Fine-Structure Constant from Golden Ratio Geometry Michael A. Sherbon Case Western Reserve University Alumnus E-mail: michael.sherbon@case.edu January 24, 2018 Abstract After a brief review of the golden ratio in history and our previous exposition of the finestructure constant and equations with the exponential function, the fine-structure constant is studied in the context of other research calculating the fine-structure constant from the golden ratio geometry of the hydrogen atom. This research is extended and the fine-structure constant is then calculated in powers of the golden ratio to an accuracy consistent with the most recent publications. The mathematical constants associated with the golden ratio are also involved in both the calculation of the fine-structure constant and the proton-electron mass ratio. These constants are included in symbolic geometry of historical relevance in the science of the ancients. Keywords fine-structure constant, dimensionless physical constants, history of mathematics, golden ratio, sacred geometry, Fibonacci sequence, mathematical constants, fundamental physics. 1. Introduction As Willem Witteveen states in his book The Great Pyramid of Giza, "Every expression of the golden mean, which includes: number, rectangle, triangle, spiral and frequency, is encoded in the design of the Great Pyramid and illustrates the importance of the ratio in the universe in which we belong." [1]. He also states, "The golden ratio, as well as the Great Pyramid as an expression of it, is an important key to our universe containing the Earth and the Moon" [1] and that "the ratio between the Earth and the Moon is in fact the basis for the mathematical concept of 'squaring the circle' ..." [1]. Marja de Vries states, "The Golden Ratio defines the squaring of a circle .... According to some, in ancient Egypt, this mathematical mystery was encoded in the measurements of the Great Pyramid of Giza." [2]. Continuing with her general theme of 1 universal laws and wholeness, de Vries says, "In short, the idea dawns that the one universal principle ... embodiment of the Principle of Least Action ... indeed seems to be the Golden Ratio Spiral." [2]. Richard Heath has another description, "The Golden Mean was considered a fundamental constant by the Egyptians and the fundamental division of the whole into two parts." [3]. Mario Livio says, "In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics." [4]. Alexey Stakhov explains that "one of the most important trends in the development of modern science ... is very simple: a return to the 'harmonic ideas' of Pythagoras and Plato (the 'golden ratio' and Platonic solids), embodied in Euclid's 'Elements'" [5], also see [6]. As stated previously, "The golden ratio is an approximate harmonic of the Planck length in meters and harmonics of fundamental units have a geometric basis in ancient metrology." [7, 8]. Further consideration on the nature of the golden section is given by Scott Olsen [9]. Fundamental modern applications are suggested by David Haight, "There is considerable evidence that the golden proportion is the foundation for the unification of mathematics and physics." [10], also see Gazalé [11]. In the description of Eckhart Schmitz, "Mathematics is a universal language and it would be fitting to examine the Great Pyramid in this language to derive its meaning. It has been suggested that the Great Pyramid is a repository of ancient knowledge." [12]. Thousands of years ago the ancients had an advanced mathematical understanding of universe that is revealed in many sources. There is a consistent link to knowledge of the golden mean, but the way in which the ancients were able to formulate and use this information speaks of a technical grasp of the subject that exceeds what we know about it in