Michael A Sherbon
email: michael.sherbon@case.edu

Wolfgang Pauli and the Fine-Structure Constant

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Introduction

Wolfgang Pauli (1900Pauli ( -1958) ) was an Austrian-Swiss theoretical physicist noted for his work on the discovery of the exclusion principle, the fourth quantum number in the theory of spin, prediction of the neutrino, and calculation of the hydrogen spectrum [START_REF] Enz | No Time to be Brief: A Scientific Biography of Wolfgang Pauli[END_REF]- [START_REF] Gustafson | His Early Physics in Jungian Perspective[END_REF]. While under the direction of Arnold Sommerfeld at the university, Pauli was influenced by Sommerfeld's search for a Platonic understanding of physics and connections implied by the mystery of the fine-structure constant. Pauli's interest in the philosophy of science was applied to the interpretation of quantum physics and magnified with the analysis of his dreams early in his career by Carl Jung [START_REF] Gustafson | His Early Physics in Jungian Perspective[END_REF]. This work then deepened his attraction toward the relationship between his dream images and the abstract concepts he encountered in physics. Richard Feynman proclaimed the fine-structure constant as one of the greatest mysteries of physics [START_REF] Miller | Deciphering the Cosmic Number: The Strange Friendship of Wolfgang Pauli and Carl Jung[END_REF]. Physicist Max Born declared, "... the explanation of this number must be the central problem of natural philosophy." [START_REF] Miller | Deciphering the Cosmic Number: The Strange Friendship of Wolfgang Pauli and Carl Jung[END_REF]. As the fine-structure constant determines the electromagnetic strength its theoretical origin was for Pauli "... the most important of the unsolved problems of ... physics." [START_REF] Enz | [END_REF][START_REF] Várlaki | Historical Origin of the Fine Structure Constant, Parts I-III[END_REF].

Wolfgang Pauli's World Clock

Both Pauli's waking vision of the World Clock and the Cosmological Circle embrace the archetypal images of space with the circle, square, triangle, and the cycles of time. In an allegorical study Plato described the geometric proportions of the Cosmological Circle [START_REF] Brumbaugh | Plato's Mathematical Imagination: The Mathematical Passages in the Dialogues and Their Interpretation[END_REF][START_REF] Sherbon | Nature's Information and Harmonic Proportion[END_REF], the ground plan for many ancient monuments and temples. The Pythagorean geometry of the 3, 4, 5 right triangle and the "squared circle" form the basis of the Cosmological Circle. The heptagon is an additional feature of the Cosmological Circle, and its relation to the cycloid curve connect it to the foundation of calculus and the history of the least action principle. Pauli's world clock has three rhythms with two orthogonal central discs, encircled by a gold colored band divided into 32 parts, and supported by a blackbird (the bird of Hermes [9]). The outer gold colored band represents 32 cubed or 32, 768 total pulses of the three rhythms, which is the harmonic of Walter Russell's carbon pressure point of the carbon octave and significant to William Conner's harmonic system [START_REF] Conner | Harmonic Mathematics[END_REF] related to the energy of the World Grid [START_REF] Farrell | The Grid of the Gods: The Aftermath of the Cosmic War and the Physics of the Pyramid Peoples[END_REF]. According to Pauli a feeling of "most sublime harmony" was produced by this abstract structure [START_REF] Miller | Deciphering the Cosmic Number: The Strange Friendship of Wolfgang Pauli and Carl Jung[END_REF]. "According to Jung, Pauli's vision of the World Clock represents the essence of space-time ..." [START_REF] Enz | No Time to be Brief: A Scientific Biography of Wolfgang Pauli[END_REF][START_REF] Jung | Psychology and Alchemy[END_REF], and was likened to Kepler's first geometrical ordering of the solar system, similar to the construction from the ancient canon that involves the geometry of the polygon circumscribing constant; to which Pauli adds his interpretation of time in his "Great Vision."

Another interpretation of Pauli's World Clock could be made comparing it to a basic yin-yang space-time model of brain-mind function describing hemispheric interactions [START_REF] Seifer | Transcending the Speed of Light[END_REF]. Pauli associated the rhythms of the World Clock with biological processes (in particular the four chambers of the heart and its average rhythm of 72 beats per minute) as well as with psychic processes [START_REF] Meier | Atom and Archetype: The Pauli/Jung Letters 1932-1958[END_REF]. In Wolfgang Pauli's visionary World Clock geometry the blackbird is a symbol for the "turning inward" at the beginning stage of alchemy and the messenger for the creative solar principle. Corresponding to this inward turnabout is the angle of 180 • = 32 • + (4 × 37 • ) 32 • + 4 tan -1 (3/4). From the Beyers-Brown drawing of Pauli's dream, half of a golden rectangle encloses the blackbird. A line extended from the left side of the golden circle through the beak to the back foot, then a right angle up to the right side of the gold circle forms a triangle of approximately 32 where R is the outer radius of the heptagon with side equal to one, R sin 32 • 3/5. On the top of this "Pauli triangle" are two smaller back-to-back triangles with approximately the same angles. The apex at the top of the vertical disk is 58 • + 58 • 116 • , approximately the regular dodecahedron dihedral angle. The dihedral angle is the internal face angle where the two adjacent faces of the polyhedron meet.

Space, time, and synchronicity

The subject of synchronicity was investigated at length by Pauli and Jung in their essay on The Interpretation of Nature and the Psyche [START_REF] Jung | The Interpretation of Nature and the Psyche[END_REF] and the geometry of this artwork could be an example of a synchronistic "meaningful coincidence" as well as intuitive clues toward something more than "numerical coincidence" for calculations that follow [START_REF] Pauli | Modern Examples of Background Physics[END_REF]. The "Pauli triangle" is also found in the symbolic form of the Emerald Tablet of Hermes, appearing at the very end of the same chapter of Jung's description of Pauli's World Clock and having a vertex on the Philosopher's Stone [START_REF] Jung | Psychology and Alchemy[END_REF]. The "Golden Chains of Homer extending from the Central Ring of Plato" [START_REF] Gardner | Genesis of the Grail Kings[END_REF] in the tablet divides the mandala into the golden angle and forms a triangle that also intersects the Philosopher's Stone [START_REF] Gardner | Genesis of the Grail Kings[END_REF][START_REF] Dobbs | Newton's Commentary on the Emerald Tablet of Hermes Trismegistus[END_REF]. The location of the Philosopher's Stone is the same area as the zero-point crossover of the infinity symbol (called the Singularity or the Primal Point of Unity) in the Rodin Coil schematic, based on the nonagon and modular arithmetic [START_REF] Rodin | Vortex Based Mathematics: Basis for the ExtraOrdinary Rodin Coil[END_REF], with parallels to Walter Russell's cosmogony [START_REF] Russell | The Secret of Light[END_REF]. The pentagon angle of 72 • minus 32 • is equal to the nonagon angle of 40 • . 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 gold [START_REF] Gadalla | Egyptian Harmony[END_REF] related to quintessence, or the "unified field" of physics.

The symbolic form of the Emerald Tablet is also described by Sir Laurence Gardner as the "Alchemical Medallion of the Hidden Stone" and he reports that Newton and Boyle's discoveries were attributed to help from the archive of the Hermetic Table [START_REF] Gardner | Genesis of the Grail Kings[END_REF][START_REF] Dobbs | Newton's Commentary on the Emerald Tablet of Hermes Trismegistus[END_REF]. Also stated by John Michell: "Newton, who laid the foundations of modern cosmology, was also one of the last of the scholars of the old tradition who accepted that the standards of ancient science were higher than the nobler, and sought, like Pythagoras, to rediscover the ancient knowledge." [START_REF] Farrell | The Grid of the Gods: The Aftermath of the Cosmic War and the Physics of the Pyramid Peoples[END_REF]. From the symbolism of Pauli's World Clock 9 × 9 × 32 = 2592, compared to the archetypal 144 × 180 = 25, 920 of the Platonic Year completing a 360 • precessional cycle [START_REF] Brumbaugh | Plato's Mathematical Imagination: The Mathematical Passages in the Dialogues and Their Interpretation[END_REF]. Also, 25, 920/504 = 360/7 85/ √ e, see Eq. (1), and e is Euler's number, base of the natural logarithm. The proportion for the classical "squared circle" construction is 8/9 = 320/360 and 8 × 9 = 72 = 32 + 40.

