 372 ≃ φA and 462 ⁄ 372 ≃ φA + 1 ≃ 2 ⁄ φ, recall discussion of Eq. (8). A gold pyramid at the tip of the Great Pyramid, with octahedral and icosahedral symmetry [26], was the "Golden Tip" described by John Michell with the support of Algernon Berriman's metrology [27]. This was represented by the height of a pyramid with 5 cubic inches, 0.152 ≃ 11.7 ⁄ (7 × 11), √(137) ≃ 11.7 and 0.152 is the tenth part of the Greek cubit of 1.52'. This pyramidion might have been similar to the legendary Golden Sun Disc of Mu. Apex angle of the regular heptagon triangle is 3π ⁄ 7 and an approximation to the apex angle of the Great Pyramid [7]. The "Golden Tip" harmonic of U ≃ φsin70○ ≃ 2πφA ≃ ln(mp ⁄ me) ⁄ ln(α − 1) ≃ 1.52 [3]. From the heptagon geometry found in the Cosmological Circle, AU ≃ R ⁄ πφ ≃ 5 ⁄ 7π ≃ cot(3π ⁄ 7) ≃ 0.227 and the tan(3π ⁄ 7) ≃ eφ ≃ 4.4. The ratio 7 ⁄ 5 ≃ φ ⁄ R ≃ Rd, the circumradius of the regular dodecahedron having the side equal to one, see the discussion following Eq. (13) and [7]. 48 International Journal of Physical Research AU ≃ 85 ⁄ 372 ≃ φ√(eα) ≃ exp(−ARK) ≃ 2 ⁄ 3√(K). (16) The sum A + U ≃ 1 ⁄ 4A, A ⁄ U ≃ R√(α) and the ARK ≃ √(7 ⁄ π) ≃ coth2R ≃ 3 ⁄ 2 ≃ 10A. Interesting parallels can be found between William Eisen's Golden Apex of the Great Pyramid design, the torus topology of Einstein's relativity and Wolfgang Pauli's World Clock Vision; with his i ring imaginal geometry of the unit circle on the complex plane and also symbolic of the unus mundus as the natural ontology of quantum theory [28]. Pauli's World Clock also has the golden ratio geometry related to the fine-structure constant together with four men swinging pendulums [28]. Ancient Egyptian architects converted celestial time periods into lengths that are equivalent to those of the pendulum measures as rediscovered by Galileo and explained by Sir Isaac Newton. Roger Newton states that "speculations concerning a long-awaited reconciliation between Einstein's general theory of relativity and the quantum, known in their various guises as superstring theory, employ as their basic element the properties of Galileo's simple pendulum." [29]. Flinders Petrie found the harmonic of the standard day when converted by the pendulum formula results in the length of the Royal Cubit, "... basis of the Egyptian land measures .... This value for the cubit is 20.617'' while the best examples in stone are 20.620''±0.005''." The Egyptian Royal Cubit is the traditionally known measure basis of the Great Pyramid "... and its base measures 440 Royal Cubits in length." [30]. R.C. ≃ 144 ⁄ 7 ≃ A ⁄ α ≃ φ + φ ⁄ √ (α) ≃ √(7π) ⁄ AU ≃ 7√(K). (17) The canonical Royal Cubit of 20.736'' is the harmonic of 1442 and the standard harmonic Royal Cubit R.C. ≃ 20.618 [27]. Also, R.C. ≃ √(7π)tan(3π ⁄ 7). Basic square perimeter of the Cosmological Circle is 44 and 442 + 1372 ≃ 1442 [28]. 22 = √ (44 + 440) ≃ 7π and A ≃ π ⁄ √(440) [3]. 372 ⁄ 440 ≃ 4√(2)A, 144 ⁄ 372 ≃ √(A) and 504 ⁄ 144 = 7 ⁄ 2. Also, 144 ⁄ 85 ≃ e ⁄ φ ≃ A + U [3]. Plato's "fusion number" 1746, described by John Michell, represented the apex of the Great Pyramid [27]. Fusion number 1746 ≃ 144√(7π ⁄ A), 372 ⁄ 1746 ≃ √(2)A and AU ≃ 396 ⁄ 1746. 528 ⁄ 372 ≃ √(2), 528 ⁄ 1746 ≃ 2A, 528 ⁄ 504 ≃ 7A, 1746 ⁄ 504 ≃ 2√(3) and R.C. ≃ 1746 ⁄ 85. The Great Pyramid Key is 528 ≃ ln(7 ⁄ A) ⁄ α [3]. 5. Conclusion From the exponential function of Euler's equation to the geometry of a fundamental form, the Golden Apex of the Great Pyramid was described, leading to a calculation of the fine-structure constant and its close relationship with the proton-electron mass ratio. Golden Apex related equations were then found for four fundamental forces of nature. Juan Maldacena restates it, "The forces of nature are based on symmetry principles." [31]. these symmetry principles were then associated with traditional physical measures. And finally, here is William Eisen's quotation again, "Just how could the builders of the Great Pyramid have been so knowledgeable of the mathematics of the universe ...?" [2]. 6. 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