Quintessential Nature of the Fine-Structure Constant Michael A. Sherbon Case Western Reserve University Alumnus E-mail: michael.sherbon@case.edu June 27, 2015 Abstract An introduction is given to the geometry and harmonics of the Golden Apex in the Great Pyramid, with the metaphysical and mathematical determination of the finestructure constant of electromagnetic interactions. Newton's gravitational constant is also presented in harmonic form and other fundamental physical constants are then found related to the quintessential geometry of the Golden Apex in the Great Pyramid. 1. Introduction The geometry of the Great Pyramid of Giza, as stated by Eckhart Schmitz in The Great Pyramid of Giza, "... demonstrates extraordinary precision in relaying highly accurate geodetic knowledge ..." and "it is evident, with a very high degree of probability, that the design parameters were expressly intent on conveying this advanced knowledge." [1]. In The Essence of the Cabalah, William Eisen describes the fundamental geometry of what was known as the Golden Apex of the Great Pyramid and the ancient pre-pharaonic science of the Agashan Masters [2]. 2. Golden Geometry of the Great Pyramid The Golden Apex of the Great Pyramid is the side of a square in the upper part of the capstone resulting from four exponential curves at the base of the Great Pyramid [2]. The Golden Apex of the Great Pyramid is A: A = eπ − 7π − 1 ' 0.1495 (1) The inverse of the Golden Apex A−1 is a harmonic of Newton's gravitational constant G. A−1 ' φ √ 2πe ' φ/ ln(4/π) and 4/π ' √ 11A. A + 1 ' R ' 1.152, approximate 1 radius of the regular heptagon with the side equal to one. A ' π/ √ 440 ' √ π/140, base of the Great Pyramid is approximately 440 cubits, and the height is 2× 140 = 280 cubits. 140/360 ' e/7 ' √ A and 4× 440 ' 2π × 280. The height of the Golden Apex, 4A/2π ' S−2, the silver constant of the heptagon, see Eq. (4). RA ' √ φ/e2 ' ln(π/ √ 7) (2) The golden ratio approximates the squaring of the circle and is also closely related to the heptagon geometry. 2R = csc(π/7) and R ' φ/ √ 2 ' 13/7φ ' lnπ ' 2/ √ 3 ' 7/ √ 37. In radians tanS−1 ' π−1. In the Timaeus, Plato "considered the golden section to be the most binding of all mathematical relationships and the key to the physics of the cosmos," quoted by Robert Schoch and Robert McNally in Pyramid Quest [3]. Quintessence is associated with the dynamic aether and embraces the four known forces of nature. Q = C/ √ 2 ' √ e/φ ' 1 + αφ2, (3) where α ' 1/137 is the fine-structure constant and Q ' 377/370 ' √ 11/S ' 1.019. C2 is the approximate inner diameter of the regular heptagon with side equal to one. C2 ' cot(π/7) ' 1/ lnφ. CQ ' √ φR ' cscα−1. The golden ratio φ ' A + CQ and φ2 ' A+ C +Q. In elementary form, 22 = 1 + (3× 7) ' 7π. Also, C = √ 2Q ' 1.441. Aristotle coined the term quinta essentia for Plato's fifth element, often represented by the dodecahedron. The quintessence is described in Malcolm Macleod's geometry of angular momentum model as the harmonic Q and also has units from the square root of Planck angular momentum [4]. 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 [5]. R is a harmonic of half the Great Pyramid base length in meters, and CQ is a harmonic of the height in meters. The lnA−1 ' secQ ' 6/π ' π/ √ e, the cube-sphere ratio. With golden ratio overlay software, a golden spiral centered on the Eye of Ra in Robert Temple's plan of the Giza Plateau, is shown to pass through both the Golden Apex of the Great Pyramid and the Great Sphinx of Giza. The ratio of the length of the sides of Temple's Perfect Square of Giza/Shadow Square of Giza is approximately equal to Q [6]. Q ' √ 7A ' ln √ R/A ' A+R−1 and ln(S/Q) ' π/e. In radians, tanQ ' φ. A ' √ 11/7π ' √ e/11 ' 2παS (4) The silver constant S from the regular heptagon, S = 4 cos2(π/7) ' 2 √ 2R ' 2 tanQ ' tan √ φ ' 3.247. CQ ' √ 7/S. 1 + 3 + 7 = 11. 