Pauli's interpretation of the World Clock as the "three permeating the four" is essential to the polemic between Fludd and Kepler that Pauli tried to resolve within himself and related to his presentation in "The Influence of Archetypal Ideas on the Scientific Theories of Kepler" of the hieroglyphic monad in Fludd's excerpt describing the quaternary [START_REF] Enz | [END_REF][START_REF] Pauli | The Influence of Archetypal Ideas on the Scientific Theories of Kepler[END_REF]. Pauli's interpretation also relates to the Pythagorean-Egyptian tradition regarding the 3, 4, 5 right triangle that is the basis for the construction of the Cosmological Circle. The various interpretations of the Cosmological Circle include the maze of nested polyhedra within the dodecahedron and their transformations. David Lindorff comments, "Pauli's sense that number in itself had a deep psychological significance is striking; it would later become of singular importance to him. ... He wrote, 'Here new Pythagorean elements are at play, which can perhaps be still further researched."' [START_REF] Lindorff | [END_REF]. Harald Atmanspacher and Hans Primas explain, "Pauli understood that physics necessarily gives an incomplete view of nature, and he was looking for an extended scientific framework." [START_REF]Recasting Reality: Wolfgang Pauli's Philosophical Ideas and Contemporary Science[END_REF]. Pauli also worked with Marie-Louise von Franz, who wrote in Number and Time, "Numbers, furthermore, as archetypal structural constants of the collective unconscious, possess a dynamic, active aspect which is especially important to keep in mind. It is not what we can do with numbers but what they do to our consciousness that is essential." [START_REF] Franz | Number and Time: Reflections Leading Toward a Unification of Depth Psychology and Physics[END_REF]. With this numerical analogy and parallels to neuroscience, Mark Morrison states in his overview, Modern Alchemy: "At this border of science and our deepest sense of our mental and even spiritual selves, alchemy is again demonstrating its relevance and durability." [START_REF] Morrisson | [END_REF]. Other examples of solving the Kepler-Fludd problem that Pauli symbolized by the numbers three and four are found in the philosophy of Joseph Whiteman [START_REF] Halsbury | Professor Whiteman's Philosophy of Space and Time[END_REF] and Franklin Merrell-Wolff [START_REF] Merrell-Wolff | Transformations in Consciousness: The Metaphysics and Epistemology[END_REF].

The fine-structure constant calculation

The fine-structure constant has a dimensionless value determined by the most recent experimental-QED calculations: α -1 = 137.035 999 173 (35) (T. Aoyama, et al [START_REF] Aoyama | Tenth-Order QED Contribution to the Electron g-2 and an Improved Value of the Fine Structure Constant[END_REF]) and α -1 = 137.035 999 173 3 (344) (T. Kinoshita [START_REF] Kinoshita | Fine Structure Constant Derived from the Electron g-2 with the QED Theory Up to the Tenth-Order[END_REF]). Approximating α -1 137.035 999 168: sin α -1 504/85κ.

(

) 1
The quantitative and qualitative reasoning for the approximation is significant to Plato's geometry, 7 × 72 = 111 + 393 = 504, proportional to the large radius of the Cosmological Circle [START_REF] Gaunt | Beginnings: The Sacred Design[END_REF], and Plato's favorite 5040 = 7!, of the larger harmonic proportion [START_REF] Sherbon | Nature's Information and Harmonic Proportion[END_REF]. cos(π/16) cot(π/16) 504/85. 6 × 85 = 6 + 504. 2 × 54 = 108 and 108 + 144 = 252. 2 × 252 = 504. The polygon circumscribing constant [START_REF] Weisstein | Polygon Circumscribing[END_REF], κ 8.700 036 625 208 e 2 sec 32 • and cot 32 • φ. Another calculation involves Pythagorean triangles related to the Cosmological Circle and the prime constant [START_REF] Weisstein | Prime Constant[END_REF], ρ 0.414 682 509 851 111 φ √ 5/κ, which has a binary expansion corresponding to an indicator function for the set of prime numbers. The inverse fine-structure constant again:

α -1 157 -337ρ/7, (2)
where α -1 137.035 999 168, approximately the same value as determined in Eq. (1) from above. The square of the diagonal of a "prime constant rectangle" is 1+ρ 2 κ/e 2 5/3 √ 2 sec 32 4) and ( 5).

An "infinitely nested set of circumscribed polygons and circles" [START_REF] Weisstein | Polygon Circumscribing[END_REF] gives the polygon circumscribing constant κ ≡ sec(π/3) sec(π/4) sec(π/5) ..., which is also formulated as a converging series involving the Riemann zeta function ζ(s) that is found in determining alpha. The Riemann zeta function is involved in the perturbative determination of the fine-structure constant from quantum electrodynamic theory and the experimentally measured value of the electron's gyromagnetic ratio [START_REF] Hanneke | New Measurement of the Electron Magnetic Moment and the Fine Structure Constant[END_REF].

In 1949 Wolfgang Pauli and Felix Villars published a paper together on Pauli-Villars regularization for the problem of infinities in quantum electrodynamics [START_REF] Pauli | On the Invariant Regularization in Relativistic Quantum Theory[END_REF]. Pauli also corresponded with Julian Schwinger, noted here for his introduction of α/2π in the corrective calculation for the anomalous magnetic moment of the electron and his zeta function regularization for the renormalization effort.

Accurate to seven places, sin α -1 8δ/7, where δ is Gompertz constant (or the Euler-Gompertz constant, which can also be expressed in relation to the Euler's number [START_REF] Finch | Euler-Gompertz Constant[END_REF]), δ ≡ -eEi(-1)

φ/e, where Ei is the exponential integral. Returning to the polygon circumscribing constant, κ 2 76, κ 2 + π 2 85, sin α -1 ≈ 32φ/κ 2 , and sin α -1 is the approximate ratio of the 32 4), with R as the radius of the regular heptagon with side equal to one. α -1 16π 2 /R and R 2γ -ln(ρ √ γ), prime constant ρ with γ, the Euler-Mascheroni constant; γ ≈ 5/κ. Also, κ + D 11, the basic diameter of the Cosmological Circle, where D is the diameter of the regular heptagon with side equal to one [START_REF] Sherbon | Nature's Information and Harmonic Proportion[END_REF]. The PDG [START_REF] Beringer | [END_REF] value for the strong coupling constant α s 0.1184 ( 7) is proportional: α s /α κ/πρ 2 . α s 1 -κ/π 2 sec 32 • /π 2 . κρφ 2 φ/ρ 2 is the diagonal of a 5 by 8 approximate golden rectangle.

Brown Landone writes that long before the Egyptians the Teleois were used by the ancient masters of Tiajura, then the Tiahuanacans and Incas of South America. "Teleois numbers form the long lost canon of Polykleitos, since they were used to determine the structures of all great temples of Greece and Egypt where Pythagoras lived ..." [START_REF] Landone | Prophecies of Melchi-Zedek in the Great Pyramid and the Seven Temples[END_REF]. "The Teleois proportions are used by the creative force because they best fit the electromagnetic energy fields of the atom." [START_REF] Hardy | Pyramid Energy[END_REF]. As part of a series based on modulo Julian Schwinger introduced α/2π to the quantum electrodynamic problem of the anomalous magnetic moment of the electron, see Eq. ( 6) [START_REF] Schwinger | On Quantum-Electrodynamics and the Magnetic Moment of the Electron[END_REF]. The harmonic of α/2π is also the value of Coral Castle builder Ed Leedskalnin's proportion, proclaimed by some of his followers as the "secret of the universe" [START_REF] Mcclure | Coral Castle: The Mystery of Ed Leedskalnin and his American Stonehenge[END_REF]. On the Teleois again, "Understand the proportions of the atom and its electromagnetic frequencies and you can understand why the proportions of the Teleois were used." [START_REF] Hardy | Pyramid Energy[END_REF]. α -1 85φ and csc α -1 √ 85/2π. More exactly, α

Quintessential questions of the golden ratio

A golden rectangle of whirling squares constructed by Bonnie Gaunt, based on the sacred geometry of the solar system, has the side of a square proportional to 504 [START_REF] Gaunt | Beginnings: The Sacred Design[END_REF]. 504/85 √ 2φ 3 ≈ 2 √ κ. Dan Winter puts two of these golden rectangles together in opposite directions, constructs a Vesica Piscis, and draws two intersecting golden ratio spirals. This forms a main Pythagorean triangle (φ, φ + 1, φ √ φ + 2) with side ratios of φ √ φ + 2/φ = √ φ + 2 ≈ csc 32 • which is also the approximate cube/sphere ratio of 6/π = π/ζ(2); a significant ratio in ancient Egyptian geometry [START_REF] Sherbon | Mathematical Constants of Natural Philosophy[END_REF]. Also shown above, cot 32 • φ.

Continuing with Winter's geometric construction, the golden ratio spirals are then shown dynamically forming a toroidal vortex that fits in a dodecahedron. Winter suggests this action is the source of gravity and the formation of mass, by "fractal charge compression" (charge is interpreted broadly here as a form of the ether, or the alchemical quintessence, in which Winter and others report an interesting relationship with the dodecahedral structure of the noble elements). The dodecahedron has been considered a three-dimensional version of the Cosmological Circle and was associated with the alchemical quintessence [START_REF] Sherbon | Nature's Information and Harmonic Proportion[END_REF][START_REF] Sherbon | Mathematical Constants of Natural Philosophy[END_REF]. Also suggestive, the expression for Newton's gravitational constant G in terms related to Planck momentum is G = c/m 2 P and from the relation for the fine-structure constant c = e 2 /α therefore, αG = e 2 /m 2 P , square of the electric charge to Planck mass ratio [START_REF] Sherbon | Mathematical Constants of Natural Philosophy[END_REF][START_REF] Macleod | Quintessence-Momentum as Link Between Mass and Charge[END_REF].

In William Eisen's Cabala the number 137 is interpreted as the MG or "image" and 37 is the CG or "center of gravity" [START_REF] Eisen | The Essence of the Cabalah[END_REF]. According to Marcus Chown [START_REF] Chown | The Golden Rule: It Links Art, Music and Even Architecture[END_REF], "Perhaps the most surprising place the golden ratio crops up is in the physics of black holes, a discovery made by Paul Davies," [START_REF] Davies | Thermodynamic Theory of Black Holes[END_REF][START_REF] Davies | Thermodynamic Phase Transitions of Kerr-Newman Black Holes in de Sitter Space[END_REF] (for other examples see [START_REF] Nieto | A Link Between Black Holes and the Golden Ratio[END_REF]- [START_REF] Keen | The Mind, Intergalactic Space, and Phi[END_REF]). This brings up one of the most fundamental concepts of alchemy, the prima materia, or "first matter" symbolized by the color black (Pauli's blackbird) and related to the aethereal quintessence of the Philosopher's Stone.

