# Fundamental Physics and the Fine-Structure Constant

Michael A. Sherbon

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Michael A. Sherbon. Fundamental Physics and the Fine-Structure Constant. International Journal of Physical Research, 2017, 5 (2), pp.46-48. 10.14419/ijpr.v5i2.8084. hal-01312695

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# Fundamental Physics and the Fine-Structure Constant

Michael A. Sherbon

Case Western Reserve University Alumnus

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.

## 1. Introduction

Leonhard Euler was one of the greatest mathematicians of the eighteenth century. In *The Legacy of Leonhard Euler* mathematician Lokenath Debnath says, “It is remarkable that Euler discovered two elegant and most beautiful formulas in mathematics.” [1].

$$e^{i\pi} + 1 = 0 \quad \text{and} \quad e^{2\pi i} - 1 = 0. \qquad (1)$$

William Eisen's description and interpretation of Euler's equation $e^{i\pi}+1 = 0$ in relation to the Great Pyramid design shows four curves of $e^x$ from $x = 0$ to $x = \pi$, one curve on each side. Dividing the sides by $\pi$ 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 ...?”

$$A = e^{\pi} - 7\pi - 1 \approx \sqrt{2}/3\pi \approx 0.1495. \qquad (2)$$

A is the side of the Golden Apex square. $\sqrt{A} \approx e/7$ and $A + 1 = e^{\pi} - 7\pi \approx R \approx 1.152$, radius of the regular heptagon with side one. The $\sin(2\pi A) \approx \phi/2$, where $\phi$ is the golden ratio, $1/2\phi \approx \sqrt{A}$. The $\tan(2\pi A) \approx 1 + \sqrt{A} \approx K/2\pi$, see Eq. (7) [3]. The polygon circumscribing constant is $K \approx 2\tan(3\pi/7) \approx \phi^2/2A$, see Eq. (7) and discussion [3]. A is also the reciprocal harmonic of the gravitational constant. Also, $A \approx \tan^2 e^{-1}$ and
$A^{-1} \approx \cosh^2(\sqrt{7/e}) \approx \phi\sqrt{2\pi e}$. $\ln(A^{-1}) \approx \pi/\sqrt{e} \approx 6/\pi$, with the cube-sphere proportion. The regular heptagon radius, $R = \csc(\pi/7)/2 \approx \phi/\sqrt{2} \approx \cot^2\alpha^{-1}$ and $2\pi R \approx 1/\phi\sqrt{\alpha}$. Also, $RA \approx 2\sqrt{\alpha}$, where $\alpha$ is the fine-structure constant, see the Eq. (7) discussion.

$$RA \approx \sqrt{\phi}/e^2 \approx \ln(\pi/\sqrt{7}) \approx \sqrt{7/\pi}/K. \qquad (3)$$

$R^{-1} \approx \sqrt{\phi} \sin \alpha^{-1}$ with $e^\pi - \pi^e \approx \sin \alpha^{-1} [3]$. The $\cosh^2(\sqrt{A}) \approx e^A - A^e \approx \pi/e$. The silver constant from the heptagon is $S \approx \sqrt{\pi/2A} \approx 2\sqrt{2}R \approx 3.247$. $\sqrt{S} = 2\cos(\pi/7) \approx 7\phi/2\pi$, $2\pi A \approx S/2\sqrt{3}$ and $S/\sqrt{\pi} \approx \sqrt{11/S} \approx \sqrt{1/2A} \approx \ln(2\pi)$. Again, with a Golden Apex A:

$$A \approx \sqrt{11}/7\pi \approx \sqrt{e}/11 \approx \sqrt{\pi\alpha} \approx 2\pi\alpha S. \qquad (4)$$

With the fine-structure constant, $2\pi\alpha$ is equal to the electron Compton wavelength divided by the Bohr radius and $\pi\alpha$ is the percentage of light absorbed by graphene [4]. $4/\pi \approx \sqrt{A}/2A \approx \sqrt{S}/2$, with Eq. (8) discussion. $K + 2R \approx 11$ and $\sqrt{e}/\phi \approx 1 + \alpha\phi^2$ [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 $\alpha = e^2/\hbar c$ in cgs units [3]. The fine-structure constant related to the Golden Apex of the Great Pyramid:

$$2A \approx 2\sqrt{\pi\alpha} \approx 4m_e/m_p\alpha \approx \phi^2/K. \qquad (5)$$

