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How YAHWEH and the Decad determine the geometrical composition of the Polyhedral Tree of Life

Imagine the 144 Polyhedron and the disdyakis triacontahedron constructed from triangles. When their vertices are joined to their centres, each of their edges is the side of an internal triangle one of whose corners is located at the centre of each polyhedron. Using their data below:

144 Polyhedron:                       Number of vertices = 74 (72 surrounding an axis that comprises three corners & two sides); number of edges = 216; number of triangular faces = 144;
Disdyakis triacontahedron:      Number of vertices = 62 (60 surrounding an axis that comprises three corners & two sides); number of edges = 180; number of trIangular faces = 120,

the following table lists the corners, sides & triangles making up the two polyhedra:

 Corners Sides Triangles Total 144 Polyhedron 72 + 3 = 75 216 + 72 + 2 = 288 + 2 = 290 144 + 216 = 360 720 + 5 = 725 Disdyakis triacontahedron 60 + 3 = 63 180 + 60 + 2 = 240 + 2 = 242 120 + 180 = 300 600 + 5 = 605 Total 132 + 6 = 138 528 + 4 = 532 264 + 396 = 660 1320 + 10 = 1330

We see that:

• the two polyhedra have 660 triangles with 138 corners. 660 = 66×10, where 66 is the 65th integer after 1. ADONAI, the Godname of Malkuth with number value 65, prescribes the number of triangles needed to shape the Polyhedral Tree of Life. The number of corners & sides surrounding both axes is also 660. The number 138 is the sum of the 10 integers on the sloping edges of the Tetrahedral Lambda (see here). It is also the number of yods outside the shared root edge of the two Type A dodecagons that are part of the inner Tree of Life. These two facts constitute evidence of the holistic nature of the 144 Polyhedron and the disdyakis triacontahedron because the two dodecagons are the single, polygonal counterpart of the outer Tree of Life, whilst the Tetrahedral Lambda is the arithmetic counterpart of the inner form of 10 overlapping Trees of Life (see here). A more direct indication of this is the fact that, of the 80 yods making up the 1-tree, six yods (coloured black in the diagram below) are Sephiroth on its side

 Disdyakis triacontahedron 144 Polyhedron

pillars that coincide with corners of polygons, namely, the pair of triangles and the pair of hexagons. This leaves 74 blue yods that are intrinsic to the 1-tree. Of the 70 corners of the (7+7) enfolded polygons, (70−6=64) corners are, likewise, intrinsic to them, where 64 is the number value of Nogah, the Mundane Chakra of Netzach. Hence, there are (74+64=138) yods or corners that are unshared between the outer and inner Trees of Life. They symbolize the 138 corners of the 660 triangles needed to construct the 144 Polyhedron and the disdyakis triacontahedron. The 20 red corners of the pair of dodecagons outside the root edge denote the 20 C vertices of the disdyakis triacontahedron, the 30 purple corners of the two squares, pentagons, hexagons & decagons outside the root edge denote its 30 A vertices and the 12 green corners of the two octagons outside the root edge denote its 12 B vertices. Kether, Tiphareth, Yesod & Malkuth denote the four vertices of both polyhedra at the ends of their axes, the centres of which are appropriately denoted by the two endpoints of the root edge (the upper endpoint corresponds to the centre of the 144 Polyhedron and the lower one corresponds to the centre of the disdyakis triacontahedron);

• the two separate polyhedra forming the Polyhedral Tree of Life have (138+532=670=67×10) corners & sides. In other words, the number value 67 of Binah, the Sephirah embodying all archetypes governing the form of the Tree of Life, determines the number of points and lines that make up its polyhedral version;
• the number of corners, sides & triangles in the two separate polyhedra = 1330

12
32  52
=        72   92  112
132 152 172 192 .

This beautiful property shows how the Decad symbolized by the tetractys determines the geometrical composition of the Polyhedral Tree of Life, for the largest odd integer in the array is 19, which is the tenth odd integer. Moreover, as pointed out in the previous comment, the two separate polyhedra have 67×10 corners & sides, where 67 is the 19th prime number!;

• the number of geometrical elements surrounding the axes of the two separate polyhedra = 1320. This is the number of yods in 26 overlapping Trees of Life whose triangles are tetractyses.* YAHWEH, the Godname of Chokmah with number value 26, prescribes the composition of the geometrical elements surrounding the axes of the two polyhedra constituting the Polyhedral Tree of Life. In fact, it prescribes all its 1330 geometrical elements, for the number of yods in the n-tree whose triangles are tetractyses = 50n +30,** so that the 26-tree has 1330 yods. The 10 extra yods needed to turn 26 overlapping Trees of Life into the 26-tree correspond to the 10 corners & sides that make up the axes of the 144 Polyhedron and the disdyakis triacontahedron;
• if we consider one polyhedron lying inside the other and sharing the same centre (for present considerations, it makes no difference which one is regarded as inside the other), their 660 triangles have 137 corners. This demonstrates how the number 137 determining the fine-structure constant α = e2/ħc ≅ 1/137 (see below) is naturally embodied in the basic geometry of the Polyhedral Tree of Life. This is not surprising, because, as indicated in the section The holistic pattern, the number 137 is a defining parameter of holistic systems. If the two polyhedra are orientated so as to share the same axis as well, (137−5=132) corners surround the latter, as well as (532−4=528=4×132) sides and 660 triangles, which comprise (144+120=264=2×132) triangular faces and (216+180=396=3×132) internal triangles. Amazingly, the 1320 geometrical elements that surround their axes (whether shared or not) can be represented by the tetractys array of the integer 132:

132
132  132
1320 =       132  132  132
132   132   132   132.