The angular momenta found in massive particles are related to quintessence, the Pythagorean harmonic of Planck momentum as described by Malcolm Macleod, Q 2 m P c/2π [START_REF] Macleod | Quintessence-Momentum as Link Between Mass and Charge[END_REF], with the fundamental geometry related to the regular heptagon (which is related to the origin of calculus and the least action principle [START_REF] Sherbon | Nature's Information and Harmonic Proportion[END_REF][START_REF] Sherbon | Mathematical Constants of Natural Philosophy[END_REF]). The quintessence Q has a Pythagorean relationship with C:

C 2 = Q 2 + Q 2 cot(π/7), (3)
where C 2 is the approximate inner diameter of the regular heptagon with side equal to one. Quintessence also has other simple relations with the pentagon geometry, tan 72 √ 3). C is also a fundamental harmonic of the "grid speed of light" found in the World Grid [START_REF] Farrell | The Grid of the Gods: The Aftermath of the Cosmic War and the Physics of the Pyramid Peoples[END_REF][START_REF] Sherbon | Mathematical Constants of Natural Philosophy[END_REF], the fundamental tone of William Conner's version of the Pythagorean Table [START_REF] Conner | Harmonic Mathematics[END_REF], and proportional to the decagon vertex angle. Also, 2C 2 √ e 2 + π 2 , and C 2 5ρ φ f /φ where φ f is the reciprocal Fibonacci constant [START_REF] Sherbon | Mathematical Constants of Natural Philosophy[END_REF]. Q 2 25/24, canonical ratio of ancient metrology related to the precessional cycle calculation [START_REF] Farrell | The Grid of the Gods: The Aftermath of the Cosmic War and the Physics of the Pyramid Peoples[END_REF]. The quintessence harmonic related to the golden ratio and inverse fine-structure constant:

csc α -1 CQ = √ 2Q 2 2γ φ. (4)
cos 32 • γCQ and R 2γ with R as the outer radius of the regular heptagon with side equal to one. Thus the fine-structure constant is shown related to the angular momentum property characteristic of massive particles, which again is related to "Pauli's triangle." Other approximations of quintessence: Q 1 + αφ 2 √ 2ρ/γ, where γ is the Euler-Mascheroni constant again. Also, Q ln(κ/π) ln(R/ρ) and Q/3 γ/5 e/8. Q 3 2/e γ φ/C. With the inverse of α/2π from Schwinger, in radians sin(2π/α) Q/φ. Another geometric relationship, recalling Euler's formula shows a connection between the golden ratio, the dodecahedron, and quintessence: e iQ (φ 2 /5)+(

√ 5i/φ 2 ) cos Q + i sin Q, thus Q 1.019 radians. The cos -1 (φ 2 /5) 58 o , sin -1 ( √ 5/φ 2 )
59 o which shows angles close to the complementary angle in "Pauli's triangle" and the sum of both is approximately the regular dodecahedron dihedral angle.

The proton/electron mass ratio m p /m e 1836.152 672 47 [START_REF] Beringer | [END_REF] and with nine place accuracy csc(m p /m e ) √ 3 -π/85. Other important mass ratios also have simple approximations involving quintessence and "Pauli's triangle." The W boson from electroweak theory and the Higgs boson, m w /m H • Q tan 32 The aspect of the dodecahedron in question here is related to Krsanna Duran's Timestar, where Winter finds the square root of phi to be significant in the proportional relation between the dodecahedron and the rotation of the enclosed tetrahedra (recall the discussion above, cos 23.5 • √ φ tan 36 • showing the decagon angle and φ = 2 cos 36 • ) [START_REF] Duran | The Human Odyssey[END_REF]. The action has parallels to Pauli's vision of a World Clock described by Jung as a mandala.

The configuration of polygons determining the polygon circumscribing constant κ was also described as a mandala, notable in the ancient canon according to Marie Franz and Elson Haas [START_REF] Franz | Number and Time: Reflections Leading Toward a Unification of Depth Psychology and Physics[END_REF]. The harmonic of the polygon circumscribing constant is also found in the diameter of Stonehenge (with an architectural design also based on the Cosmological Circle) and 87/54 φ. Also from the Cosmological Circle, the Earth-Moon radius divided by the Earth radius, 504/396 √ φ 108/85, with the proportional Moon radius of 108 [START_REF] Gaunt | Beginnings: The Sacred Design[END_REF][START_REF] Sherbon | Mathematical Constants of Natural Philosophy[END_REF].

A more precise approximation for quintessence reveals a factor of 7 and 85 again, Q exp(42/(56P + 85e)) 2 [START_REF] Sherbon | Nature's Information and Harmonic Proportion[END_REF][START_REF] Sherbon | Mathematical Constants of Natural Philosophy[END_REF]. P 1.324 795 724 47 5Q 3 /4, the silver number P [8] is the root of x 3 -x -1 = 0, also the limiting ratio of successive terms in the Padovan sequence and Perrin sequence. The silver number has a property similar to the golden ratio, P + 1 = P 3 .

Electron magnetic moment anomaly

In modern quantum electrodynamics the g-factor of the electron is represented as a series expansion in powers of α/2π yielding the value g/2 1.001 159 652 180 73 [START_REF] Merrell-Wolff | Transformations in Consciousness: The Metaphysics and Epistemology[END_REF] [START_REF] Hanneke | New Measurement of the Electron Magnetic Moment and the Fine Structure Constant[END_REF], also one of the most precise experimental measurements [START_REF] Hanneke | New Measurement of the Electron Magnetic Moment and the Fine Structure Constant[END_REF]. A general approximation of the anomalous magnetic moment of the electron a e ≡ (g -2)/2 can be made from explanations of spin-orbit coupling and reasoning from alternative theory [START_REF] Winter | One Crystal's Dance -The Star Mother: Geometric Keys to the Resonant Spirit of Biology[END_REF], [START_REF] Dennis | A Framework Linking Non-living and Living Systems[END_REF]- [START_REF] Miller | Early Quantum Electrodynamics: A Source Book[END_REF]:

g/2 1 + (α/2π) -(φα/2π √ 2) 2 + (α/2πγ) 3 -(α/2πφ) 4 . (6)
g/2 1.001 159 652 180 75 with the value of alpha from Eqs. ( 1) and (2). The harmonic formula above involves a convergence of Julian Schwinger's work with Green's functions [START_REF] Schwinger | On Quantum-Electrodynamics and the Magnetic Moment of the Electron[END_REF], the classical geometry behind Pauli's World Clock vision, and the corresponding golden ratio φ aspect of the Cosmological Circle (from the center to the mid-edge of the dodecahedron equals φ 2 /2 = π/golden angle in radians and the chord of the pentagon face is φ). From the regular heptagon, the radius with side equal to one, R 2γ. The Euler-Mascheroni constant is the finite part of the harmonic series in the Riemann zeta function of ζ(1) [START_REF] Zeidler | Quantum Field Theory II[END_REF], used in quantum electrodynamic calculations [START_REF] Drell | Form Factors in Quantum Electrodynamics[END_REF][START_REF] Drell | Anomalous Magnetic Moment of the Electron, Muon, and Nucleon[END_REF]. Euler's constant γ 0.577 215 664 901 532 [START_REF] Havil | Gamma: Exploring Euler's Constant[END_REF], for γ and the Riemann zeta function ζ(s) see Julian Havil's Gamma [START_REF] Havil | Gamma: Exploring Euler's Constant[END_REF]. The heptagon of the Cosmological Circle is related to the torus. With volume of a torus, or the hypersphere surface area: 2π 2 R 3 32R 3 /φ, and R = φ sec 32 • , α -1 2π 2 (φ sec 32

• ) 3 . R 3 360 • / tan -1 ( √ φ)
, with an approximate heptagon angle. The cube/sphere ratio, 6/π φ sec 32 • .

As Malcolm Mac Gregor explains in The Power of Alpha, the essence of the finestructure constant is the ratio of "... the 'spherical geometry' of the electrostatic energy E = e 2 /r e = mc 2 to the 'spherical geometry' of mechanical energy, c/r m = mc 2 , which is defined by the Compton radius of a relativistically spinning sphere." [START_REF] Gregor | The Power of Alpha[END_REF]. Then r e /r m = e 2 / c = α.

Another interpretation of the fine-structure constant related to Pauli's World Clock geometry is the standard perspective of action, product of energy and time. Two elementary particles separated by a distance r have an electrostatic energy of e 2 /r and the time for light to travel a distance r is r/c, so the action is (e 2 /r) × (r/c) = e 2 /c. Since the unit of quantum action for light is (from E = ω), the ratio of the electrical action to the quantum action is e 2 / c = α.

Conclusions

Since Arnold Sommerfeld's introduction of the fine-structure constant and the discovery of Planck's constant by blackbody radiation, attempts were made to find relationships between them. They were found connected by prime numbers, particular values of the Riemann zeta function, the Boltzmann constant, and even a dimensionless blackbody radiation constant [START_REF] Xiao | Dimensionless Constants and Blackbody Radiation Laws[END_REF]. Pauli displayed elegance in his own mathematical techniques for calculating the hydrogen spectrum [START_REF] Valent | The Hydrogen Atom in Electric and Magnetic Fields: Pauli's 1926 Article[END_REF].