Also, $2A \approx \tanh S^{-1} \approx \tan^2(1/2) \approx \sqrt{K}/\pi^2$ and $\sqrt{2A} \approx \sqrt{\pi}/S$. When substituting the fine-structure constant value and approximate value for the proton-electron mass ratio $\alpha m_p/4m_e \approx \phi + \sqrt{3}$ and $\ln(m_p/m_e)/\ln(\alpha^{-1}) \approx 2\pi\phi A$, see the discussion of Eq. (14) [3]. The Wilbraham-Gibbs constant is $G_w$ and the sinc function is the sinc $x = \sin x/x$ [5]:

$$G_w = \int_0^\pi \mathrm{sinc} \, x \, dx \approx e \sin \alpha^{-1} \approx K/\sqrt{7\pi}. \qquad (6)$$

The Wilbraham-Gibbs constant $G_w \approx \phi \ln \pi \approx \phi^2/\sqrt{2} \approx 1.852$. The Wilbraham-Gibbs constant is related to the overshoot of Fourier sums in the Gibbs phenomena [5] and other approximations: $G_w \approx \sec(1) \approx \exp(\phi^{-1}) \approx 7A\sqrt{\pi} \approx 5/7\sqrt{A}$, Eq. (17) discussion. The inverse Kepler-Bouwkamp constant is the polygon circumscribing constant $K$ [3]:

$$K = \frac{\pi}{2} \prod_{n=1}^{\infty} \mathrm{sinc} \left( \frac{2\pi}{2n+1} \right) = \prod_{n=3}^{\infty} \sec \left( \frac{\pi}{n} \right). \qquad (7)$$

Also, the polygon circumscribing constant $K \approx 2\pi(1 + \sqrt{A}) \approx \sqrt{7/4\pi\alpha} \approx \sqrt{11}\phi^2$ and $K \approx \phi^2/2A \approx 5A/\sqrt{\alpha} \approx 4\pi\sqrt{SA} \approx 3/2RA \approx 2\tan(3\pi/7) \approx 8.7$. $KA \approx \sqrt{e/\phi} \approx R+A$.

Half the face apex angle of the Great Pyramid plus half the apex angle is approximately $70^\circ$ and $\sin \alpha^{-1} \approx 2 \cos 70^\circ$ [3]. $\csc \alpha^{-1} \approx R\sqrt{\phi} \approx \sqrt{7/S} \approx \sqrt{85}/2\pi \approx \ln(372/85)$, see below. $528/504 \approx 7A$, see discussion of Eq. (17) [3]. First level sum of Teleois proportions is 85, foundational in Great Pyramid design [3]. $85/11 \approx R/A$ and $528/85 \approx$
$2\pi$, see Eq. (17). The latest value for the inverse fine-structure constant by Aoyama et al, $\alpha^{-1} \simeq 137.035999157(41)$, from the most recent experimental results and quantum electrodynamics [6]. Eq. (8) derives the approximate value for $\alpha^{-1} \simeq 137.035999168$ [7]:

$$ \sin \alpha^{-1} \simeq 504/85K = 7!(713 + 137)K. \qquad (8) $$

Sum of the eight main resonant nodes on the Turenne Rule is $372 \simeq e/\alpha$, part of a spectrum analysis also related to crystallographic groups [8]. $K \simeq \phi \ln(372 - 137)$, $\ln 372 \simeq 1 + \ln 137 \simeq \sqrt{\pi}/2A$ and $\ln(137 \times 372) \simeq \phi/A$. $137^2 + 372^2 \simeq 396^2$ and from the harmonic radii of the Cosmological Circle, $108 + 396 = 504$ [3]. $504/396 = 14/11 \simeq 4/\pi$. Pyramid base angle $\theta_B \simeq \tan^{-1}(4/\pi)$. The $\ln(4/\pi) \simeq A\phi$ and $\pi/4 \simeq \csc \alpha^{-1} - \sin \alpha^{-1} \simeq \sin \theta_B$. The pyramid apex angle $\theta_A$, $\sin \alpha^{-1} \simeq \theta_B/\theta_A \simeq \sqrt{2}\tan(\pi/7) \simeq \sqrt{S/7} \simeq 2\pi/\sqrt{85}$ [3]. $528/396 = 4/3$ and $396/85 \simeq \sqrt{S/A}$. The quartz crystal harmonic is $Q_c = 786432$ and $1/\sqrt{\alpha} \simeq \ln(AQ_c)$. Base octave harmonic of $Q_c$, $192 \simeq 7\pi K$ and $\phi/\pi \simeq 192/372$ [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 of Eq. (4) discussion.