2πα is the ratio between the Compton wavelength of the electron in hydrogen and the Bohr radius. From the hyperdimensional aether, elementary charge and spin angular momentum (determining the fine-structure constant) is manifested through the proportions of the golden ratio. A ' √ S/12, AS2 ' 11/7 ' π/2 and α−1 ' A−2 tan 72◦, with the regular pentagon angle. S ' √ 7π/C and C2 ' R √ S ' cot(π/7). Jean-Paul and Robert Bauval describe in the Secret Chamber Revisited how prime numbers 7 and 11 are significant keys to the Great Pyramid [7]. The number seven was 2 "especially dedicated to Sirius," [1, 8, 9]. Half the face apex angle of the Great Pyramid is approximately 32◦ [10]. 11 × 11 = 121 and 121/137 ' √ 2 tan 32◦. Ga ' 1/ √ RA ' 2 √ C ' ln 11 ' 2.4, where Ga is the golden angle in radians. From the pyramid face apex angle, tan 64◦ ' √ 11/φ ' 137/67. The special number 528 translates as the Key to the Pyramid of Light [11, 12]. The harmonic 5.28 ' 37/7 ' 2 √ 7 ' eπ/φ. Also, 2π ' ln(528) and 5.28 ' 2π/ √ C ' π/4A. Found in several places, 2.64′ is the width and height of the Ark of the Covenant, the proportioned Golden Apex height and section lengths above the 5.28′ rise in the Grand Gallery. 2.64 ' Q/ √ A ' √ 7. 3. Mathematics of the Fine-Structure Constant The fine-structure constant was introduced by Arnold Sommerfeld with the addition of elliptic orbits to Bohr's atomic model [13]. The fine-structure constant determines the strength of the electromagnetic interaction and being related to quintessence it is involved in all aetheric phenomena. The usual definition for alpha, the fine-structure constant is α = e2/~c in cgs units. Mark Rohrbaugh's review of Nassim Haramein's work motivated the question of recombining the formula for the classical electron radius re = e2/mec2 with the formulation by Haramein for the proton charge radius, from the modified Eq. (31) rp = 4~/mpc [14]. α = 4/(mp/me)(rp/re) (5) When substituting the reference value for alpha and using the latest reference values for the proton and electron mass with the classical radius of the electron, gives a value for the proton radius rp ' 0.8412 fm. The proton/electron radius ratio: rp/re ' tanhS−1 ' 2A, with the Golden Apex of the Great Pyramid. The inverse ratio: re/rp ' 5π/ √ 7π. The inradius of the regular pentagon, √ 25+10 √ 5/10, is approximately equal to −eπ+7π+ln 2π. Also, the heptagon diameter divided by the pentagon inradius is approximately equal to re/rp. The approximate value for the inverse fine-structure constant from Eqs. (9− 11): α−1 ' 137.035 999 168 (6) Latest value by Aoyama et al, α−1 ' 137.035 999 157 (41), determined from quantum electrodynamic theory and experiment [15]. M. Temple Richmond says that in the esoteric tradition 137 is a representation of the Law of One, the Three Cosmic Laws and the Seven Rays [8]. Given Gw is the Wilbraham-Gibbs constant and the sinc function sincx = sinx/x, then: Gw =  π 0 sincx dx ' φ lnπ ' e sinα−1 (7) The Wilbraham-Gibbs constant Gw ' sec(1) ' φ2/ √ 2 ' 1.852. The Wilbraham-Gibbs constant is related to the overshoot of Fourier sums in Gibbs phenomena [16]. lnGw ' 3 1/φ and 137 ' (37 + 37)Gw. Ga/Gw ' KA ' 3/2R ' √ e/φ, where K is the polygon circumscribing constant. Again with K, the Wilbraham-Gibbs constant Gw ' π/(K−7). The polygon circumscribing constant K [13]: K = π 2 ∞∏ n=1 sinc ( 2π 2n+ 1 ) = ∞∏ n=3 sec (π n ) (8) The polygon circumscribing constant K ' 2 + 3 √ 5 ' √ 11φ2 ' 8.7. KA ' R + A '√ e/φ ' φ2/2. Also, 1 + φ−2 ' K/2π ' √ 6/π ' sinh2(1). KA + RA ' CQ and K ' 10/R. A curious relationship between the Key harmonic 528 and the polygon circumscribing constant K, 528/140 ' K/D, where D = 2R is the heptagon diameter. The internal angle of the nonagon is 140◦ and is also the central angle in the hieroglyph for gold [10]. D ' 85/37 ' √ φ2 + φ2. 85/11 ' R/A and 528/85 ' 2π. 440/85 ' π √ e, 137/85 ' φ and 12× 44 = 528. 