The apex denotes the number of corners, the second row denotes the number of faces, the third row denotes the number of internal triangles and the fourth row denotes the number of sides! The crucial property that makes this possible is that the number of vertices of the two polyhedra that surround their axes is 132, which happens to be exactly half the total number (264) of their faces. This can be seen by writing Euler's formula for two polyhedra P1 & P2 with, respectively, C1 & C2 vertices, E1 & E2 edges and F1 & F2 faces:

C1 − E1 + F1 = 2

and

C2 − E2 + F2 = 2.

C1 + C2 − (E1 + E2) + F1 + F2 = 4.

Then, as F1 + F2 = 264, C1 = c1 + 2 and C2 = c2 + 2, where c1 & c2 are the numbers of vertices surrounding the axes of P1 & P2,

E1 + E2 = c1 + c2 + 264.

If c1 + c2 = 132, i.e., half the number of faces of P1 & P2, then E1 + E2 = 396 = 3×132, so that the number of sides of the triangles surrounding their axes = E1 + c1 + E2 + c2 = (E1 + E2) + c1 + c2 = 3×132 + 132 = 4×132. ADONAI prescribes this common factor of the various numbers determining the geometrical composition of the Polyhedral Tree of Life because 132 is the 65th even integer after 2.

In one of his books, the great physicist Richard Feynman said:

"There is a most profound and beautiful question associated with the observed coupling constant, e — the amplitude for a real electron to emit or absorb a real photon. It is a simple number that has been experimentally determined to be close to -0.08542455. (My physicist friends won't recognize this number, because they like to remember it as the inverse of its square: about 137.03597 with about an uncertainty of about 2 in the last decimal place. 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.) Immediately you would like to know where this number for a coupling comes from: is it related to pi or perhaps to the base of natural logarithms? Nobody knows. It's one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the "hand of God" wrote that number, and "we don't know how He pushed his pencil." We know what kind of a dance to do experimentally to measure this number very accurately, but we don't know what kind of dance to do on the computer to make this number come out, without putting it in secretly! ........Richard P. Feynman (1985). QED: "The Strange Theory of Light and Matter". Princeton University Press. p. 129.

The magnitude of the coupling constant α = e2/ħc for the electromagnetic field has yet to be calculated from first principles. The closeness of its reciprocal to the number 137 has led physicists to speculate that this number has a place in a "grand, unified theory" that truly unifies the forces of nature. We can now explain why it should do so. Like numbers such as 248 and 496, which are encountered in superstring theory, the number 137 is a primary parameter of holistic systems. For example, it is:

1. the number of tetractyses whose yods fill up the inner form of the Tree of Life (see #5);
2. the number of sides of basic triangles in the 144 Polyhedron constructed from Type A triangles (#5);
3. the number of yods on average that are needed to construct from tetractyses the faces of the Platonic solids that ancient philosophy associated with the Elements Earth, Water, Air & Fire (see here);
4. the number of points marking out in space the two polyhedra that constitute the Polyhedral Tree of Life (see above);
5. the sum of the powers of 2, 3 & 4 on three edges of the Tetrahedral Lambda, the tetrahedral array of 20 integers that is the 3-dimensional generalisation of Plato's "Lambda" (see here).

There is, therefore, no more mystery about why the "magic number" 137 is present in quantum electrodynamics, for it is a characteristic parameter of all sacred geometries (see here under the heading "137") embodying the divine blueprint, as manifested in the physical universe of matter and its forces — in particular, the electromagnetic field described by QED. This statement is not a speculation about the metaphysical status of the number 137 based upon religious dogma or New Age philosophy. Instead, it is a hard, mathematical fact, as provable as any theorem of Euclid's geometry. Contemplate this, marvel at it and reflect on its profound implications. Only if your humanistic or materialistic philosophy has no room for the notion of the 'sacred,' as manifested in sacred geometries, will you continue — like Prof. Feynman and "all good theoretical physicists" — to worry about it.

* Proof: the number of yods in n overlapping Trees of Life whose (12n+4) triangles with (6n+4) corners and (16n+6) sides are tetractyses = 6n + 4 + 2×(16n+6) + 12n + 4 = 50n + 20. Therefore, 26 overlapping Trees have 1320 yods.

** Proof: the number of yods in the n-tree whose (12n+7) triangles are tetractyses with (6n+5) corners and (16n+9) sides = 6n + 5 + 2×(16n+9) + 12n + 7 = 50n + 30. Therefore, the 26-tree has (50×26 + 30 = 1330) yods.

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