the present day. –Alison Primrose [13]. David Haight says, "The golden proportion is the only one in which its (legato) addition and (staccato) multiplication of itself are equivalent. It is both an arithmetic and a geometrical progression, two sides of the same coin (another "two that are one"), and is the basis of logarithms and exponentials (logarithms transform multiplication into addition, and exponentials transform addition into multiplication)." [10]. In The Essence of the Cabalah, William Eisen describes the fundamental geometry of what he described as the "Golden Apex of the Great Pyramid" [14]. Eisen's description and interpretation of Euler's identity [15]-[17], exp(iπ)+1 = 0 ("this most compact and mysterious formula" that "Richard Feynman referred to as 'the most remarkable formula in mathematics,'"[18]) in relation to the Great Pyramid shows a drawing of four curves of ex from x = 0 to x = π , one curve on each side and labelled the "Graphical Representation of the Exponential Function to the Base e." ex = exp(x) = ∞ ∑ n=0 xn x! = lim n→∞ ( 1+ x n )n . (1) In addition to the exponential function [19, 20], Euler's formula is exp(ix) = cosx+ isinx and in hyperbolic terms exp(x) = coshx+ sinhx. Eisen was asking himself how the ancient Egyptians could know so much about this and how it supports his effort to understand the role of imaginary and complex numbers in the geometry of the Great Pyramid [14, 15]. The measures he found in this model with exponential curves align with other traditional measures found in the Great Pyramid design, including the golden ratio, also given as φ = exp(iπ/5)+ exp(−iπ/5). 2 These exponential curves could be considered as having the esoteric properties of an alchemical vessel as Steven Rosen explains that they "possess the feature of curving back into themselves." [21]. Wolfgang Pauli's i ring, part of his World Clock vision [22], was "used in his description of microphysical spin" [21]. Rosen quotes Pauli, "The ring with the i is the unity beyond particle and wave, and at the same time the operation that generates either of these." [21]. Also see the following references for Pauli and the spin concept development [23]-[29]. The exponential growth function is the simplest possible differential equation, the unique solution to dy/dx = y; y(0)1 and the most primitive (prime) derivative in which state and rate, form and function, symmetry and dynamism, being and becoming, structure and process, the integral and the derivative, the evolute and the involute, the 'splice' and the 'slice' and the squaring of a root and the extraction of a root are the same. –David Haight [30]. Dividing the sides of his mathematical model for the Great Pyramid by π lengths results in a small square in the center called the Golden Apex, the geometry and symmetry thought to be associated with the generation of the four fundamental forces of nature [14, 15]. Eisen provides a description of the dimensions formed by the exponential function and the Golden Apex square, A = eπ −7π−1' √ 2/3π ' 0.1495. (2) A is the side length 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 heptagon is traditionally associated with the geometry of 'squaring the circle'. The sin(2πA) ' φ/2, where φ is the golden ratio [31]. The tan(2πA) ' 1 + √ A ' K/2π, see Eq. (7) [32]. The polygon circumscribing constant is K ' 2tan(3π/7) ' φ 2/2A, see Eq. (7) discussion and 2A ' cos √ φ ' √ 2/7π [7, 32, 33]. A is also the reciprocal harmonic of Newton's gravitational constant. Also, A ' ln( √ φ)/φ ' 3/e3 ' tan2(e−1) and ln(A−1) ' π/ √ e ' √ 1+ √ 7 ' 6/π ' √ 1+φ 2. The regular heptagon radius, R = csc(π/7)/2 ' φ/ √ 2 ' cot2 α−1 and 2πR ' 1/φ √ α. RA ' 2 √ α, where α is the fine-structure constant, see the Eq. (7) discussion below [7, 32, 33]. RA' √ φ/e2 ' ln(π/ √ 7)' √ 