From an overview by Domenico Giulini, "In his last paper on the subject of discrete symmetries ... Pauli comes back to the question which bothered him most: how is the strength of an interaction related to its symmetry properties?" [START_REF] Giulini | Concepts of Symmetry in the Work of Wolfgang Pauli[END_REF]. This question is associated with the manifestation of charge from alchemical quintessence, the rotations associated with the symmetry groups of Platonic solids, and part of the archetypal process behind Pauli's World Clock.

In "Science and Western Thought" from Wolfgang Pauli: Writings on Philosophy and Physics Pauli asks the question, "Shall we be able to realize, on a higher plane, alchemy's old dream of psychophysical unity, by the creation of a unified conceptual foundation for the scientific comprehension of the physical as well as the psychical?" [START_REF] Pauli | Science and Western Thought[END_REF]. Kalervo Laurikainen responds in The Message of the Atoms, "If we are not bound to materialistic presuppositions, the psycho-physical unus mundus of Pauli and Jung appears, in fact, to be the natural ontology of quantum mechanics." [START_REF] Laurikainen | The Message of the Atoms: Essays on Wolfgang Pauli and the Unspeakable[END_REF]. This leads to a complementary representation of reality similar to Bohr's philosophy, rather than that of Cartesian dualism.

Thus Wolfgang Pauli considered, "Concepts created by mathematics, such as Riemann's surfaces, lend themselves very well to a symbolic representation of the relativization of the concept of time ...." and concluded that "... most modern physics lends itself to the symbolic representation of psychic processes." [START_REF] Pauli | Modern Examples of Background Physics[END_REF]. Pauli's World Clock vision involves the symbolic images and archetypal stages of alchemical transformation at the root of both mental and physical worlds [START_REF] Wolf | Towards a Quantum Field Theory of Mind[END_REF].

From Pauli's imaginal geometry the unit circle on the complex plane was his i ring, represented by e iθ , and yielding the "Pauli triangle" for θ Q, quintessence in radians.

For Pauli the ring embraced both intellect and intuition symbolic of the unus mundus.

The World Clock geometry combined with the self-recursive property enabled by the golden ratio and an alchemical interpretation of its symbolism is also descriptive of the pine cone shaped energy field of an activated pineal gland, often depicted in ancient Egyptian art along with the legendary bennu bird that is known for its association with the cycles of time and the final phase of the alchemical process (an awakening of the "golden" crown chakra in the aetheric field of mfkzt [START_REF] Gardner | Genesis of the Grail Kings[END_REF]). The pineal gland biology is an active area of research in modern biophysics and pineal activation might explain the "Pauli effect" reported by experimental physicists. Pauli knew of Descartes ideas about the pineal gland and Plato suggested that mathematical discipline such as Pauli engaged in could help awaken this special gland. Pauli wrote several unpublished papers considering the symbolic representation of mental process from the mathematics of the electromagnetic field. From the geometry of neb, the hieroglyph for gold [START_REF] Gadalla | Egyptian Harmony[END_REF], Q 140α. From the previous force field analogy with the four basic constants of "recovered science," Q πe 2 /κφ 2 [START_REF] Sherbon | Mathematical Constants of Natural Philosophy[END_REF]. From the torus, tan -1 ( √ φ) is the approximate heptagon angle, with √ φ 4/π. Beyond the alchemical interpretations of Carl Jung, the torus topology that relates the dynamics of Einstein's relativity with classical mechanics and its geometry is found in Egyptian art, especially their representation of anatomical proportions. Some depictions of Tehuti-Thoth from ancient Egyptian art are similar to Pauli's World Clock having the bird space-form of the alchemical tradition with the archetypal signs and symbols of time-cycles [START_REF] Wilcock | The Source Field Investigations[END_REF]- [START_REF] Cox | Egyptian and Form Energy Head-Dress of the Ancients[END_REF].

In The Sphinx and the Rainbow David Loye states, "Wolfgang Pauli felt there could be a correspondence between the wave-particle mystery in physics and the mind-body problem in philosophy ..." [START_REF] Loye | The Sphinx and the Rainbow[END_REF]. F. David Peat also writes in Synchronicity, "Pauli, as a physicist, was also seeking to discover an inner unity between the elementary particles and their abstract symmetries." [START_REF] Peat | Synchronicity: The Bridge Between Matter and Mind[END_REF].

Beverley Zabriskie writes in her introduction to Atom and Archetype, the Pauli/Jung letters edited by C.A. Meier, "For Pauli, symmetry was the archetypal structure of matter. Just as the alchemists looked for the substratum of reality beneath matter, he came to the view that the elementary particles were not themselves the ultimate level of realty." [START_REF] Zabriskie | Atom and Archetype: The Pauli/Jung Letters 1932-1958[END_REF]. Then she explains, "As he became more familiar with alchemy as a psychophysical unity, Pauli saw the same lumen naturae, the light of nature, or the 'spirit in matter,' glimpsed by Paracelsus and Jung." [START_REF] Zabriskie | Atom and Archetype: The Pauli/Jung Letters 1932-1958[END_REF].

Writing on "Wolfgang Pauli's Philosophical Outlook" in Across the Frontiers, Werner Heisenberg says; "Among the studies to which Pauli was prompted by the philosophical labors just referred to, it was those on the symbolism of the alchemists which left particularly lasting traces behind ..." [START_REF] Heisenberg | Wolfgang Pauli's Philosophical Outlook[END_REF]. Many of Pauli's ancestors were from Prague, a traditional center for alchemical activity [START_REF] Várlaki | Historical Origin of the Fine Structure Constant, Parts I-III[END_REF].

In an article about Pauli by his former assistants Marcus Fierz and Victor Weisskopf, "If we ask ourselves what above all was Pauli's calling the answer would be: he was a natural philosopher in the classical sense of the phrase -as it applies to Kepler, Galileo, and Newton." [START_REF] Fierz | In Commemoration of Wolfgang Pauli[END_REF]. A later retrospective by Victor Weisskopf shows Pauli "... developed a deep friendship with Gershon Scholem, the great scholar and world authority on Jewish mysticism, the Kabalah. (The Kabalah ascribes a number to each word of the Hebrew language, a number that has a deep symbolic significance. The number corresponding to the word Kabalah happens to be 137.)" [START_REF] Weisskopf | Personal Memories of Pauli[END_REF].

In his essay, "Pauli (Wolfgang) 1900-1958," Charles Enz (Pauli's last assistant) writes; "This number 137 symbolized for Pauli the link with the magic world of the alchemists which has so much fascinated him." [START_REF] Enz | Of Matter and Spirit[END_REF]. Recalling his preoccupation with the finestructure constant and research on synchronicity with Carl Jung, Pauli was moved upon finding his room number was 137 at the Red Cross hospital during his last days; when pineal activation can happen in transition [START_REF] Pauli | Modern Examples of Background Physics[END_REF][START_REF] Enz | Of Matter and Spirit[END_REF].

  -1 137 tan(tanh(302/285)) where 302/285 = 1 + 17/2852/e γ δ √ π, see Eq. (4) discussion. From the harmonic braiding shown by William Conner[START_REF] Conner | Harmonic Mathematics[END_REF], 370 -285 = 85, 5 × 17 = 85, 360 -58 = 302, and 32 + 58 = 90.From William Eisen's[START_REF] Eisen | The Essence of the Cabalah[END_REF] construction of the "All-Seeing Eye" 60 • + 77 • = 137 • and sin 137 • = cos 47 • . 47 • /2 = 23.5 • , the approximate tilt of the Earth axis, considered by Scott Creighton and Gary Osborn in The Giza Prophecy as "the most important of the precessional angles encoded in the Giza pyramids ..."[START_REF] Creighton | The Giza Prophecy[END_REF]. With tetrahedral angle, 19.5 • + 23.5 • +47 • = 90 • . Additionally, the northern shaft in the King's Chamber pointed to Alpha Draconis, circa 2450 B.C., at an angle of approximately 32 • . Corresponding to the construction of the Great Pyramid layering of stones and Pauli's World Clock symbolism is the connection between paramagnetism (including Pauli-paramagnetism) and the diamagnetism found in the golden ratio geometry of organic growth[START_REF] Winter | One Crystal's Dance -The Star Mother: Geometric Keys to the Resonant Spirit of Biology[END_REF].

  • , 90 • , and 58 • respectively. The cot 32 • + 45 • = 77 • 3π/7 radians, apex angle of the large triangle in the heptagon. One radian, 360 • /2π

	the golden ratio. The csc 2 32 • ≈ 1 + φ 2 = φ	√	sec(2π/7) 5, square of the diagonal of a golden 16/π 2 φ,
	rectangle, and 360 • /φ 2	137.5 • the golden angle. 7φ divides the circle into approxi-
	mately 32 • segments, 360 • /7φ	32 • (φ = 2 cos 36 •	1.618 033 988 749, see Eq. (6)),
	32 •

  • + 180 • /7. The World Clock has three rhythms or pulses and 32 partitions dividing the circle, 360 • /32 = 45 • /4 = π/16 radians and cos(π/16) ≈ R cos 32

  , proportional to the Moon radius of the Cosmological Circle. 7 2 + 108 = 157. Related to this is the main Pythagorean triangle 108, 144, 180. The triangles 85, 132, 157, and 175, 288, 337 are primitive Pythagorean triples. 60 + 72 = 132 and 72+85 = 157. 85+90 = 175, 4×72 = 288, and 85+504/2 = 157+180 = 337. 6 2 +7 2 = 85 and another triangle is 36, 77, 85. 36 is the basic multiplier for the 3, 4, 5 right triangle geometry, while 72 is the next. With the two basic radii 7 and 11, 7 × 11 = 77, and 5 × 36 = 180.

