The “pauli Triangle”

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.​