Golden Ratio Geometry and the Fine-Structure Constant Michael A. Sherbon Case Western Reserve University Alumnus, United States E-mail: michael.sherbon@case.edu October 4, 2019 Abstract The golden ratio is found to be related to the fine-structure constant, which determines the strength of the electromagnetic interaction. The golden ratio and classical harmonic proportions with quartic equations give an approximate value for the inverse fine-structure constant the same as that discovered previously in the geometry of the hydrogen atom. With the former golden ratio results, relationships are also shown between the four fundamental forces of nature: electromagnetism, the weak force, the strong force and the force of gravitation. Keywords fine-structure constant, fundamental constants, history of physics, history of mathematics, golden ratio, quartic equation. 1. Introduction From John Rigden, "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. ... Since Sommerfeld expressed the energy states of the hydrogen atom in terms of the constant [alpha], it came to be called the fine-structure constant." [1]. Arnold Sommerfeld states, "In our theory of the fine structure there is a confluence of the three main currents of modern research in theoretical physics, namely, the theory of electrons, the theory of quanta, and the theory of relativity" and the fine-structure constant "... could be interpreted more physically as the ratio of an electron's velocity in the first Bohr orbit to the speed of light." [2]. From previous work, "The electromagnetic coupling constant determining the strength of its interaction is the fine-structure constant α = e2/hc in cgs units with the elementary charge e, the reduced Planck's constant h = h/2π and the speed of light c." [3]. The fine-structure constant has a variety of physical interpretations from which measurements and determinations have been made. Even though the fine-structure constant is often defined as 1 a low energy limit, in high energy physics it has become commonplace to regard it of less importance because the gauge coupling runs with energy. As Toichiro Kinoshita explains, "... the detailed high energy structure of the ultimate theory is irrelevant to the analysis of low energy phenomena except insofar as it determines these parameters." [4]. What actually determines the fine-structure constant remains questionable for high energy theorists. A reminder from Jesse DuMond's work that is still of significance today: 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 [5]. From the writings of Wolfgang Pauli, "The theoretical determination of the fine structure constant is certainly the most important of the unsolved problems of modern physics." [6, 7]. Richard Feynman said it was the greatest mystery of physics, perhaps the most often quoted physicist on the fine-structure constant: "It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it. ... You might say the 'hand of God' wrote that number, and 'we don't know how He pushed his pencil'" [8]. 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," as quoted by Robert Schoch and Robert McNally in Pyramid Quest [9]. Peter Champoux and William Buehler state that, "The capstone of the Pyramid generates the golden spirals which creates the whole form." [10]. The golden ratio φ = (1+ √ 5)/2 and is a root of its minimal polynomial x2− x− 1. Both the golden ratio and its conjugate are roots of the quartic equation x4− 2x3− x2 + 2x+ 1 = 0. For "... a particular quartic plane curve, different combinations of the coefficients of the general curve give rise to the lemniscate of Bernoulli. ... 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." [11]. The inflection points of the quartic are also related to the golden ratio [12]. 2. Calculation of the ne-structure constant An approximation from previous work [3], the inverse fine-structure constant α−1 is a root of: x4−137x3−10x2 +697x−365 = 0. (1) This equation gives a value of for x as α−1 ' 137.035999168. The latest determination from the Gabrielse group [13] is α−1 ' 137.035999150(33), from experimental measurement and quantum electrodynamics [14]. This calculation of the inverse fine-structure constant gives the same approximate value as ancient geometry combined with the extension of Raji Heyrovska's 2 work on the golden ratio structure of the hydrogen atom [15]. 697− 137 = 280+ 280. 