#34 The Tetrad expresses the yod population of the inner Tree of Life with Type B polygons

When they are Type B polygons, the (7+7) enfolded polygons of the inner Tree of Life have 1370 yods, where

and 137* is the nearest integer approximation to the reciprocal of the fine-structure constant α = e²/ħc ≅ 1/137. This is the number of yods in 27 overlapping Trees of Life,** whose highest Malkuth (Kether of the 25th Tree) is the 155th SL prescribed by ADONAI MELEKH, the Godname of Malkuth with number value 155. The significance of the 25-tree is that, when Trees are interpreted as dimensions of space, it is the Tree of Life map of the 25 dimensions of space predicted by quantum mechanics for spinless strings — hence the association with Malkuth, the final Sephirah, which denotes the material form of the Tree of Life. Its 10-tree section has 65 SLs prescribed by ADONAI because its number value is 65.

The two side pillars of the outer Tree of Life are the vertical axes of the two hexagons of the inner Tree of Life. This means that the 14 yods lining these two axes are shared with the outer Tree when its 16 triangles are tetractyses. One of the seven yods on each axis is a corner of the triangle that is the centre of the hexagon. There are (7+6=13) yods in each set of seven enfolded polygons that are either shared or their centres. Hence, (2×13=26) yods in the (7+7) enfolded polygons are either their centres or shared. This is how YAHWEH, the Godname of Chokmah with number value 26, quantifies these two types of yods. The top corners of the two hexagons coincide with the bottom corners of the two hexagons in the 14 polygons that are enfolded in the next higher Tree of Life. 24 centres or shared yods are intrinsic to the 14 polygons enfolded in each Tree, so that the number of centres or shared corners in the 14n polygons enfolded in n Trees of Life = 24n + 2. The Tetrad determines the number 24 because 24 = 4! = 1×2×3×4 = 3 + 5 + 7 + 9 (Fig. A). It is the number of yods surrounding the centre of a Type A square, its four corners symbolising the Tetrad.

The number of yods in the 14n Type B polygons enfolded in n Trees of Life = 1368n + 2. They include

1368n + 2 − (24n+2) = 1344n

yods that are intrinsic to these polygons and surround their centres. Intrinsic to the set of 14 polygons enfolded in each Tree are 1344 yods that surround their centres. This is the number generated by a cross pattée (Fig. A) whose four arms are 2nd-order tetractyses, each of the 336 yods surrounding its centre being weighted with the number 4 (Tetrad): 1344 = 336×4.

The Tetrad arithmetically expresses the number 1344 because

1344 = 4² + 12² + 20² + 28² = 4³ + 4⁴ + 4⁵.***

As

1370 = 26 + 1344

and

26 = 10 + 16,

where 16 = 4²,

1370 = 10 + 4² + 4³ + 4⁴ + 4⁵ = 1 + 2 + 3 + 4 + 4²(4¹+4²+4³+4⁴) = 1 + 2 + 3 + 4 + 85×4²,

where 85 is the number of yods in the 2nd-order tetractys:

This demonstrates how the Tetrad, symbolised by the square, represents through the 2nd-order tetractys representation of a holistic system the yod population of the inner Tree of Life with Type B polygons. The number of yods other than the six yods located at the Sephirothic corners (denoted by • in Figure B) of triangles in the Tree of Life = 1370 − 6 = 1364 = 4¹ + 4² + 4³ + 4⁴ + 4⁵. This beautiful property of the inner Tree of Life can be represented by a pentagram with the first five powers of 4 assigned to its points (Fig. B). The first power of 4 (4¹) denotes the four yods in the root edge. Outside it are (4²+4³+4⁴+4⁵) non-Sephirothic yods. The number of such yods in each set of seven enfolded polygons = 2³ + 2⁵ + 2⁷ + 2⁹ = 680. Noting that two yods in the root edge can be associated with each set of seven polygons, the total number of non-Sephirothic yods associated with each set = 2 + 680 = 682 = 2¹ + 2³ + 2⁵ + 2⁷ + 2⁹. The total number of yods outside the root edge in each set of polygons = 3 + 680 = 683 = 2⁰ + 2¹ + 2³ + 2⁵ + 2⁷ + 2⁹. This can be represented by a pentagram array of the first five odd powers of 2, with 2⁰ at its centre denoting the sole yod in each set of seven polygons (the top of the hexagon) that is shared with the set of polygons enfolded in the next higher Tree of Life.