	360 -23 = 337. 23 + 37 = 60, 60/φ 37, and φ ≡ (1 +	√	180-23 = 157, and 5)/2. see [34]. 37 + 120 = 157.
	23 + 85 = 108Other related approximations include 1+ρ the reciprocal Fibonacci constant [34], and 360/φ 3 85. Also, α ρ/32 √ 2, cos 32 • 2ρ, 5ρ φ f /φ where φ f is √ π. The outer radius of the regular dodecahedron ( √ 3 + √ 15)/4 γ/ρ where γ is the Euler-Mascheroni
	constant, see Eqs. (

• , with the angle from "Pauli's triangle" found above.

  • × φ 51.8 • base angle of the Great Pyramid of Giza to the apex angle of approximately 76 • . Vertex angle of the pentagon 108 • -32 • = 76 • and csc α -1

R

√

φ, see Eq. (

  • 1 + 2Q2 , and Q √ 3 cos 54 • 2 tan 27 • (54 • is half the vertex angle of the pentagon and also noted, ln 32 2

  • . The Z boson from electroweak theory and the Higgs boson, m z /m H • cos 2 32 • Q/ Also, the Higgs boson and the top quark from quantum chromodynamics, m H • /m t Q cos 2 32 • [38] In later graphical representations Winter shows how racheting a cube five times by 32 • forms the dodecahedron, and explains how this is related to the opening of the pineal gland. The main supporting reference is the 32 • head tilt of the Great Sphinx of Giza noted by John Anthony West.

	√	2.	(5)

arithmetic, the Teleois proportions are easily noticeable in the Queen's Chamber of the Great Pyramid of Giza, designed with seven Teleois spheres that also correspond to the geometry included in the Cosmological Circle. "Within the great sphere of 31represented by a circle -six other Teleois circles exactly contact or intersect each other in perfect Teleois proportions."[START_REF] Landone | Prophecies of Melchi-Zedek in the Great Pyramid and the Seven Temples[END_REF]. The diameters are 1,

 4, 7, 10, 13, 19, and 31. The sum of these seven diameters is 85, harmonic of the √ α. The sum of the first six diameters is 54.William Conner also referenced the Teleois as a "cosmic formula behind form in the physical world" and modifies this series with a 144 multiplier beginning with 4 as 144, giving a culminating Teleois diameter of 11, 664 or 108 2 ("a number of extraordinary interlocking potential" determining the root tone generators of his Fibonacci-harmonic Quadrispiral and also found in the Great Pyramid proportions)[START_REF] Conner | Harmonic Mathematics[END_REF]. 11, 664 is also the proportional harmonic of α/2π, equal to the classical electron radius divided by its Compton wavelength.

Acknowledgments

Special thanks to Case Western Reserve University, Franklin Merrell-Wolff Fellowship, Social Science Research Network, Pauli Archives, MathWorld, and WolframAlpha.


Michael A Sherbon
email: michael.sherbon@case.edu

Fundamental Nature of the Fine-Structure Constant

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In addition to introducing the fine-structure constant [START_REF] Barrow | The Constants of Nature: From Alpha to Omega-the Numbers That Encode the Deepest Secrets of the Universe[END_REF]- [START_REF] Sommerfeld | On the Quantum Theory of Spectral Lines[END_REF], Arnold Sommerfeld added elliptic orbits to Bohr's atomic model deriving the Bohr-Sommerfeld model [START_REF] Sommerfeld | Atomic Structure and Spectral Lines[END_REF][START_REF] Kragh | Niels Bohr and the Quantum Atom[END_REF]. Then Wolfgang Pauli was influenced by Sommerfeld's search for the Platonic connections that were implied by the mystery of the fine-structure constant [START_REF] Miller | Deciphering the Cosmic Number[END_REF][START_REF] Enz | No Time to be Brief: A Scientific Biography of Wolfgang Pauli[END_REF][START_REF] Miller | Early Quantum Electrodynamics: A Source Book[END_REF]. As the fine-structure constant determines the electromagnetic strength its theoretical origin was for Pauli a key unsolved physical problem, also considered especially significant by Max Born, Richard Feynman and many other physicists [START_REF] Feynman | The Strange Theory of Light and Matter[END_REF].

The fine-structure constant, alpha, has a variety of physical interpretations from which various determinations have been made; atom interferometry and Bloch oscillations, the neutron Compton wavelength measurement, AC Josephson effect, quantum Hall effect in condensed matter physics [START_REF] Yoshihiro | Determination of the Fine Structure Constant Based on the Quantum Hall Effect[END_REF], hydrogen and muonium hyperfine structure, precision measurements of helium fine-structure, absorption of light in graphene [START_REF] Nair | Fine Structure Constant Defines Visual Transparency of Graphene[END_REF], also the topological phenomena in condensed matter physics [START_REF] Maciejko | Topological Quantization in Units of the Fine Structure Constant[END_REF], the relative optical transparency of a plasmonic system [START_REF] Kravets | Fine Structure Constant and Quantized Optical Transparency of Plasmonic Nanoarrays[END_REF], elementary particle lifetimes [START_REF] Gregor | The Power of Alpha[END_REF] and the anomalous magnetic moment of the electron in quantum electrodynamics [START_REF] Sturm | Electron g-Factor Determinations in Penning Traps[END_REF]. Slightly different values of the fine-structure constant are found from different experimental measurements [START_REF] Karshenboim | Recent Progress in Determination of Fundamental Constants and Fundamental Physics at Low Energies[END_REF] and finally there is also the question of its variation from the subatomic to the cosmological scale [START_REF] Truppe | A Search for Varying Fundamental Constants using Hertz-Level Frequency Measurements of Cold CH Molecules[END_REF]. The fine-structure constant is viewed in this work from both a mathematical and a physical perspective, as Hemmo and Hagar maintain that "... in current spacetime physics there can be no dynamical derivation of primitive geometrical notions such as length." In several cases, "... geometrical notions are assumed rather than derived." [START_REF] Hagar | The Primacy of Geometry[END_REF].

Max Born considered the idea of a fundamental length to be associated with the mystery of the fine-structure constant, see Eq. ( 25) [START_REF] Born | The Mysterious Number 137[END_REF]. The "primacy of geometry" is a mathematical assumption made in the work that follows, with the aim of progress toward additional physical understanding of the fine-structure constant [START_REF] Blaum | Precision Experiments and Fundamental Physics at Low Energies -Part I[END_REF]- [START_REF] Bouchendira | State of the Art in the Determination of the Fine Structure Constant[END_REF]. Relevant to this, Wolfgang Pauli's World Clock involved basic geometric constructions depicting the cycles of time. His World Clock was likened to Kepler's first geometrical ordering of the solar system found in ancient geometry [START_REF] Enz | No Time to be Brief: A Scientific Biography of Wolfgang Pauli[END_REF] and related to geometric constructions involving the Pythagorean right triangle [START_REF] Sherbon | Wolfgang Pauli and the Fine-Structure Constant[END_REF].

Calculating the fine-structure constant

The fine-structure constant was proposed by Sommerfeld as the ratio of the speed of the electron in the ground state of Bohr's hydrogen atom model to the speed of light [START_REF] Sommerfeld | On the Quantum Theory of Spectral Lines[END_REF]:

α = v/c = e 2 / c, ( 1
)
with the elementary charge e, Planck's constant h = 2π and the speed of light c in cgs units; where the Coulomb constant or the permittivity factor (found in the fine-structure formula with SI units) is one and dimensionless. An interpretation of the fine-structure constant related to Wolfgang Pauli's World Clock geometry is the standard perspective of action, product of energy and time. Two elementary particles separated by a distance r have an electrostatic energy of e 2 /r and the time for light to travel a distance r is r/c, so the action is (e 2 /r) × (r/c) = e 2 /c. Since the unit of quantum action for light is (from E = ω), the ratio of the electrical action to the quantum action is then e 2 / c = α.

The fine structure constant is in the formula for the energy levels of the hydrogen atom first given by Sommerfeld. Before the discovery of the Schrödinger equation, Pauli applied the elliptical eccentricity invariant from the classical Kepler problem with the "Pauli vector" (Laplace-Runge-Lenz vector) in calculating the hydrogen spectrum [START_REF] Pauli | On the Hydrogen Spectrum from the Standpoint of the New Quantum Mechanics[END_REF]. In the alternative Rydberg formula for the energy levels of the hydrogen atom E n = -hcR ∞ /n 2 with the Rydberg constant R ∞ = α 2 /2λ e , having the fine-structure constant and Compton wavelength of the electron.

Julian Schwinger was noted for his introduction of α/2π in the corrective calculation for the anomalous magnetic moment of the electron and his zeta function regularization in the renormalization effort [START_REF] Schwinger | On Quantum Electrodynamics and the Magnetic Moment of the Electron[END_REF]. α/2π is equal to the classical electron radius divided by its Compton wavelength, see Eq. ( 24) discussion.