7/π/K. (3) R−1 ' √ φ sinα−1 and e2+φ 2 ' RK. The cosh2( √ A)' π/e. A' √ α cosh(π/e). The √ 7/π ' coth2 R ' ARK [7, 32, 33]. The silver constant from the heptagon [34, 35] is approximated by S' 2 √ 2R' tan √ φ ' 3.247. The √ S = 2cos(π/7)' 7φ/2π and 2πA' S/2 √ 3' sin70◦. Also relevant, 2A' S/11' 1/ √ 11, see the discussion of Eq. (9). More approximations with the Great Pyramid's Golden Apex A [7, 32, 33]: A' √ 11/7π ' √ e/11' √ πα ' 2παS. (4) As Jean-Paul and Robert Bauval describe in the Secret Chamber Revisited how prime numbers 7 and 11 are significant keys to the Great Pyramid, 22/7' π [36]. Also noted by David Haight, "When the Fibonacci number sequence is based on the number seven and its multiples, the Fibonacci sequence self-reflexively reappears when differences are calculated between it and this new number-seven-based Fibonacci sequence. The same thing happens with Lucas numbers." 3 [10]. With the fine-structure constant, 2πα is equal to the electron Compton wavelength divided by the Bohr radius [21] and πα is the percentage of light absorbed by graphene [37]. The√ α ' √ R/4π ' 3A/2φ 2 and √ 2A' √ π/S. Also, 4/π ' √ A/2A' √ S/2 and see the Eq. (10) discussion. Finally, √ e/φ ' 1+αφ 2 [7, 32, 33]. 2. The golden ratio and ne-structure constant From David Haight, "prime numbers, the 'atoms' of mathematics, are necessarily related to the atoms of nature because of the well-known Rydberg rule that follows the same pattern as Euler's harmonic zeta power series (derived from the self-derived exponential growth function)." [30]. The Euler product formula for the Riemann zeta function [38, 39]: ζ (2) = ∞ ∑ n=1 1 n2 = ∏ p prime 1 1− p−2 = π2 6 ' 1.645. (5) Approximations for ζ (2) ' 11A ' √ 7− 1 and π2 ' √ K/2A ' 6 √ e. The ln(A−1) ' Rζ (2) ' 6/π, with the cube-sphere proportion. Quoting David F. Haight again, "There is geometry in the humming of the primes, there is music in the spacing of the primes." [30]. The sinh √ φ ' ζ (2). [32, 33]. The polygon circumscribing constant K is the reciprocal of the Kepler–Bouwkamp constant [40]-[42], related to "Pauli's triangle" [22], with sides approximately proportional to 1, φ , √ φ √ 5 with the golden ratio φ = (1+ √ 5)/2 with √ 5 = 2cosh(lnφ).[31]. The prime constant [32, 33], [43] is described as a binary expansion corresponding to an indicator function for the set of prime numbers. Defining the prime constant P for p(k) as the k-th prime: P = ∞ ∑ k=1 2−p(k) ' ζ (2) √ αK ' √ RA' 0.4147. (6) The prime constant, P ' √ RA ' φ 2/2π ' √ 3/2K ' √ 2− 1. Again, K is the inverse Kepler–Bouwkamp constant [40]-[42]. Introduced by Arnold Sommerfeld, the fine-structure constant determines the strength of the electromagnetic interaction [44]-[49]. John S. Rigden states, "The fine-structure constant derives its name from its origin. It first appeared in Sommerfeld's work to explain the fine details of the hydrogen spectrum." [50]. Arnold Sommerfeld generalized Bohr's model to include elliptical orbits in three dimensions. He treated the problem relativistically (using Einstein's formula for the increase of mass with velocity) .... According to historian Max Jammer, this success of Sommerfeld's fine-structure formula "served also as an indirect confirmation of Einstein's relativistic formula for the velocity dependence of inertia mass. –Stephen G. Brush [51]. Charles P. Enz writes, "For Pauli the central problem of electrodynamics was the field concept and the existence of an elementary charge which is expressible by the fine-structure constant ... 