Such a mathematical pattern of powers of 2 can leave no doubt about the transcendental status of the geometrical object called the "inner form" of the Tree of Life.

As

 683 = 2⁰ + 2¹ + 2³ + 2⁵ + 2⁷ + 2⁹,

the number of yods outside the root edge in both sets of polygons = 2×683 = 2¹ + 2² + 2⁴ + 2⁶ + 2⁸ + 2¹⁰ = 2 + 4¹ + 4² + 4³ + 4⁴ + 4⁵. Including the four yods in the root edge, the total number of yods = 1370 = 4 + 2 + 4¹ + 4² + 4³ + 4⁴ + 4⁵. This can be represented by a pentagram array of the first five powers of 4 enclosed by a square with yods just at its corners and with the number 2 at its centre representing the two yods that are shared with the inner form of the next higher, overlapping Tree of Life, namely, the apices of the two hexagons:

The yods at the corners of the square correspond to the four yods in the root edge. Alternatively, they are the four yods in addition to the apices of the two hexagons that are shared with the outer Tree of Life as Sephirothic corners of triangles. This basic geometrical structure of a square enclosing a pentagram is the archetypal geometry of the Godname ADONAI (אדני), the values of its letters being the numbers of different types of yods (see diagram above). The sum of the first two powers of 4 = 4¹ + 4² = 20. This is the number of yods that are either centres of the polygons or yods shared with the outer Tree of Life. The sum of the next three powers of 4 = 4³ + 4⁴ + 4⁵ = 1344. This is the number of yods intrinsic to the inner Tree of Life that surround the centres of its polygons. Its significance for the 4₂₁ polytope, whose 240 vertices represent the 240 roots of the exceptional Lie group E₈ at the heart of E₈×E₈ heterotic superstring theory, is discussed in the section 4-d sacred geometries.

We found above that 1370 = 10 + 4² + 4³ + 4⁴ + 4⁵ = 10 + 1360. The Decad is the sum of the first four positive integers:

10 = 1 + 2 + 3 + 4

symbolised by the 1st-order tetractys, whilst 1360 is, remarkably, the sum of the numbers of Malkuth (496), its Godname ADONAI (65), its Archangel Sandalphon (280), its Order of Angels Ashim (351), and Cholem Yesodoth (168), its Mundane Chakra:

 
1360 is the yod population of the inner Tree of Life outside the root edge and excluding the six Sephirothic points shared with its outer form. The Tetrad expresses this parameter as the sum of the powers of 4², 4³, 4⁴ & 4⁵. Also, 1360 = 136×(1+2+3+4), where

All these properties are amazing signs of the beautiful design inherent in the outer and inner forms of the Tree of Life. The number 136 is the number of yods outside the root edge that surround the centres of the two joined dodecagons when they are Type A:

 As 1370 = 10 + 136×10 and the dodecagon is the 10th type of regular polygon, we see how the Decad determines the yod population of the inner form of the 10-fold Tree of Life.

* According to the well-known physicist Max Born: "If alpha were bigger than it really is, we should not be able to distinguish matter from ether [the vacuum, nothingness], and our task to disentangle the natural laws would be hopelessly difficult. The fact however that alpha has just its value 1/137 is certainly no chance but itself a law of nature. It is clear that the explanation of this number must be the central problem of natural philosophy." (Quoted by A.I. Miller in his Deciphering the Cosmic Number: The Strange Friendship of Wolfgang Pauli and Carl Jung (W.W. Norton & Co. (2009)), p. 253). The number 137 is one of the parameters that define the holistic paradigm, as expressed in the sacred geometries of the mystical traditions of religions analysed in this website. Its presence in physics is no longer a mystery once the holistic nature of matter is recognised. See the entry under "137" in The holistic pattern.
** Proof: the number of yods in n Trees of Life ≡ Y(n) = 50n + 20 (see Table 7 here). Therefore, Y(27) = 1370.
*** Proof: 1344 = 4×336 = 4×4×84, where

84 = 1² + 3² + 5² + 7² = 4¹ + 4² + 4³.

Therefore,

1344 = 4²×84 = 4²(1²+3²+5²+7²) = 4² + 12² + 20² + 28² = 4²(4¹+4²+4³) = 4³ + 4⁴ + 4⁵.

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