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The Tetrahedral Lambda as the arithmetic counterpart of the inner Tree of Life

Imagine the 47 sectors of the seven enfolded polygons of the inner Tree of Life turned into tetractyses (Fig. 6). They have 88 sides lined by (88×2=176) hexagonal yods. Table 2 displays their numbers in each polygon. 174 hexagonal yods are outside the two hexagonal yods on their shared root edge. The two sets of seven enfolded polygons have (2 + 2×174 = 350) hexagonal yods. This is the inner Tree of Life counterpart of the Tetrahedral Lambda whose 20 integers add up to 350.

Table 2. Numbers of hexagonal yods on sides of tetractys sectors of the 7 enfolded polygons.

 Triangle Square Pentagon Hexagon Octagon Decagon Dodecagon Total Hexagonal yods on sides of polygon 2+4=6 6 8 10 14 18 22 84 Internal hexagonal yods on sides of tetractyses 6 8 10 8 16 20 24 92 Total = 12 14 18 18 30 38 46 176

Figure 6. 176 hexagonal yods line the 88 sides of the 47
tetractyses making up the seven enfolded, regular polygons.

The sum of the 10 integers in the first face is 90 and the sum of the 10 integers in the remaining three faces is 260. Are there combinations of polygons that have 90 and 260 hexagonal yods? As the former is the measure of the Lambda Tetractys, it is, itself, a parameter of holistic systems. This means that any set of polygons with 90 hexagonal yods on the sides of their tetractyses must consist of two subsets, one the mirror image of the other. Hence, each subset must have 44 hexagonal yods outside their shared side, which has two hexagonal yods. Inspection of Table 2 shows that only the square and octagon have this property. Figure 7 displays the 90:260 division.

 Figure 7. The generalisation of Plato's Lambda is the arithmetic counterpart of the inner Tree of Life because its 20 integers add up to the number (350) of hexagonal yods lining the sides of the 94 tetractyses that make up the 14 enfolded, regular polygons.

The following correlations indicate that this is not a coincidence:

• Number "6" at centre of 1st face → six black hexagonal yods on sides of square;
• Sum of integers at corners of tetractys in 1st face = 36 → 36 white yods on sides of second square & two octagons;
• Sum of integers at corners of hexagon in 1st face = 48 → 48 blue hexagonal yods on internal sides of tetractyses in squares & octagons;
• Sum of integers in 2nd, 3rd & 4th faces = 260 → 260 red yods in triangles, pentagons, hexagons, decagons & dodecagons.

Table 3 lists the numbers of triangular sectors in the seven enfolded polygons and their corners and sides (Fig. 8a). Remarkably, they, too, have 176 corners, sides & triangles. They have 87 sides outside their shared side. This is the number value of Levanah, the Mundane Chakra of Yesod. They have 129 corners & sides, where 129 is the number value of YAHWEH SABAOTH, the Godname of Netzach. Both sets of seven enfolded polygons have 80 corners, where 80 is the number value of Yesod (notice the

Table 3. Number of corners, sides & triangles in the 7 enfolded polygons.

 Triangle Square Pentagon Hexagon Octagon Decagon Dodecagon Total Number of corners 4 3 4 4 7 8 11 41 Number of sides 6 7 9 9 15 19 23 88 Number of triangles 3 4 5 5 8 10 12 47 Total = 13 14 18 18 30 37 46 176

Figure 8a. The 7 enfolded polygons divided into their 47 sectors.

conjunction here of two gematria numbers associated with the same Sephirah). They have 175 sides & 94 triangles, i.e., 349 corners, sides & triangles. This is the sum of the 19 integers below the integer 1 at the top of the Tetrahedral Lambda. Starting with 1, the Pythagorean Monad, the 19 integers below it add up to the number of geometrical elements needed to construct the complete inner Tree of Life! 349 is the 70th prime number. This means that, if successive prime numbers 2, 3, 5, etc were assigned to the 70 yods in the outer Tree of Life with its 16 triangles turned into tetractyses (see here), the final prime number assigned to Malkuth at its nadir would be the number of geometrical elements composing its inner form, namely its Malkuth aspect! This is an amazing property that is highly indicative of "divine design."

349 = 137 + 312,

where 137 is the sum of the nine powers of 2, 3 & 4 below 1 on the three edges of the Tetrahedral Lambda:

137 = 21 + 22 + 23 + 31 + 32 + 33 + 41 + 42 + 43.

This is how the integer 137, which is well-known to physicists for determining the approximate value of the fine-structure constant: α = e2/ħc ≈ 1/137, finds expression in this archetypal array of numbers that is the arithmetic counterpart of sacred geometries. Table 3 shows that the octagon & decagon have 67 geometrical elements outside the root edge, where 67 is the number value of Binah. Therefore, the pair of octagons and the pair of decagons have (3 + 2×67 = 137) geometrical elements when their shared side is included. They correspond to the nine powers of 2, 3 & 4, whilst the remaining 10 polygons making up the inner Tree of Life with 312 geometrical elements outside their shared side correspond to the 10 remaining integers below the integer 1 at the apex of the tetrahedral array of integers (Fig. 8b).

 Figure 8b

Figure 8b. Correspondence between the Tetrahedral Lambda and the geometry of the inner Tree of Life.

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