1/137. This fundamental pure number had greatly fascinated Pauli .... For Pauli the explanation of the number 137 was the test of a successful field theory, a test which no theory has passed up 4 to now." [52]. And again, Pauli is quoted by Varlaki, Nadai and Bokor concerning his evaluation and regard for the fine-structure constant, "The theoretical interpretation of its numerical value is one of the most important unsolved problems of atomic physics." [53]. Michael Eckert, "Even among physicists of the twenty-first century, the 'Bohr-SommerfeldAtom' and the 'Sommerfeld fine-structure constant,' remain current concepts." [49]. David Haight states, "Since the ideal divergence angle in nature is the golden proportion of 137.5 degrees, could this unique proportion be the reason why 137 is the 'prime' prime number or paradigm prime in nature, beginning with hydrogen?" [10]. Harald Fritzsch recalls that "Richard Feynman, the theory wizard of Caltech in Pasadena, once suggested that every one of his theory colleagues should write on the blackboard in his office: 137 –how shamefully little we understand!" [54]. Alpha,the electromagnetic coupling constant is α = e2/hc in cgs units [32, 33]. Calculation of the inverse fine-structure constant as an approximate derivation from prime number theory: α −1 ' 157−337P/7' 137.035999168. (7) This equation gives the same approximate value for the inverse fine-structure constant as determined in Eqs. (10-12), having three prime numbers and the prime constant [55]. Also, α−1 '√ 7tan−1( √ φ) ' 3/A2. The ln(A−1) ' 1/P √ φ ' √ 1+ √ 7 ' √ 1+φ 2 and 117+ 157 ' 2× 137, see discussion of Eq. (17). 337−157= 180 and the ratio 528/337' π/2 [7], see discussion of Eq. (10). The fine-structure constant is described by David Haight as the "basis or boundary condition of the 'pre-established harmony' at the prime 'atomic' level of mathematics, physics, and cosmology, for the reason that the essences of things are numbers, especially primes." [30]. Other approximations with the Golden Apex of the Great Pyramid: √ α ' Ae/φ √ K ' √ R/4π and 2πφR ' 1/ √ α. Also, ln(A−1) ' √ KP ≈ S √ P and 2A ' 2 √ πα ' φ 2/K ' √ 2/7π. The heptagon is a feature of the Cosmological Circle, a geometric template for the Great Pyramid and many ancient architectural designs; related to the cycloid curve and the history of the least action principle [8, 22]. From the Golden Apex A and the silver constant S, 2A' tanhS−1 ' tan2(1/2) , see Eq. (4). The Wilbraham-Gibbs constant is Gw and the sinc function, sincx = sin(x)/x [56]: Gw = ∫ π 0 sincxdx' e sinα−1 ' K/ √ 7π. (8) The Wilbraham-Gibbs constant Gw ' φ lnπ ' φ 2/ √ 2' 1.852. The Wilbraham-Gibbs constant is related to the overshoot of Fourier sums in the Gibbs phenomena [56]. Additional approximations: Gw ' sec(1)' exp(φ−1)' √ φ/πA, see the discussion of Eq. (18). The inverse Kepler-Bouwkamp constant is the polygon circumscribing constant K [40]-[42]: K = π 2 ∞ ∏ n=1 sinc ( 2π 2n+1 ) = ∞ ∏ n=3 sec ( π n ) . (9) The polygon circumscribing constant, K ' 3/2RA ' 2tan(3π/7) ' 8.700 and KA ' √ e/φ '√ 5/3 ' R+A. Half the face apex angle of the Great Pyramid plus half the apex angle is approximately 70◦ and sinα−1 ' 2cos70◦ [7]. The cscα−1 ' R √ φ ' √ 7/S ' √ 85/2π and 528/504 ' 7A ' π/3, see discussion of Eq. (10). First level sum of Teleois proportions is 85, foundational in Great Pyramid design [7] and 85 ' 360/φ 3, see Eq. (11) discussion. Also, 5 85/11 ' R/A and 528/85 ' 2π, see Eq. (10) discussion below. The latest value for the inverse fine-structure constant by Aoyama et al, α−1 ' 137.035999157(41), from experimental results and quantum electrodynamics [57]. The inferred value [58] determined from quantum electrodynamic theory and experiment [59] with the least standard uncertainty in CODATA results is α−1 ' 137.035999160(33) [58]. sinα−1 ' 504/85K = 7!/(713+137)K. (10) Again, this equation gives α−1 ' 137.035999168 [60]. √ 13/7 ' 2sinα−1 and from the harmonic radii of the Cosmological Circle, 108+ 396 = 504 [7]. 504/396 = 14/11 ' 4/π. Additionally, sinα−1 ' ζ (2)P ' 1−π−1 ' e/π √ φ ' 2π/ √ 85. 