The most recent value of the inverse fine-structure constant determined by theoretical and experimental work: α -1 = 137.035 999 173 [START_REF] Weisstein | MathWorld-A Wolfram Web Resource[END_REF] in research by T. Aoyama, et al. "Tenth-Order QED Contribution to the Electron g-2 and an Improved Value of the Fine Structure Constant," [START_REF] Aoyama | Tenth-Order QED Contribution to the Electron g-2 and an Improved Value of the Fine Structure Constant[END_REF]. Another form of calculation involves traditional Pythagorean triangles [START_REF] Sherbon | Wolfgang Pauli and the Fine-Structure Constant[END_REF] together with the prime constant [START_REF] Weisstein | Prime Constant[END_REF], described as a binary expansion corresponding to an indicator function for the set of prime numbers. Calculation of the inverse fine-structure constant as an approximate derivation from prime number theory:

α -1 ≃ 157 -337ρ/7, (2)
with an approximate value of α -1 ≃ 137.035 999 168, with three prime numbers and the prime constant. The square of the diagonal of a "prime constant rectangle" is

1 + ρ 2 ≃ κ/e 2 ≃ 5/3 √ 2.
The polygon circumscribing constant κ is the reciprocal of the Kepler-Bouwkamp constant [START_REF] Finch | Kepler-Bouwkamp Constant[END_REF]

ρ = ∞ ∑ k=1 2 -p(k) ≃ ϕ √ 5/κ, (3)
with the prime constant ρ ≃ 0.414 682 509 851 111 and κ again as a reciprocal of the Kepler-Bouwkamp constant [START_REF] Finch | Kepler-Bouwkamp Constant[END_REF]. Also,

ρ ≃ √ 2 -1, ϕ √ 5 ≃ 1 + ϕ 2 = 3ϕ 3 -5ϕ 2 + 4 = (5 + √ 5)/2 ≃ 3/2ρ and √ 5 = 2 cosh(ln ϕ). Additionally, sin α -1 ≃ 504/85κ ≈ 2π/ √ 85. sin α -1 ≃ 7!/(713 + 137)κ, (4)
with the approximate value of α -1 ≃ 137.035 999 168, same value as determined in Eq. ( 2) from above. Plato's favorite symbolic number 5040 = 7!. The polygon circumscribing constant, the reciprocal of the Kepler-Bouwkamp constant [START_REF] Finch | Kepler-Bouwkamp Constant[END_REF], is also formulated as a converging series involving the Riemann zeta function ζ(s) found in the perturbative determination of the electron magnetic moment anomaly from quantum electrodynamics.

κ = ∞ ∏ n=3 sec( π n ) ≃ ϕ √ 5/ρ. (5)
Polygon circumscribing constant κ ≃ 8.700 036 625 208. Again approximating, κ ≃ 14/ϕ and κ -1 ≃ ρ 2 λ, where λ is the Laplace limit of Kepler's equation, defined in Eq. [START_REF] Gregor | The Power of Alpha[END_REF].

ϕ = (1 + √ 5)/2 = 2 cos(π/5) ≃ ρκ/ √ 5, (6)
with the approximate value of ϕ ≃ 1.618 033 988 749 ≃ √ 2πρ and sin

α -1 ≃ 4/ sinh 2 ϕ. Also, √ ρk ≃ 5/ϕ 2 ≃ 6/π ≃ 3 √ 7
, which is the cube-sphere ratio (also the approximate diagonal of a golden rectangle with sides 1 and ϕ, see references in [START_REF] Sherbon | Wolfgang Pauli and the Fine-Structure Constant[END_REF]) and ρκ ≃ 3ζ(3) ≃ πR ≃ (π/2) csc(π/7) with heptagon radius R and Apéry's constant ζ(3) for the value of the Riemann zeta function ζ(s), generalization of the harmonic series. Kepler showed the golden ratio ϕ [START_REF] Weisstein | MathWorld-A Wolfram Web Resource[END_REF] as the limit of the ratio of consecutive Fibonacci numbers [START_REF] Stakhov | The Golden Section and Modern Harmony Mathematics[END_REF].

sin α -1 ≃ cosh ρ ˆ∞ 0 cos x 2 dx ≃ (κ/8) √ π/8, (7)
with the Fresnel integral [START_REF] Weisstein | Fresnel Integrals[END_REF] from optical physics. From hyperbolic [START_REF] Stakhov | The Golden Section, Fibonacci Series and New Hyperbolic Models of Nature[END_REF] and heptagon geometry [START_REF] Sherbon | Wolfgang Pauli and the Fine-Structure Constant[END_REF], see the discussion below of Eq. ( 12), cosh ρ = cos iρ ≃ κ/8 ≃ sec(π/7).

Euler's constant and Kepler's equation

The Euler-Mascheroni constant is γ, also known as Euler's constant [START_REF] Lagarias | Euler's Constant: Euler's Work and Modern Developments[END_REF]. The numerical value of Euler's constant γ ≃ 0.577 215 664 901 532. With the harmonic number H n :

γ = lim n→∞ (H n -ln n) = ∞ ∑ k=2 (-1) k ζ(k)/k ≃ ρκ/2π. ( 8
)
Related approximations include γ ≃ sech R ≃ 4/7 and

D = 2R = csc(π/7) ≃ κ/πζ(3) ≃ S/ √ 2 ≃ 4/ √ 3
, which is the approximate diameter of the circumscribing circle of the regular heptagon with side equal to one; also Apéry's constant again from the Riemann zeta function ζ(s) and the silver constant S ≃ 3.247 [START_REF] Weisstein | MathWorld-A Wolfram Web Resource[END_REF] basic to the geometry of the regular heptagon. The heptagon radius is R ≃ 2γ ≃ cot 2 α -1 and 2ρκ ≃ π csc(π/7).

From optical physics, α/2π ≃ exp(-πϕn) where n is the index of refraction for water. Snell's Law: n 1 sin θ 1 = n 2 sin θ 2 , where θ 1 is the angle between the ray and the surface normal in the first medium, θ 2 is the angle between the ray and the surface normal in the second medium and n 1 and n 2 are indices of refraction (n 1 ≃ 1 in a near vacuum and n 2 > 1 in a transparent substance). With angle of incidence 45 • the angle of refraction is 32 • (Pauli's World Clock [START_REF] Sherbon | Wolfgang Pauli and the Fine-Structure Constant[END_REF]) for water and

n 2 ≃ 1.33 ≃ R 2 ≃ √ 7/2 ≃ 1/ρ √ S ≃ γ csc(π/7). Also approximating, csc α -1 ≃ √ ϕn, S ≃ 1 + √ πϕ and ϕπ 2 ≃ S ln(α -1 ). sin α -1 ≃ 2γ sin(π/5) ≃ 2/γπϕ, (9)
relating the golden ratio and inverse fine-structure constant. Other approximations, sin

α -1 ≃ π/2D ≃ (π -1)/π ≃ γ sec 32 • . Also, 2 cos(π/7) = √ S and csc(2π/7) ≃ √ ϕ. sin α -1 ≃ √ 2 tan(π/7) ≃ 1/2γ √ ϕ. ( 10
)
The eccentricity of a golden ellipse [START_REF] Huntley | The Golden Ellipse[END_REF] is [START_REF] Kravets | Fine Structure Constant and Quantized Optical Transparency of Plasmonic Nanoarrays[END_REF]. The constant µ is the real fixed point of the hyperbolic cotangent and λ is the Laplace limit of Kepler's equation [START_REF] Weisstein | Laplace Limit[END_REF]. Kepler's equation for calculating orbits: M = E -ε sin E, where M is the mean anomaly, E is the eccentric anomaly and ε is the eccentricity [START_REF] Cariglia | Dynamical Symmetries of the Kepler Problem[END_REF]- [START_REF] Wulfman | On the Dynamical and Geometrical Symmetries of Keplerian Motion[END_REF]. With suggestive elliptic connections from the study of polarized light, Tse and MacDonald [START_REF] Tse | Magneto-Optical Faraday and Kerr Effects in Topological Insulator Films and in Other Layered Quantized Hall Systems[END_REF], in their theory of "magneto-optical Faraday and Kerr effects of thin topological insulator films," find a "Faraday angle equal to the fine structure constant" and an approximate π/2 Kerr rotation [START_REF] Shimano | Quantum Faraday and Kerr Rotations in Graphene[END_REF][START_REF] Burinskii | The Dirac-Kerr-Newman Electron[END_REF]. If the eccentricity ε = λ, which is the Laplace limit of Kepler's equation, and E = tan -1 (α -1 ) ≃ π/2 in the parametric form of Kepler's equation with radians; then the mean anomaly M is:

ε = 1/ √ ϕ ≃ csc α -1 -sin α -1 ≈ µλ, see Eq.
M = E -λ sin E ≃ Eγ ≃ 2π/7. ( 11
) M ≃ √ S/2 ≃ 2 √ 2/π ≃ πD/8 ≃ cos(π/7
) and the sec(2π/7) ≃ ϕ [START_REF] Sherbon | Wolfgang Pauli and the Fine-Structure Constant[END_REF]. From Kepler's triangle (1, √ ϕ, ϕ) and the heptagon, tan(2π/7) ≃ √ π/2 and tan -1 ( √ ϕ) is an approximate heptagon angle 2π/7, with √ ϕ ≃ 4/π. The diameter of the circumscribing sphere of the regular dodecahedron with side equal to one is ρκ ≃ 2πγ. Also, the outer radius of the dodecahedron (

√ 3 + √ 15)/4 ≃ γ/ρ and csc α -1 ≃ R √ ϕ.
The square root of phi is described by the dodecahedron proportions. The silver constant from the heptagon:

S = 2 + 2 cos(2π/7) = 4 cos 2 (π/7) ≃ πϕ/eγ. ( 12
)
The regular heptagon is related to the origin of calculus, the cycloid curve, the least action principle and the squaring of the circle; see the historical references in [START_REF] Sherbon | Wolfgang Pauli and the Fine-Structure Constant[END_REF].