365+ 365−10 = 280+440 = 720. Also stated previously, "The scaling factor for the second pyramid on the Giza Plateau is 137, having a height of 2× 137 = 274 Royal Cubits and a base of 3× 137 = 411 Royal Cubits." [3]. "The Great Pyramid has a combined height and base of 280+ 440 = 720 Royal Cubits, 280/220 = 14/11 ' √ φ and 220/136 ' φ ." [3]. Colonel R. S. Beard states that, "Sir William Petrie himself was thoroughly convinced that the Egyptians constructed the pyramid with a height-to-width-of-base ratio of seven to eleven." [16]. Ernest Pecci quotes Thoth: "The Pyramid is a representation of all created things. All matter consists of cosmic frequencies held in place by the angulations contained within specific harmonic triangles that are found within the Platonic solids. ... the principle of proportion determines and maintains all forms." [17]. Another approximation from our previous work [3] also gives the same approximate value for the inverse fine-structure constant. α−1 ' φx, where x is a root of: 3x4−250x3−346x2 +48x−36 = 0. (2) 280 = 36−3−3+250, 440 = 36−346+(3×250) and 346 = 48+48+250. Another quartic equation having the Eq. (1) value with α−1 ' π7/7π √ x, where x' 4/π √ φ ' 1 is a root of: 138x4 +83x3−70x2 +7x−161 = 0. (3) 137 = 7+70+83+138−161, 280 = 83+83+137+138−161 and 440 = 7+70+83+280. π7/7π is from the Hebrew inverted heart Tetractys or Tetragrammaton from 'Tengri 137' [18]. The π7 is the perimeter of the Great Pyramid base and 7π is found in the Golden Apex of the Great Pyramid, see Eq. (9) discussion. Jerry Gin states, It is also interesting to note that the name of God in Hebrew is YHWH (or Yahweh) which is translated as 'Tetragrammaton' from Greek. Tetra refers to four and may refer to the four faces of a tetrahedron and grammaton may refer to weight or gravity [19]. Ernest Pecci explains, "One importance of the Tetractys lies in its expansion to 3-dimensional and multi-dimensional levels. The Star of David, The Tree of Life, and the Tetragrammaton of the heart are all based upon the simple equilateral triangle divided into the ten dots of the Tetractys." [20]. Gordon Plummer says, "Perhaps of paramount importance is that of the Tetraktys. This was the symbol probably most revered by the Pythagoreans." [21, 22]. For Manly P. Hall: "The tetrad -4was esteemed by the Pythagoreans as the primogenial number, the root of all things, the fountain of nature and the most perfect number. ... Wherein the Pythagoreans expressed God as a tetrad is explained in a sacred discourse ascribed to Pythagoras, wherein God is called the Number of Numbers." [23]. Leon Marvell writing about Robert Fludd: "Interestingly, Fludd's conviction that the Tetragrammaton or deity was the infinite dimension within and between all things is an early precursor of late nineteenth-century conceptions (such as those of William Clifford) that saw the fourth dimension as precipitating phenomena such as the appearance of matter." [24]. 3 Another quartic polynomial also gives the same approximate value for α−1. The inverse finestructure constant α−1 ' 4Rsinh(x), with R as the Reality Number from the Great Pyramid [25] and x is a root of: 38x4 +313x3−273x2−44x+50 = 0. (4) This quartic has a root x ' φ/e, with the golden ratio and Euler's number. 4 = 44+ 273− 313, 280 = 4(38 + 38 + 44− 50), 160 = 4(313− 273), and 440 = 160 + 280. Also, α−1 ' Rcosh−1(2π)' R √ 2π, see the reference to Alfred Landé in our previous work [3]. The Reality Number, R, "... is an angle by which the least amount of energy is needed to accomplish the most amount of good when rotating from one dimension into the next." [25]. R = 54.277 degrees. This value is encoded into the structure in this way: since the Pyramid has the apothem angle of 51.853... degrees, and since the Ascending and Descending Passages needed a specified angle between them ... the value exactly midway between the apothem angle and the R number was chosen. Step one is from 51.853... degrees to 53.065 degrees. Step two is from 53.065 degrees to 54.277 degrees. ... and the three angles honor the ascension principle [25]. R ' 7AB, where A is the Golden Apex of the Great Pyramid and B = tan−1(4/π) is the base angle of the Great Pyramid [26]. θg ' 26.57◦ ' tan−1(1/2) is the ancient Golden Angle of Resurrection noted by Robert Temple [27]. θg' (R+B)/4 and 7= φ 4+φ−4' sinh √ 7. Another approximation for the inverse fine-structure constant α−1 ' exp(4x), where x≈ 2/φ is a root of: 7x4−28x3 +9x2−11x+36 = 0. (5) This equation also gives the same approximate value for the inverse fine-structure constant as Eq. (1). 280= 28(11+36−9−28). 440= (9+11)(11+11). The inverse fine-structure constant α−1 ' (4×4)x, where x' 5φ 4/4 is a root of: 2x4 +44x3−550x2 +233x−56 = 0. (6) This equation also gives the same approximate value for the inverse fine-structure constant as Eq. (1). 