504/144 = 7/2 and φ sinα−1 ' 3/e ' √ φ/R ' Ae2. From Eq. (7), sin2 α−1 ' 157/337. Mamombe demonstrates the link between 137 and 117 in his analysis of the Fibonacci sequence and describes "its significance to the theory of the golden section and its relationship with the fine-structure constant," [61], also see the discussion of Eq. (17) and [62]. Also, 144/85 ' e/φ ' 1/4A. 528/504 ' 7A, and the Great Pyramid Key is 528 ' ln(7/A)/α [33]. Pyramid base angle θB ' tan−1(4/π) ' 51.85◦. The ln(4/π)' Aφ and π/4' cscα−1− sinα−1 ' sinθB. 792/5280 = 54/360 = 3/20' A and φ = 2sin54◦ [7]. Helmut Warm found a significant ratio between the Venus/Earth synodic period and the Mars orbital period to be 17/20 = 1−3/20' 1/ ln(2φ)' √ 2π/K. This had the strongest cycle resonance found among the planets [63]. Also, 17/20= 0.85, the fraction contained in the base angle of the Great Pyramid. The pyramid apex angle θA, sinα−1 ' θB/θA ' √ 2tan(π/7) ' √ S/7 ' 2π/ √ 85 [33]. Apex angle of the regular heptagon triangle is 3π/7 and an approximation to the apex angle of the Great Pyramid. 528/396 = 4/3 and 396/85' √ S/A. While twentieth-century physicists were not able to identify any convincing mathematical constants underlying the fine structure, partly because such thinking has normally not been encouraged, a revolutionary suggestion was recently made by the Czech physicist Raji Heyrovska, who deduced that the fine structure constant ... really is defined by the [golden] ratio .... –John Calleman [64]. Raji Heyrovska says, "On noticing the closeness of the fine structure constant ... to the ratio of the angles, 360◦/φ 2 ... the author suggested that the small difference ... could be due to the Sommerfeld's relativity correction factor." [65]. "It was also pointed out that the ratio 360◦/φ 2 ... which is a Golden section of 360◦, differs from the inverse fine structure constant by ... 2/φ 3 ... probably due to the difference in the g-factors for the electron and proton ... arising from the magnetic momenta of the two particles," with the result of α−1 ' (360/φ 2)− (2/φ 3) [66]. α −1 ' 360 φ 2 − 2 φ 3 + A2 Kφ 4 − A 3 K2φ 5 + A4 K3φ 7 . (11) This extension of Heyrovska's equation also gives α−1 ' 137.035999168 [67, 68]. Raji Heyrovska found it a "surprise to find for the first time that the Bohr radius is divided into two unique sections at the point of electrical neutrality, which is the Golden point. The Golden ratio, which manifests itself in many spontaneous creations of Nature, was thus found to originate right in the core of atoms." [66]. In degrees, the modern golden angle [69], θG = 360◦/φ 2 and related 6 θg' 26.57◦' tan−1(1/2) (twice the Cabibbo angle) is the ancient Golden Angle of Resurrection noted by Robert Temple [7, 70]. Along with the Golden Angle of Resurrection, Robert Temple states the Pythagorean Comma, Pc was one of the greatest secrets of the ancient Egyptians [59]. Highly complex numbers like the Comma of Pythagoras, Pi and Phi (sometimes called the Golden Proportion) ... lie deep in the structure of the physical universe, and were seen by the Egyptians as the principles controlling creation, the principles by which matter is precipitated from the cosmic mind. –Jonathan Black [71]. Aubrey Meyer finds that the "Pythagorean Comma and the Fibonacci series converge on the Golden Section." [72]. For the same approximate value of α−1 ' 137.035999168, the three terms on the right of Eq. (11) can be reduced to √ 2(SA)2/ √ 3K2φ 4 ≈ 4A3/φ 6 and the last two terms can be reduced to one term with the Pythagorean Comma,−A3/ √ PcK2φ 5. Pc' √ 2φ/π ' 1+ √ α/2π ' 2ζ (2)/S and Pc = (3/2)12/27 ' 1.014 [7]. The fourth approximation of the inverse fine-structure constant with this particular value: α −1 ' 1/ tanh−1 τ−1 ' 137.035999168, (12) where τ is the root of x4−137x3−6x2+98x+73 = 0, "which represents a particular quartic plane curve, different combinations of the coefficients of the general curve give rise to the lemniscate of Bernoulli." [32]. Number 7 also appears as 2× 72 = 98 and 73 = 343 = 6+ 337. The root τ ' 137.0384316101 [73]. The other root is approximately 1.523' πSA' π−φ . In another representation of Eq. (12), cothα ' τ . Other relations include τ/coshφ ' 7π/P' 117/ √ 5 (see the discussion of Eq. (18) below) and τ/cosh2 φ ' 7Gw + 7 ' 7L2r − 7, also from the discussion of Eq. (18). The coshφ ' 1/2A √ φ ' Pc/ √ A ' KA(Pc + 1) ≈ φ 2. "Gauss's and Euler's study of the arc length of Bernoulli's lemniscate, a polar curve having the general form of a toric section, led to later work on elliptic functions." [32]. Also, "the lemniscate, inverse curve of the hyperbola with respect to its center, has the lemniscate constant L which functions like π does for the circle." [32, 74]. The lemniscate constant L ' coshφ ≈ φ 2 and sinα−1 '√ 6L/3π, where L/π is Gauss's constant and √ 6/3 is the diameter of the inscribing sphere of the octahedron [32]. 3. Proton-electron mass ratio approximations With the A and K constants shown above, approximate values for the proton-electron mass ratio including the golden ratio and fine-structure constant: mp/me ' Aφ 2/4α2 ' 4K/αφ 2. (13) Other approximations of proton-electron mass ratio with the golden ratio and Golden Apex [7]: mp/me ' 4(φ + √ 3)/α ' 2/Aα. (14) Also, φ sin70◦ ' 2πφA' ln(mp/me)/ ln(α−1) [3, 7] and ln(αmp/me)' φ 2 cosA. Malcolm H. Mac Gregor has stated, "The bridge between the electron and the other elementary particles is 7 provided by the fine structure constant .... An expanded form of the constant leads to equations that define the transformation of electromagnetic energy into electron mass/energy ...." [75]. The expζ (2)' 4π cot2(1) and replacing expζ (2) in mp/me' expζ (2)/α √ A with 73/(102−191P) gives a more accurate value, 1836.152673485 [76]. This is approximate to the value, mp/me ' 1836.152673346(81), that has been derived from the most recent experimental measure of the proton mass by Heisse, F. et al. [77]. Additionally, 73+191 = 102+162 = 528/2 = 264, related to Great Pyramid geometry in several ways [7, 33]. The half-wavelength associated with a frequency resonant to quartz crystal is 191 meters [33] and the height of the Great Pyramid without the capstone is about 137 meters: 191/137 ' 7/5 ' P/2A ' K/2π ' √ 5/φ ' 528/382. From the discussion of Eq. (7) 528− 337 = 191, 191/73 ' φ 2 and 102/85 = 6/5 ' π/φ 2. The length of the horizontal passage in the Great Pyramid is 102 feet [78]. 117/102 ' R and 117/73 ' φ ' 191/117. The electron/pion mass ratio, me/mπ ' α/2 and mπ/mp ' A, where mπ is the pion mass [32]. The omega constant [79] is defined as Lambert W (1) [80]-[82], the attractive fixed point of e−x. W (1) = Ω = exp(−Ω)' sinh−1(4A)' 0.5671 and 4A' φ/e. Ω = exp(−Ω)' A ln(mp/me)/2. (15) W (x), the Lambert W -function, is an analog of the golden ratio for exponentials and is expressed as exp[−W (1)] =W (1). Also, Ω ' 2A ln(A−1)' √ α/A' 1/ √ π and ln(α−1)'KΩ . From the zeta function, ζ (2)/Ω ' 1+ ln(A−1). In the Foundation Stone of classical harmonics the alpha harmonic is equal to the sum of the golden ratio harmonic and the omega constant harmonic, relating the ancient Greek Pythagorean form of metaphor for the fine-structure constant to golden ratio geometry and architecture [32]. 