Silver constant S ≃ 2/λγϕ ≃ 2µ/γ √ ϕ ≃ √ ϕ coth ρ ≃ coth π -1 ≃ √ 2D.
The real fixed point of the hyperbolic cotangent is µ and γ ≃ µ/λπ [START_REF] Weisstein | MathWorld-A Wolfram Web Resource[END_REF], with the Laplace limit of Kepler's equation again, see Eq. ( 14). The cosh ρ ≃ sec(π/7) = 2/ √ S. In another reference relating to relevant geometry Li, Ji and Cao discovered that Fibonacci spirals found on conical patterns in nature can be effectively modeled as a least energy configuration [START_REF] Li | Stressed Fibonacci Spiral Patterns of Definite Chirality[END_REF].

Electron magnetic moment anomaly

In quantum electrodynamics the g-factor of the electron is represented in theory as a series expansion in powers of α/2π, with the latest experimental value by the Gabrielse Research Group: g/2 ≃ 1.001 159 652 180 73 (28) [START_REF] Hanneke | New Measurement of the Electron Magnetic Moment and the Fine Structure Constant[END_REF][START_REF] Hanneke | Cavity Control of a Single-Electron Quantum Cyclotron: Measuring the Electron Magnetic Moment[END_REF]. Anomalous magnetic moment: [START_REF] Sherbon | Wolfgang Pauli and the Fine-Structure Constant[END_REF][START_REF] Schwinger | On Quantum Electrodynamics and the Magnetic Moment of the Electron[END_REF][START_REF] Pfister | The Gyromagnetic Factor in Electrodynamics, Quantum Theory and General Relativity[END_REF].

a -1 e = 2/(g -2) ≃ 26(34 -µ -1 ), (13)
µ = coth µ = √ λ 2 + 1 ≃ λ √ S ≃ π/ϕ 2 . ( 14
)
Laplace limit of Kepler's equation λ ≃ 0.662 743 419 349 181 and "... the Laplace limit is the maximum value of the eccentricity for which the series solution to Kepler's equation converges." [START_REF] Weisstein | Laplace Limit[END_REF]. The sin α -1 ≈ µ -2 , µ ≈ 12 √ κ, λ ≈ 12 √ α, α ≃ λ/µκ 2 and µ ≃ κ/πD; see the Pythagorean references in [START_REF] Sherbon | Wolfgang Pauli and the Fine-Structure Constant[END_REF]. Also, µ/λ ≃ √ S and S ≃ √ 2D. ϕ -2 ≃ γλ and µ ≃ cosh(ϕ -1 ) ≃ cosh(π/5) ≃ 6/πϕ ≃ DS/2π, see Eq. ( 6) discussion. The coth x also appears in the Brillouin function and the Langevin function, L(x) = coth x -x -1 , in the statistical mechanics of magnetic moments [START_REF] Greiner | Thermodynamics and Statistical Mechanics[END_REF]. L(µ) ≃ R/π ≃ µ/S ≃ 1/e. R is the radius of the regular heptagon with side equal to one. The golden rhombus has angles similar to Pauli's World Clock geometry. A golden rhombus of unit edge length has long diagonal length 2ϕ/ √ ϕ √ 5 ≃ D 2 /π ≃ √ 2µ ≃ (8/9)(6/π), with the squared-circle ratio.

λ = csch µ = √ µ 2 -1 ≃ µ/ √ S ≃ 1/γϕ 2 , ( 15
)
showing the Laplace limit of Kepler's equation as the hyperbolic cosecant [START_REF] Weisstein | Hyperbolic Cosecant[END_REF]

of µ. λ ≃ γR ≃ sinh(π/5) ≃ ρϕ ≃ γ cosh(1/ √ π). µ ≃ ζ(3) ≃ γ/ ln ϕ ≃ π/ϕ 2 ≃ πγλ ≃ ρκγ 2 ≃ λ csc(π/7
) and ρκ ≃ 2µ/λ. The 7/5 ratio, relevant to Pauli's World Clock geometry, is the approximate outer radius of the dodecahedron. The golden ratio, ϕ ≃ 7π/5e ≃ 5e/7µ and µ ≃ S tanh(e/7) ≃ S/e, see Eq. ( 24) discussion. Also, the sinh ϕ ≃ 5ϕ f /7≃ 2µ and ϕ f is the reciprocal Fibonacci constant [START_REF] Sloane | The On-Line Encyclopedia of Integer Sequences[END_REF], see the parabolic constant following Eq. ( 24).

The Laplace limit of Kepler's equation and a 5th degree polynomial having coefficients with a factor of 5 also gives another good approximation for the fine-structure constant: α -1 ≃ ηπλ, where η is a root of ( 16)

x 5 -65x 4 -55x 3 + 80x 2 -5x + 40 = 0. ( 17
)
The value for α -1 ≃ 137.035 999 168 and root η ≃ 65.817 195 651 9 for Eq. ( 16). The eccentricity of a golden ellipse ε = 1/ √ ϕ ≃ sin(2π/7) ≈ µλ, see discussion of Eq. ( 10).

µ -1 ≃ 2ρ ≃ 2/ √ κλ and ρϕ ≈ λ. Also, λ -2 ≃ 1 + √ ϕ ≃ √ πϕ ≃ ϕ 2 /R. With x = α/2π: g/2 ≃ 1 + x -(ϕx/ √ 2) 2 + (x/γ) 3 -(λx) 4 + (ϕx) 5 , ( 18
)
with alpha value from Eq. ( 2), g/2 ≃ 1.001 159 652 180 69. ϕ/ √ 2 ≃ 3/L ≃ 3µ/π ≃ ln π where L is the lemniscate constant [START_REF] Weisstein | Lemniscate Constant[END_REF]. L ≃ π/µ ≃ 8 √ κ/9, the 8/9 ratio relates to the classical construction of squaring the circle; see Eq. ( 14) discussion and references in [START_REF] Sherbon | Wolfgang Pauli and the Fine-Structure Constant[END_REF]. Also, λ ≃ 1/µ √ ϕ. Eq. ( 18) involves a convergence of Julian Schwinger's work with Green's functions [START_REF] Schwinger | On Quantum Electrodynamics and the Magnetic Moment of the Electron[END_REF]. Julian Schwinger introduced α/2π to the problem of the anomalous magnetic moment of the electron in quantum electrodynamics [START_REF] Schwinger | On Quantum Electrodynamics and the Magnetic Moment of the Electron[END_REF]. In the dodecahedron related to Pauli's World Clock geometry, from the center to the mid-edge the mid-radius equals ϕ 2 /2 = (3 + √ 5)/4 ≃ 34/26 ≃ πρ [START_REF] Heyrovska | Golden Ratio Based Fine Structure Constant and Rydberg Constant for Hydrogen Spectra[END_REF], see Eq. ( 6). The silver constant

S = 2 + 2 cos(2π/7) ≃ 1 + √ πϕ ≃ √ 2D ≃ 2L/ϕ ≃ 2 ln(πϕ) ≃ eµ.
A golden ellipse of unit length minor radius and major radius of length ϕ has an area equal to πϕ [START_REF] Huntley | The Golden Ellipse[END_REF] and the sin α -1 ≃ 6γ/πϕ with the cube-sphere ratio, see the discussion of Eq. ( 6). coth α ≃ τ, where τ is a root of ( 19)

x 4 -137x 3 -6x 2 -98x + 343 = 0, (20)
also gives the value for α -1 ≃ 137.035 999 168 with Eq. ( 19) and the value for τ ≃ 137.038 431 610. The factor of 7 also appears as 2 × 7 2 = 98 and 7 3 = 343. Eq. ( 20) represents 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. Some forms for Gauss's constant [START_REF] Weisstein | Gauss's Constant[END_REF]:

G = L/π = 2 π ˆ1 0 (1 -x 4 ) -1 2 dx = ˆ∞ 0 (cosh(πx)) -1 2 dx. ( 21
)
Lemniscate constant L = πG ≃ 2.622 057 554 292 119 ≃ cosh ϕ and G ≃ 0.8346 is Gauss's constant, the reciprocal of the arithmetic-geometric mean of 1 and √ 2, the basis for his exploration of the lemniscate function [START_REF] Weisstein | Lemniscate Constant[END_REF]. Gauss's constant is also linked with the beta function, the gamma function at argument 1/4 and Jacobi theta functions.

G -1 ≃ µ ≃ λ √ S ≃ πϕ/3 √ 2, see Eq. (14) above. The sin α -1 ≃ D i G. D i = √ 6/3
, the diameter of the inscribing sphere of the octahedron. 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 [START_REF] Weisstein | Lemniscate Constant[END_REF]. The lemniscate L ≃ Sϕ/2 ≃ cosh ϕ ≈ 1/γλ ≈ ϕ 2 . With coefficients 3/L, Lλ, 5/L and S in place of ϕ/ √ 2, 1/γ, λ and ϕ there is for x = α/2π:

g/2 ≃ 1 + x -(3x/L) 2 + (Lλx) 3 -(5x/L) 4 + (Sx) 5 . ( 22
)
With alpha value from Eq. ( 2), g/2 ≃ 1.001 159 652 180 69. Also, ρκ ≃ L + 1, see Eq. ( 8).

