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 [12]. The “Golden Chains of Homer extending from the Central Ring of Plato” [17] in the tablet divides the mandala into the golden angle and forms a triangle that also intersects the Philosopher’s Stone [17, 18]. 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 [19], with parallels to Walter Russell’s cosmogony [20]. 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 [21] 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 [17, 18]. 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.” [11]. 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 [7]. 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 [3, 22]. 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.”’ [23]. 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.” [24]. Pauli also worked with Marie-Louise von Franz, who wrote in Number and Time, “Numbers, furthermore, as archetypal structural constants of the collective 3 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.” [25]. 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.” [26]. 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 [27] and Franklin Merrell-Wolff [28]. 4. 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 [29]) and α −1 = 137.035 999 173 3 (344) (T. Kinoshita [30]). 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 [31], and Plato’s favorite 5040 = 7!, of the larger harmonic proportion [8]. 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 [32], 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 [33], 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/e2 ' 5/3 √ 2 ' sec 32◦ , with the angle from “Pauli’s triangle” found above. 180−23 = 157, and 360 − 23 = 337. 23 + 37 = 60, 60/φ ' 37, and φ ≡ (1 + √ 5)/2. see [34]. 37 + 120 = 157. 23 + 85 = 108, proportional to the Moon radius of the Cosmological Circle. 72 + 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. 62+72 = 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. Other related approximations include 1+ρ ' √ 2, cos 32◦ ' 2ρ, 5ρ ' φf /φ where φf is the reciprocal Fibonacci constant [34], and 360/φ3 ' 85. Also, α ' ρ/32√ π. The outer radius of the regular dodecahedron (√ 3 + √ 15)/4 ' γ/ρ where γ is the Euler-Mascheroni constant, see Eqs. (4) and (5). https://hal.archives-ouvertes.fr/…/Wolfgang_Pauli_and_the_F… ps.pdf

Accurate to seven places, sin α −1 ' 8δ/7, where δ is Gompertz constant (or the EulerGompertz constant, which can also be expressed in relation to the Euler’s number [37]), δ ≡ −eEi(−1) ' φ/e, where Ei is the exponential integral. Returning to the polygon circumscribing constant, κ 2 ' 76, κ2 + π 2 ' 85, sin α −1 ≈ 32φ/κ2 , and sin α −1 is the approximate ratio of the 32◦ × 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. (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 [8]. The PDG [38] value for the strong coupling constant αs ' 0.1184 (7) is proportional: αs/α ' κ/πρ2 . αs ' 1 − κ/π2 ' sec 32◦/π2 . κρφ2 ' φ/ρ2 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 ...”

[39]. “The Teleois proportions are used by the creative force because they best fit the electromagnetic energy fields of the atom.” [40]. As part of a series based on modulo 3 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 31 – represented by a circle – six other Teleois circles exactly contact or intersect each other in perfect Teleois proportions.” [39]. 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 1082 (“a number of extraordinary interlocking potential” determining the root tone generators of his Fibonacci-harmonic Quadrispiral and also found in the Great Pyramid proportions) [10]. 11, 664 is also the proportional harmonic of α/2π, equal to the classical electron radius divided by its Compton wavelength.