528/440 ' π/φ2, see [13]. 504/280 ' √ S and 504/440 ' lnπ. 528/504 ' 7A. With the angular harmonic, sinα−1 ' 504/85K = 7!/(713+137)K. sinα−1 ' 7!/(713 + 137)K, (9) with the same approximate value for the inverse fine-structure constant as Eq. (6). Plato's favorite symbolic number 5040 = 7!. 504/396 ' 108/85 ' √ φ. 7920 represents the canonical harmonic of the Earth's diameter. 7920/5040 = 11/7 and 25920/7920 = (14+ 22)/11. 25920 represents the precession of the equinoxes. 2592/1370 ' 6/π. 792/528 = 3/2, 792/396 = 2 and 792/264 = 3. The harmonic of Newton's gravitational constant, A−1 ' ln 792. 14 × 20 = 280 and 22 × 20 = 440. DK ' 20. 504 = 7 × 8 × 9 and √ 14 ' SR. Also, sinα−1 ' 2 sin(π/9). CQ is the approximate circumradius of the nonagon. Other relationships include cscα−1−sinα−1 ' π/4 and tan−1(4/π) ' 51.85◦ ' 2×25.92◦, the Great Pyramid base angle. 4/π ' √ S/2 ' √ φ and RA ' 3/2K. 2.592 ' 1/ √ A ' √ 7/Q. D √ A ' 8/9, the proportion for "squaring the circle." D = 2R ' csc(π/7) and R ' cot2 α−1. In degrees, the modern golden angle, Ga = 360◦/φ2 and the related 26.57◦ ' tan−1(1/2) is the ancient Golden Angle of Resurrection according to Robert Temple [6]. Along with the Golden Angle of Resurrection, Robert Temple states the Pythagorean Comma Pc was one of the greatest secrets of the ancient Egyptians [6], Pc ' √ 2φ/π ' √ 7/φ2 ' 370/365. Q5 ' P 7c and Pc = (3/2)12/27 ' 1.0136. Also, 27/12 ' 3/2 ' ARK. Another form of calculation involves the prime constant [13], described as a binary expansion corresponding to an indicator function for the set of prime numbers. The inverse fine-structure constant: α−1 ' 157− 337P/7, (10) with the same approximate value for the inverse fine-structure constant as determined in Eq. (6), having three prime numbers and the prime constant. The prime constant P ' √ RA ' φ2/2π ' 220/528 ' 0.4147. PQ ' 2 √ 2A and PK ' 2 √ S. KA ' Pπ ' e lnφ. 852 + 1322 = 1572. Grand Gallery length in the pyramid is 157′. 137 − 90 = 47 and 47 + 85 = 132. 132/90 = 528/360 ' CQ. 157/288 ' 288/528. 180/157 ' R and 4 337/180 ' φR. With the Key, 528/337 ' π/2 and 157/528 ' 2A, proton/electron radius ratio. With pyramid base, 440/337 ' φ2/2 and 440/157 ' √ 3φ. The sin 140◦ = sin 40◦ '√ P . 528/140 ' K/D ' 7/Gw and 140/360 ' e/7. The sec 40◦ ' P √ 3S ' KA. α−φ ' 2867.2867 + 28672−η, (11) which also gives the same approximate value for the inverse fine-structure constant as Eq. (6), where η ' 365/365.24 and ηφ ' − sin(14/3) ' 286.7/287 ' ( √ 2− 1)/P, inverse of the proposed "pyramid inch" by Taylor and Smyth [17]. The Solar Year harmonic is 365.24. The proton/neutron mass ratio is approximately mp/mn ' η2. From the capstone height, 8× 286.7 ' 2π × 365 [18]. 2.867/2.592 ' √ 11/3 and 25920/2867 ' 2πC. The ascending and descending passages of the Great Pyramid are displaced 287′′ to the east [3]. 365/287 ' √ φ. 365/286.7 ' 4/π ' √ S/2 and 528/287 ' S/ √ π ' √ 11/S. (3× 37)/287 ' √ A. P φc ' Q and PcQ ' (64/63)2 ' sec(1/4). The inverse of the Eye of Horus fractions is 64/63 [6]. With the polygon circumscribing constant, √ 7π −K ' √ 2 + √ 5 ' 3.65, again from the harmonic 365 of the capstone height [18]. 504/365 ' √ 5/φ and (286.7×8)/2π ' 365. 2.867 ' 3/7A ' 9/π ' φ √ π. Another expression with alpha, α−φ ' 3π223/35, having prime factors of 3, 5, 7 and 11 with a value approximately the same as Eq. (6). Also, α−1 ' (π/A)φ and again, 2.867 ' 2 √ 2Pc ' Gw √ Ga ' (4 + √ 3)/2. The Golden Apex A ' (1 + √ 5)−φ ' 432/2867. The Wilbraham-Gibbs constant Gw ' 528/286.7 and 2867/528 ' S/4A ' 2e. The harmonic of the Solar Year again, 365.24 ' 2πK/A, 365.24/140 ' φ2 and 365.24/37 ' π2. 