280 = 550−56−56+2 and 440 = 44(56−44−2) = 2(44−56+233)−2. The inverse fine-structure constant α−1 ' 372/x,where x' e' φ √ 2 √ 2 : 16x4 +25x3−37x2−97x−833 = 0. (7) 4× 4 = 16 is the main harmonic number of the Great Pyramid according to Karim [28] and Witteveen [29]. Karim states that, The proportion derived from that number (1:1.6) is the famous Golden Proportion used in Ancient architecture and now the basis of 'Sacred Geometry'. Sixteen brings out the sacred quality hidden within the One to manifest in our time-space dimension [28]. The ln372' 1+ ln137 and sinh−1(372)' A−1, inverse Golden Apex of the Great Pyramid [30]. The harmonic sum of 372 is part of a spectrum analysis related to crystallographic groups [30]. 4 37/7π ' √ 2 √ 2 ' 1+ sinα−1 and φ/ lnφ ' 2 √ 2 √ 2. 280 = 16(37− 25)+ 97− 25+ 16 and 440 = 16+97+97−25−25+280. 440+440 = 97−25−25+833. From Robert Lawlor describing harmonic division with the square root of two, "Here is a means of drawing a beautifully proportioned cup or grail-shaped vessel, using only the harmonic division to establish its curves and measures. We may speculate that this is the geometric essence of the Holy Grail." [31]. This grail geometry includes √ 2− 1 ' φ 2/2π and 2− √ 2 ' 4/φ 4 ' 4A. Also, in radians cot( √ 2− 1) ' 1+ √ φ , linked to Plato's beautiful scalene triangle found in the Cosmological Circle extension known as Plato's Wheel [20]. Ernest Pecci says: It is of significance that √ 2–1, 2– √ 2, 1 creates a 3-term harmonic proportion that places the baseline of the Pyramid at the harmonic mean between the floor of the Subterranean Chamber and the floor of the King's Chamber! [20]. From the Pythagorean perspective the Foundation Stone number of 729 is a harmonic of the fine-structure constant value. Another equation from previous work [3] that also gives the same approximate value for the inverse fine-structure constant as Eq. (1) is α−1 ' 729/x, where x is a root of: 10x4−37x3−113x2 +73x+371 = 0. (8) The root x ' φφ f , where φ f is the reciprocal Fibonacci constant. 280 = 10+ 10− 37− 73+ (10×37) and 440 = 73+371− ((113−73)/10). Czech physicist 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." [33]. "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 ...," with the result of α−1 ' (360/φ 2)− (2/φ 3) [34]. α −1 ' 360 φ 2 − 2 φ 3 + A2 Kφ 4 − A 3 K2φ 5 + A4 K3φ 7 . (9) This extension of Heyrovska's equation found previously also gives α−1' 137.035999168 [14]. 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 [34]. William Eisen describes the geometry of what he called the 'Golden Apex of the Great Pyramid' where dividing the sides of his mathematical model for the Great Pyramid by π lengths along with four curves of the exponential function 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 [30]. The Golden Apex A is the side length of the resulting square. A = eπ −7π−1' √ πα ' π/ √ 440' 3/20. 5 The inverse Kepler-Bouwkamp constant is the polygon circumscribing constant K ' φ 2/2A' 8.7 and sinα−1 ' 7!/(713+137)K ' (π−1)/π [11]. Also of interest, ln(φ/α)' φ/2A'K/φ and 11/37' cos √ φ ' 2A [3]. The polygon circumscribing constant "has interesting geometric connections with the torus topology relating the relativity of Einstein with the geometry of classical and quantum mechanics." [7]. The Reality Number R ' 7AB ' 2πK ' 360A, see Eq. (4) discussion. Approximations from previous work [30] involving the four fundamental forces: A' 4πφα ' αw/sin2 θw ' πφαs/4' K √ π/ lnα−1G . (10) With sin2 θw as the Weinberg angle or the weak mixing angle, αw is the weak nuclear force, αs is the strong nuclear force and αG is the gravitational coupling constant [35]. The inclusion of the four fundamental forces in the Golden Apex design is part of the symbolic interpretation presented by William Eisen [36]. The geometry of the Tetraktys has been associated with the four elements and the four fundamental forces. 3. Conclusion The inverse fine-structure constant calculated with quartic polynomials and the main harmonic parameters of the Great Pyramid have suggested the golden ratio as the mathematical basis for the fine-structure constant. From previous work "... several more formulations for the finestructure constant with the same approximate value have connections with prime number theory, the real fixed point of the hyperbolic cotangent, anomalous magnetic moment of the electron, Laplace limit of Kepler's equation and harmonic proportions of the Cosmological Circle" [3, 30, 37]. Acknowledgments Special thanks to Case Western Reserve University, MathWorld and WolframAlpha. References [1] Rigden, J.S. Hydrogen: The Essential Element, Cambridge, MA: Harvard University Press, 2003, 55. [2] Sommerfeld, A. Atomic Structure and Spectral Lines, New York, NY: Dutton, 1934. 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