$$ \alpha_E \simeq \exp(-2/Ae) \simeq A/2\pi S \simeq 7.29 \times 10^{-3}. \qquad (9) $$

The electromagnetic coupling constant is $\alpha_E = \alpha = e^2/\hbar c$ in cgs units [7]. Together with Eq. (12) the ratio $\alpha_W/\alpha_E \simeq A\pi^3 \simeq \sqrt{\pi}\phi^2$. $\sqrt{\alpha} \simeq Ae/\phi\sqrt{K} \simeq \sqrt{R}/4\pi$ and $2\pi\phi R \simeq 1/\sqrt{\alpha}$.

Gravitational constant $G = \hbar c/m_{Pl}^2 \simeq 6.67191(99)\AA\AA^10^{-11} m^3kg^{-1}s^{-2}$ and Planck mass $m_P$ [9, 10]. Gravitational coupling $\alpha_G = Gm_e^2/\hbar c = (m_e/m_P)^2$, Golden Apex A:

$$ \alpha_G \simeq \exp(-K\sqrt{\pi}/A) \simeq 1.752 \times 10^{-45}. \qquad (10) $$

Also, $-\ln(\alpha_G) \simeq 2R/A^2 \simeq \exp(A\pi^3) \simeq 2\pi R_d/\sqrt{\alpha}$, where dodecahedron circumradius $R_d = \phi\sqrt{3}/2$. $A\pi^3 \simeq K \sin\theta_C$, where $\theta_C \simeq 32^\circ$, half the Great Pyramid face apex angle.

An approximation with the Golden Apex A for the strong force coupling constant [11]:

$$ \alpha_S \simeq \exp(A - \sqrt{2}\phi) \simeq A/\sqrt{\phi} \simeq 1.177 \times 10^{-1}. \qquad (11) $$

The tetrahedral angle $\theta_t \simeq 109.5^\circ$, $\csc^2(\theta_t) = 9/8$, the whole tone. $\alpha_S \simeq \ln(9/8) \simeq \sqrt{\alpha} \tan(2\pi A) \simeq \sqrt{2/A}/\pi^3 \simeq K\sqrt{\alpha}/2\pi$, $\sqrt{2}\phi - A \simeq e/\sqrt{\phi}$ and $\alpha_S^{-1} \simeq 7\pi\sqrt{A} \simeq \pi^2/R$ [7].

Approximation with the Golden Apex for the coupling constant of the weak force [12]:

$$ \alpha_W \simeq \exp(-1/2A) \simeq 2A/K \simeq 3.4 \times 10^{-2}. \qquad (12) $$

Also, $\alpha_W/\alpha_E \simeq A\pi^3 \simeq \sqrt{\pi}\phi^2$. Coupling constant $\alpha_W = g_w^2/4\pi \simeq A/\sqrt{2}\pi$, where $g_w$ is the coupling constant mediating the weak interaction. The Fermi coupling constant $G_F$ determines the strength of Fermi's interaction that explains the beta decay caused by the weak nuclear force. $G_F = g_w^2/4\sqrt{2}m_W^2 = \alpha_W\pi/\sqrt{2}m_W^2$ [13], $m_W$ is the mass of the W boson. The weak interaction is mediated by the exchange of W and Z gauge bosons. Weinberg angle $\theta_W$, $\sin^2\theta_W \simeq \sqrt{\alpha}\phi^2 \simeq 3A/2$ [12, 14], Eq. (17); $\cos\theta_W = m_W/m_Z \simeq \sqrt{2}/\phi$. $K^2 \simeq \cos(2\theta_W)/\alpha \simeq \sqrt{6}/\pi$. The pyramid base angle $\theta_B$, $\theta_W/\theta_B \simeq \sqrt{2A} \simeq \sqrt{\pi}/S$.
## 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 $q = \exp(2\pi iτ)$ has the Fourier expansion [21]:

$$j(\tau) = q^{-1} + 744 + 196884q + 21493760q^2 + .... \quad (13)$$

The $\ln(21493760/196884) \approx \sqrt{7\pi} \approx \sqrt{2}/2A$. The $\ln 196884 \approx \sqrt{2/A}/2A \approx KR_d$ and $R_d = \phi\sqrt{3}/2$ 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 $\ln 196883$ [22].