4. Conclusion The height of the gold pyramidion thought to be on the original Great Pyramid, as described by John Michell (with the metrology of Algernon Berriman) [19], is 0.152 ' AQ ' 11.7/(7 × 11), √ 137 ' 11.7 and the tenth part of the Greek cubit of 1.52 ' π − φ. The modern golden angle in radians, Ga ' 365/152. This small pyramidion was supposed to be the final top and possibly made of something like transmuted gold, similar to the legendary Golden Sun Disc of Mu. Reports about stargates, resurrection and the replication of golden pyramids in the hyperdimensional aether suggest the ancient Egyptian initiates were well-informed by Thoth, "Architect of the Great Pyramid." They knew about the mathematical constants, fundamental physical constants and advanced mathematical functions inherent in the analysis and applications of the golden ratio vortices of the quintessential dynamic aether [20]-[23]. According to Manly P. Hall, the capstone of the Great Pyramid was associated with the Eye of Horus. "The exact science of human regeneration is ... when the Spirit Fire is lifted up through the ... spinal column ... passes into the pituitary body (Isis), where it invokes Ra (the pineal gland) and ... the Eye of Horus is opened." [24]. Acknowledgments Special thanks to Case Western Reserve University, MathWorld and WolframAlpha. 5 References [1] Schmitz, E.R. The Great Pyramid of Giza: Decoding the Measure of a Monument, Ontario: Roland Publishing, 2012, 177. [2] Eisen, W. The Essence of the Cabalah, Camarillo, CA: DeVorss, 1984. [3] Schoch, R.M. & McNally, R.A. Pyramid Quest: Secrets of the Great Pyramid and the Dawn of Civilization, New York: Jeremy P. Tarcher/Penguin, 2005, 168. [4] Macleod, M. Plato's Code: Fifth Element – A Model of the Universe in Terms of the Geometry of Momentum, Minsk: Plato's Code, 2007. [5] Sherbon, M.A. "Nature's Information and Harmonic Proportion," SSRN Philosophy of Science eJournal, 5, 3 (2011) SSRN:1766049. [6] Temple, R.G. Egyptian Dawn: Exposing the Real Truth Behind Ancient Egypt, London: Century, 2010. [7] Bauval, R. Secret Chamber Revisited: The Quest for the Lost Knowledge of Ancient Egypt, Rochester, VT: Bear & Company, 2014. [8] Richmond, M.T. Sirius, Mariposa, CA: Source Publications, 1997. [9] Hope, M. Ancient Egypt: The Sirius Connection, Shaftesbury: Element, 1990. [10] Sherbon, M.A. "Wolfgang Pauli and the Fine-Structure Constant," Journal of Science, 2, 3, 148-154 (2012) SSRN:2147980. [11] Gaunt, B. The Magnificent Numbers of the Great Pyramid and Stonehenge, Jackson, MI: Gaunt, 1985. [12] Horowitz, L.G. The Book of 528, Las Vegas, NV: Tetrahedron Publishing, 2011. [13] Sherbon, M.A. "Fundamental Nature of the Fine-Structure Constant," International Journal of Physical Research, 3, 2(1):1-9 (2014) SSRN:2380218. [14] Haramein, N. "Quantum Gravity and the Holographic Mass," Physical Review & Research International, 3, 4, 270-292 (2012). [15] Aoyama, T., Hayakawa, M., Kinoshita, T. & Nio, M. "Tenth-Order Electron Anomalous Magnetic Moment - Contribution of Diagrams without Closed Lepton Loops," Physical Review D, 91, 3, 033006 (2015) arXiv:1412.8284v3. [16] Zi-Xiang, Z. "An Observation of Relationship Between the Fine Structure Constant and the Gibbs Phenomenon in Fourier Analysis," Chinese Physics Letters, 21.2, 237-238 (2004) arXiv:0212026. [17] Smyth, C.P. The Great Pyramid: Its Secrets and Mysteries Revealed, New York: Bell Publishing, 1978. 6 [18] Pecci, E.F. The Sacred Geometry of the Great Pyramid, Walnut Creek, CA: Pavior Publishing, 2010. [19] Michell, J. The New View Over Atlantis, New York: Thames and Hudson, 1995. [20] Coppens, P. The Canopus Revelation: The Stargate of the Gods and the Ark of Osiris, Enkhuizen, Netherlands: Frontier/Adventures Unlimited Press, 2004. [21] Henry, W. Starwalkers and the Dimension of the Blessed, Kempton, IL: Adventures Unlimited Press, 2007. [22] Hills, C.B. Rays from the Capstone, Boulder Creek, CA: University of the Trees, 1976. [23] Ellis, R. Thoth, Architect of the Universe, Cheshire: Edfu Books, 1997. [24] Hall, M.P. The Secret Teachings of All Ages, Los Angeles, CA: Philosophical Research Society, 1969.