Ratios of fundamental physics

Some fundamental mass ratios have approximations with hyperbolic functions. The 80.4

GeV W boson from electroweak theory and 125 -126 GeV Higgs boson [START_REF] Beringer | Review of Particle Physics -Gauge and Higgs Bosons[END_REF], m w /m H • ≃ ν/µ ≃ 2/π, see Eq. ( 28). The fixed point of the hyperbolic secant ν = sech ν, Eq. ( 27). Higgs boson and the 136.5 GeV Higgs-like resonance [START_REF]Properties of the Observed Higgs-like Resonance using the Diphoton Channel[END_REF],

m H • /m H * ≃ νµ ≃ 1/ √ µ.
m z /m H • ≃ (λ + 1/2)/ϕ. m H • /m z ≃ m t /m H • ≃ κ/2π. ( 23
)
Again, κ is the reciprocal of the Kepler-Bouwkamp constant [START_REF] Finch | Kepler-Bouwkamp Constant[END_REF], see Eq. ( 5). Also, the Higgs boson and 172.9 GeV top quark from quantum chromodynamics,

m t /m H • ≃ πλ 2 ≃ √ 6/π ≃ γ/ρ ≃ 7/πϕ ≃ √ µϕ ≃ µR ≃ ρκ/ϕ 2 ≃ 1 + ϕ -2
, see Eq. ( 6) discussion and Eq. ( 8). The Weinberg angle of the electroweak interaction cos θ W = m w /m z ≃ √ ϕ/µ 2 . The proton/electron mass ratio [START_REF] Mohr | CODATA Recommended Values of the Fundamental Physical Constants[END_REF]: m p /m e ≈ exp(2πµ) and 2µ ≃ 2π/ϕ 2 with the golden angle in radians [START_REF] Weisstein | MathWorld-A Wolfram Web Resource[END_REF][START_REF] Munteanu | From Golden Spirals to Constant Slope Surfaces[END_REF]. m p /m e ≃ exp(σπ)/α ≃ 1836.152 672 37 where σ ≃ 0.826 072 743 917 8 is a root of 13x 3 + 28x 2 + 37x -57. This is compared to the latest determination of m p /m e = 1836.152 672 45 (75) [START_REF] Mohr | CODATA Recommended Values of the Fundamental Physical Constants[END_REF]. Also, σ ≈ µ -1 and σ 2 ≈ sin α -1 . Suggesting a squared-circle for classical electron radius/charge radius of the proton [START_REF] Antognini | Proton Structure from the Measurement of 2S-2P Transition Frequencies of Muonic Hydrogen[END_REF]:

r e /r p ≃ S sec(1/4). ( 24
)
Radius r e = αλ e /2π ≃ 2.81794 × 10 -15 m, where λ e ≃ 2.4263 × 10 -12 m is the Compton wavelength of the electron. Eq. ( 24) gives r p ≃ 0.84089 fm and the latest experimental result is r p ≃ 0.84087 [START_REF] Nucci | The Harmony in the Kepler and Related Problems[END_REF] fm [START_REF] Antognini | Proton Structure from the Measurement of 2S-2P Transition Frequencies of Muonic Hydrogen[END_REF]. The ratio r e /r p ≃ πDS/7, πD is the circumference of a regular heptagon with side equal to one and D = 2R. The ratio r e /r p ≃ coth

S -1 ≃ 7µ/ √ 2π ≃ π/ √ 2λ ≃ λπϕ. Also, R ≃ L/ √ πϕ ≃ 5ϕ/7 ≃ π/e,
another approximation with Bernoulli's lemniscate constant. Other suggestive numerical coincidences r p /r e ≃ δ s /5ϕ ≃ tanh S -1 ≃ tan 2 (1/2), where δ s is the silver ratio [START_REF] Weisstein | MathWorld-A Wolfram Web Resource[END_REF].

δ s = 1+ √ 2 = cot(π/8) ≃ 2µ and cosh(ln δ s ) = √ 2.
The silver ratio is the inner diameter of a regular octagon with side equal to one. The ln δ s ≃ √ ϕ/µ 2 . The universal parabolic constant [64] P = √ 2 + ln δ s ≃ S/ √ 2 and sin α -1 ≃ P /ϕ f , where ϕ f is the reciprocal Fibonacci constant [START_REF] Sloane | The On-Line Encyclopedia of Integer Sequences[END_REF]. Also, ϕ f ≃ λπϕ and ln(α -1 ) ≃ S √ P . The ln δ s is an inflection point of sech x, the hyperbolic secant function [START_REF] Weisstein | MathWorld-A Wolfram Web Resource[END_REF], which is a special case of the Jacobi elliptic function.

The fundamental length g M is from the work of Mendel Sachs [START_REF] Sachs | Quantum Mechanics from General Relativity[END_REF] on the spinorquaternion formulation for the wave function of the electron in hydrogen and prediction for the Lamb shift of the hydrogen spectrum. Sachs presents another viewpoint on the anomalous magnetic moment of the electron, developing a continuous field concept without assuming point charges and their problems with infinities. The explanation has some correlations with the golden ratio geometry of Wolfgang Pauli's World Clock [START_REF] Sherbon | Wolfgang Pauli and the Fine-Structure Constant[END_REF].

g M /r e ≃ 2πϕα. (25)
g M /λ e ≃ ϕα 2 and g M ≃ 2.087×10 -16 m. Also, g M /r e ≃ 2π/85 and the sin α -1 ≈ 2π/ √ 85. The omega constant [START_REF] Weisstein | MathWorld-A Wolfram Web Resource[END_REF] is defined as Lambert W (1) [START_REF] Valluri | Some Applications of the Lambert W Function to Physics[END_REF][START_REF] Weisstein | Lambert W-Function[END_REF], the attractive fixed point of e -x : W

(1) = Ω = exp(-Ω) ≃ 0.567 ≃ 1/ √ π ≃ ϕ 2 /2D and 17/13 ≃ πρ ≃ ϕ 2 /2 ≃ DΩ. Ω = exp(-Ω) ≃ sinh -1 (µ/2). (26)
W (x), the Lambert W -function, is an analog of the golden ratio for exponentials as exp[-W (1)] = W (1). The sin α -1 ≈ 6Ω/πϕ. The cosh Ω ≃ λ/Ω and the csch ν ≃ µ, where µ again is equal to the coth µ and ν is the real fixed point of the hyperbolic secant: ((e x -x) 2 + π 2 ) -1 dx. [START_REF] Weisstein | Prime Constant[END_REF] Also, m w /m H • ≃ ν/µ ≃ √ ϕ/2 ≃ 2/π, an approximation with the golden ratio geometry. From the Foundation Stone of classical harmonic theory, the alpha harmonic is equal to the sum of the golden ratio harmonic and the omega constant harmonic, relating the Greek Pythagorean form of the fine-structure constant to the golden ratio geometry [START_REF] Sherbon | Wolfgang Pauli and the Fine-Structure Constant[END_REF].

ν = sech ν ≃ 1/DΩ ≃ µ √ ϕ/2. ( 27

Conclusion

From Arnold Sommerfeld's introduction of the fine-structure constant and Wolfgang Pauli's search for an explanation, approximate values for the fine-structure constant have been determined. With an extension of the Keplerian intuition regarding the fundamental geometry of basic polygons, conic sections and Platonic polyhedra included in Wolfgang Pauli's World Clock geometry; the mathematical and physical model for the calculations is an alternative to accounting for individual contributions of the interactions between field quanta and begins to address some of the questions raised by Richard Feynman, Freeman Dyson, Paul Dirac and others about quantum electrodynamics [START_REF] Schweber | QED and the Men Who Made It: Dyson, Feynman, Schwinger and Tomonaga[END_REF].

The polygon circumscribing constant, reciprocal of the Kepler-Bouwkamp constant, has interesting geometric connections with the torus topology relating the relativity of Einstein with the geometry of classical and quantum mechanics. The nature of this topology was pursued by Wolfgang Pauli and is suggested in the Keplerian paradigm of Pythagorean harmonic proportions from ancient geometry [START_REF] Sherbon | Wolfgang Pauli and the Fine-Structure Constant[END_REF]. Quantum mechanics is thus found to be an approximation theory based on the classical mathematical problem of squaring the circle, with all of its complex analogies and philosophical implications.

  23 = 337. 23 + 37 = 60 and 60/ϕ ≃ 37[START_REF] Sherbon | Wolfgang Pauli and the Fine-Structure Constant[END_REF]. The triangles 85, 132, 157 and 175, 288, 337 are primitive Pythagorean triples. Defining ρ for p(k) as the k-th prime:

	proportional to 1, ϕ, √	ϕ	related to "Pauli's triangle" with sides approximately 5 with the golden ratio ϕ = (1 + √ √ 5)/2 [30]. 180 -23 = 157
	and 360 -

  The 91.2 GeV Z boson from electroweak theory and Higgs boson,

  With the Weinberg angle of the electroweak interaction cos θ W = m w /m z ≃ 1/2Ω ≃ λ/ν. From Adamchik's integral for the omega constant[START_REF] Weisstein | MathWorld-A Wolfram Web Resource[END_REF] the value is(1 + Ω) -1 ≃ R/ √ S ≃ 0.638.

) Again, csch ν ≃ µ and ν ≃ 13/17 ≃ δ s /π ≃ 1 -ζ(2)/7 ≃ 1/πρ ≃ 2/ϕ 2 ≃ 0.765. ν/µ ≃ (1 + Ω) -1 = ˆ+∞ -∞

Acknowledgments

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