4. Lucas numbers and golden ratio geometry The reciprocal Lucas constant is Lr ' 1.9628 [81, 82] with a derivation from the golden ratio: Lr = ∞ ∑ n=1 1 Ln = ∞ ∑ n=1 1 (−φ)−n +φ n . (16) Approximately, Lr ' 1/ √ A √ 3 ' 2 √ K/3 ' √ φ cosh(1) ' √ e/φ/λ , where λ is the Laplace limit [85] of Kepler's equation described previously with the fine-structure constant [32]. Also, cothλ−1 ' φ sinα−1, see the discussion of Eq. (10) and Eq. (18). Lr ' P/ √ 2A' S/4P' √ 2/φ/Ω . (17) The cosh−1 Lr ' √ e/φ ' KA and ln(mp/me) ' ΩLr/A. The reciprocal Lucas constant Lr ' − tan117◦ ' 3φ 2/4, the square of the radius of the circumscribing sphere of the dodecahedron. The golden ratio φ ' A √ 117 and sinh−1(117) ' φ/2A. Also, 137 ' 117sinh(1) and 117 ' 2K/A ' 10/ √ α. The approximate dihedral angle of the dodecahedron 117◦ ' 180◦− tan−1(2) ' Lr tan−1(φ−1)/T, see Eq. (18) for Tau T . Also, the Pythagorean relation 1172 + 1372 ' 1082 + 1442 = 1802 and 144/117 ' 2/φ [22]. With the Great Pyramid Key 528 again, 528/137 ' L2r and 137/21 ' L2r/4A; related to the hydrogen emission resonance of the King's Chamber [1]. 8 Regarding the golden ratio again, Boeyens and Thackeray [86] are quoted by Mamombe [61], "We suggest that there is a strong case that the so-called, 'Golden Ratio' (1.61803 ...) can be related not only to aspects of mathematics but also to physics, chemistry, biology and the topology of space-time." [86]. Xu and Zhong state, "... we would like to draw attention to a general theory dealing with the noncommutativity and the fine structure of spacetime which comes to similar conclusions and sweeping generalizations about the important role which the golden mean must play in quantum and high energy physics." [87]. In the Fibonacci inspired sequences studied by Mamombe the number 117 marks a transition point, which is also a harmonic of the square root of the inverse fine-structure constant [48] and he describes a "geometrical basis for the fine-structure constant in the golden section." [61]. The Egyptian Royal Cubit is a traditionally known measure basis for the Royal Cubit and the numerical harmonic of the meter is also found in the Great Pyramid geometry, π/6' φ 2/5' π− φ 2 ' 0.5236 meters, which is approximately equal to a Royal Cubit [33]. Lr ' (T + √ 7)/φ ' 2T φ lnπ, coshT ' lnπ and Tau T is described as: T = α +π/6' 2A/Ω ' ln(Lr)/ √ φ . (18) T ' √ 7/5 ' 2π √ α ' 0.5309. From the Great Pyramid, with a height of 280 Royal Cubits, 280/117' √ 7π/Lr ' 2.4 is the golden angle in radians or 360◦/φ 2 ' 137.5◦. 117◦ ' θG(1−A) with θG in degrees. The modern golden angle, θG ' √ φ/T ' ln(A−1)+1/2 ' 2.4 radians, see the discussion of Eq. (11). Also, T ' 3 √ A ≈ 2A √ π ' Lr/2Gw ' Lr(1−A)/π ' φ 2/ ln(α−1)' Lr √ φ/7π ' cos58◦. The base of the face angle on the Great Pyramid is approximately 58◦. Manu Seyfzadeh explains that "The pyramid's core design is based on a Kepler Triangle whose hypotenuse (356 cubits) is in the golden ratio (1.618. . . ) to its short leg (220 cubits) incidentally producing a side-slope angle of 51◦50′." [88]. T/φ ' 117/356. The ratio of angles, 117◦/51.85◦ ' √ φπ; φπ is another frequently encountered measure in the Great Pyramid. 356/70' φπ, from another Kepler Triangle. Seyfzadeh says, "Two classic interrelated geometric relationships can be approximated from an observable measure of Earth's curvature and a sidereal lunar year." [88]. With another significant number from the Great Pyramid measures 432, 432/117' 6/φ ' Lr/T. Also, R/T ' 2S/3' 2ζ (4) and RT ' 1/ζ (2). 70/117' 4A and KA' T √ 6. The Royal Cubit again, π/6 = ζ (2)/π ' T/Pc, where Pc is the Pythagorean Comma from the discussion of Eq. (11). Also, 6/π ' PKT ' √ 1+ √ 7. From tetrahedral geomery, cos19.5◦ ' sin(6/π), in radians. From Great Pyramid geometry, sin−1 φ−2 ' 69.4◦+ 90◦− 137◦ and the tan69.4◦ ' φζ (2)' √ 2/T ' tan(φ 2θg), see the discussion of Eq. (11). The ln(Lr/T )' φ 2/2. T−1 ' 4πA ' ζ (2) lnπ ' √ φ +φ−1 ' L2r −Lr, √ T ' 1/Lr(1− 2A) ' 2π/K and 3 √ T ' φ/2. T/Ω ' 2πA and ΩT ' 2A' KA−1. The Royal Cubit in inches, 144/7 ' Rc [89]. Alpha, α ' A/Rc, related to the symbolism in historical references concerning the fine-structure constant [90]. Multiplying, α×Ω × T ' 1/7πRc. Adding, α +Ω + T ' √ φ/R, see the discussion of Eq. (10). Also, A + Ω + T ' 2 √ 140/360 = √ 14/3 ' φ tan(π −φ) ' φ √ φ/e ' √ 2/2Ω ' Rc/2λH , with Rc and λH (21cm hydrogen emission wavelength) in meters and T ' PcRc. From the hieroglyph for gold, 14/9' tan(1) and A+Ω ' 9/4π ' √ φ/π [7, 22]: "The vertex angle of the nonagon, 140◦, when considered as the central angle of a triangle in a circle forms the Egyptian hieroglyph for neb or 9 gold related to quintessence, or the 'unified field' of physics." [22]. These suggestive references to the quintessential aether are interpretations of Herschel's AlphaOmega-Taurus Star Gate [91], the Ark of Osiris from Coppens [92] and Hardy's Pyramid Energy [93]. Inverse Golden Apex, A−1 as a harmonic of the Earth frequency utilized by Tesla of 6.67 Hz [94], divided by Tesla's magnifying transmitter frequency of 11.78 Hz: 6.67/11.78 ' Ω . With Parr's 51.5 Hz pyramid orb frequency, the ratio is 51.5/11.78' 2π−(6/π)' 6 √ T 'K/2 [95, 96]. 5. Conclusion Euler's equation and the exponential function applied to the geometry of the Golden Apex of the Great Pyramid were part of four different calculations of the inverse fine-structure constant with the same approximate value. The fine-structure constant and the Golden Apex were related to the proton-electron mass ratio, golden ratio and other fundamental constants; finally speculating briefly on the metaphorical physics and mathematical metaphysics of the Great Pyramid [97, 98]. These results illustrate clearly the highly interrelated nature of the fundamental constants. Dependent as our knowledge of them is upon many different fields of physics, we have here a good example of the importance of making occasional analyses of the consistency situations of sufficiently inclusive scope to serve as valuable guides to further research. The present example emphasizes especially that a better knowledge of the Sommerfeld constant, α , would be of great value to physics at the present time. –Jesse DuMond [99]. David Haight quotes "Physicist John Wheeler, who coined the term 'geometrodynamics' put it this way, 'Physics is really geometry .... Some profound connection exists between the fundamental constants of microphysics and the geometry of the cosmos.'" [10]. This unification of mathematics through Phi should not come as a complete surprise to us since Phi is related to all three means that are essential to mathematics- -the arithmetic, the geometric and the harmonic. (These three means are the result of the calculus of differences, just as the harmonic intervals in music are the result of the calculus of variations.) –David Haight [10]. Eckhart Schmitz emphasizes, "The Great Pyramid of Giza may clearly be regarded as a repository of ancient knowledge." [12]. In conclusion, Witteveen is quoted again on Pythagoras, "All is number, everything is movement, everything is music in the harmony of the spheres." [1]. This 'harmony of the spheres' is exemplified by the Golden Apex [7] in a relation which also 'squares the circle' [1], √ A/2 ' 3/11. As Robin Heath explains, "This ratio of 3:11 is exactly the ratio between the Moon's diameter ... and the diameter of the mean Earth (the first major treatment of this was in City of Revelation by John Michell [100], who was responsible for its rediscovery)." [3]. With reference to the revelation of this harmony Mary Anne Atwood states, "And here the external and internal worlds were seen to blend together in confluent harmony, proving and establishing each other, and leaving reason nothing more to doubt of, or the senses to desire, but a fulfillment under the universal law. [101]. 10 Acknowledgments Special thanks to Case Western Reserve University, PhilPapers, MathWorld and WolframAlpha. 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