$G.f.$ is a Fourier series which is the convolution square root of $j(τ)$, see Eq. (13) [23].

$$G.f. = q^{-1} + 372q + 29250q^3 - 134120q^5 + .... \quad (14)$$

The $\ln(29250/372) \approx \phi^2/4A \approx K/2$ and the $\ln(2 \times 372) \approx A^{-1}$. The $\ln 372 \approx \phi\sqrt{2/A} \approx \sqrt{\pi}/2A \approx 2K \cos 70^\circ$ and $\sin 70^\circ \approx 372/396 \approx 2\pi A$, 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 $M_{24}$.” 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 $e(q)$:

$$e(q) = 90q + 462q^2 + 1540q^3 + 4554q^4 + 11592q^5 + .... \quad (15)$$

The $\ln(462/90) \approx \sqrt{7}/\phi \approx \sqrt{A}\phi^3$ with the $\ln 90 \approx \sqrt{3}\phi^2 \approx 2/3A \approx U\sqrt{K}$, see Eq. (16). Also correlated, $90/372 \approx \phi A$ and $462/372 \approx \phi A + 1 \approx 2/\phi$, 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, $0.152 \approx 11.7/(7 \times 11)$, $\sqrt{137} \approx 11.7$ 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 $3\pi/7$ and an approximation to the apex angle of the Great Pyramid [7]. The “Golden Tip” harmonic of $U \approx \phi \sin 70^\circ \approx 2\pi\phi A \approx \ln(m_p/m_e)/\ln(\alpha^{-1}) \approx 1.52$ [3]. From the heptagon geometry
found in the Cosmological Circle, $AU \simeq R/\pi\phi \simeq 5/7\pi \simeq \cot(3\pi/7) \simeq 0.227$ and the $\tan(3\pi/7) \simeq e\phi \simeq 4.4$. The ratio $7/5 \simeq \phi/R \simeq R_d$, the circumradius of the regular dodecahedron having the side equal to one, see the discussion following Eq. (13) and [7].

$$AU \simeq 85/372 \simeq \phi\sqrt{e\alpha} \simeq \exp(-ARK) \simeq 2/3\sqrt{K}. \qquad (16)$$

The sum of $A+U \simeq 1/4A$, $A/U \simeq R\sqrt{\alpha}$ and the $ARK \simeq \sqrt{7/\pi} \simeq \coth^2 R \simeq 3/2 \simeq 10A$.

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].

$$R.C. \simeq 144/7 \simeq A/\alpha \simeq \phi + \phi/\sqrt{\alpha} \simeq \sqrt{7\pi}/AU \simeq 7\sqrt{K}. \qquad (17)$$

The canonical Royal Cubit of 20.736" is the harmonic of $144^2$ and the standard harmonic Royal Cubit $R.C. \simeq 20.618$ [27]. Also, $R.C. \simeq \sqrt{7\pi} \tan(3\pi/7)$. Basic square perimeter of the Cosmological Circle is $44$ and $44^2 + 137^2 \simeq 144^2$ [28]. $22 = \sqrt{44+440} \simeq 7\pi$ and $A \simeq \pi/\sqrt{440}$ [3]. $372/440 \simeq 4\sqrt{2}A$, $144/372 \simeq \sqrt{A}$ and $504/144 = 7/2$. Also, $144/85 \simeq e/\phi \simeq A+U$ [3]. Plato's “fusion number” 1746, described by John Michell, represented the apex of the Great Pyramid [27]. Fusion number $1746 \simeq 144\sqrt{7\pi}/A$, $372/1746 \simeq \sqrt{2}A$ and $AU \simeq 396/1746$. $528/372 \simeq \sqrt{2}$, $528/1746 \simeq 2A$, $528/504 \simeq 7A$, $1746/504 \simeq 2\sqrt{3}$ and $R.C. \simeq 1746/85$. The Great Pyramid *Key* is $528 \simeq \ln(7/A)/\alpha$ [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].
Acknowledgments

Special thanks to Case Western Reserve University, MathWorld and WolframAlpha.

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