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Outer & inner Tree of Life representation of CTOL

The outer Tree of Life contains 70 yods when its 16 triangles become tetractyses. Seven yods lining each side pillar and two hexagonal yods on the Path connecting Chesed and Geburah (that is, the 16 black yods in the diagram opposite) are shared with the (7+7) enfolded polygons, which have 524 yods. Therefore, (70−16=54) yods belonging to the outer Tree are intrinsic to it. They include four yods located at the Sephiroth Kether, Tiphareth, Yesod & Malkuth, leaving 50 intrinsic yods that are not located at Sephiroth, where 50 is the number value of ELOHIM, the Godname of Binah. The (7+7) enfolded polygons have (524−16=508) yods that are intrinsic to them. (4+4=8) of these yods are yellow centres of polygons (the centres of the two decagons are not included because they are also corners of the pentagons and we want to count here only intrinsic yods that surround the centres of their own polygons; the centres of the triangles belonging to the inner Tree coincide with the two hexagonal yods on the Chesed-Geburah Path and are not among the 508 yods, being shared with the outer Tree). Therefore, (508−8=500=50×10) intrinsic yods surround the centres of the polygons. The outer and inner Trees of Life have (50+500=550) intrinsic yods that are neither Sephiroth nor centres of polygons. Outside the root edge are 260 yods in each set of seven enfolded polygons. Of these, (260−8−4=248) yods are intrinsic yods that surround their centres, that is, both sets have 496 such yods. 248 is the number value of Raziel, the Archangel of Chokmah, and 496 is both the number value of Malkuth (the physical universe, in the cosmic context of this Sephirah) and the dimension of the two possible superstring gauge symmetry groups SO(32) and E8×E8. The mirror symmetry of the two sets of seven enfolded polygons of the inner Tree of Life is responsible for the existence of two identical groups E8 present in one unified symmetry group  as a direct product.

The 550 yods other than Sephiroth and centres of polygons that are intrinsic to the outer and inner Trees of Life symbolize the 550 SLs of CTOL. The 50 intrinsic yods other than Sephiroth in the outer Tree denote the lowest 50 SLs up to Yesod of the ninth Tree. It is the 501st SL from the top of CTOL. The complete Godname of Kether is:

Its traditional translation is "I am that I am." The gematria number value 543 of this Godname measures the 543 SLs in CTOL down to Yesod of the second Tree — the last SL before the first Sephirah of Construction of the lowest Tree (see here). The number value 501 of ASHER is the number of SLs down to Yesod of the ninth Tree. The 500 SLs above it are symbolized by the 500 intrinsic yods surrounding the centres of all the polygons except the decagons, whilst the 50 SLs up to this SL are symbolized by the 50 intrinsic yods other than Sephiroth that belong to the outer Tree of Life and are coloured green in the adjacent diagram. The interpretation of what the number value of EHYEH ASHER EHYEH means in terms of CTOL is consistent with the geometries of the outer and inner Trees of Life because the 50 intrinsic yods of the former correspond to its lowest 50 SLs and the 500 intrinsic yods of the latter correspond to its highest 500 SLs.

Through its letter values, EHYEH (Hebrew: AHIH) specifies the ten Sephiroth (H=10) of the Tree of Life, Daath (A=1) and the five centres of each set of seven polygons (H=5) that do not coincide with any of their corners (see here). This means that the number 550 refers to the yods unshared by the combined outer and inner Trees of Life other than those specified by EHYEH.

What does the switch between the outer and inner Trees of Life mean in terms of CTOL? The 50th SL (Yesod of the eighth Tree), counting from its lowest point, is the 26th tree level (see Article 2). The lowest 25 tree levels signify the 25 spatial dimensions of the 26-dimensional space-time predicted by the quantum mechanics of spinless strings. The 26th tree level signifies the dimension of time. The switchover, therefore, represents the transition from the space-time continuum to non-temporal realms — from that part of CTOL where time exists to those levels of reality beyond space and time. The 248 intrinsic yods outside the root edge of one set of seven enfolded polygons correspond to the 248 SLs in CTOL down to Binah of the 50th Tree — the very Sephirah whose Godname number specifies the Tree to which it belongs! The 496 intrinsic yods outside the root edge of both sets of enfolded polygons correspond to the 496 SLs in CTOL down to Chesed of the ninth Tree. The remaining 54 yods consist of the four yods on the root edge and the 50 intrinsic yods in the outer Tree other than Sephiroth; they correspond to the four SLs from Geburah of the ninth Tree to Hod of this Tree and to the lowest 50 SLs up to Yesod of this Tree.

An alternative view
The fact that the outer Tree of Life has 54 intrinsic yods unshared with its inner form and that

54 + 496 = 550

allows an alternative analysis that retains as part of the set of 550 yods the four unshared Sephirothic points (Kether, Tiphareth, Yesod & Malkuth) and excludes the four yods making up the root edge from the set of 496 yods. Outside the root edge in each set of seven enfolded polygons are 260 yods that comprise eight shared yods, four centres of polygons and the centre of the decagon that coincides with a corner of the pentagon. Because of this dual character, it should be included in any count of the total number of yods that surround the centres of the polygons to which they belong. This means that the number of unshared yods outside the root edge that truly surround centres of each set of seven polygons = 260 − 8 − 4 = 248, both sets having (248+248=496) such yods. Hence, the number 550 is embodied in the combined Trees of Life as the number of yods unshared between them that are outside the root edge and surround centres of polygons. This viewpoint seems more natural than the one discussed above because it does not exclude from the set of 550 yods for no good reason the four Sephiroth of the outer Tree of Life that are unshared with its inner form. Instead, it excludes the four yods of the root edge. This is easier to justify because of their unique status in being the root source of the outer Tree that connects it to the microbiological level of its "DNA" — its inner form.

The Tetrad expresses the outer & inner Trees of Life
The total number of yods in the combined Trees of Life = 54 + 524 = 578 = 2 + 576 = 2 + 242 = 2 + 2×288, where

288 = 1!× 2!×3!×4!.

It comprises the two endpoints of the root edge and 242 yods, of which 1!×2!×3!×4! yods belong to each half of the combination. This illustrates the Pythagorean character of its mathematical beauty as an archetypal object. Surrounding the centres of the seven separate, regular Type A polygons in each half of the inner Tree of Life are 288 yods. The permutational meaning of this number is as follows: arranged in a tetractys, ten different objects have 1! (= 1) permutations of its apex object, 2! (=2) permutations of the two objects in the second row, 3! (=6) permutations of the three objects in the third row and 4! (=24) permutations of the four objects in the fourth row. Every permutation generates a different tetractys. The total number of tetractyses that can be generated by rearranging the objects in each row = 1×2×6×24 = 288. The 288 yods surrounding the centres of the seven polygons can be regarded as corresponding to every possible tetractys that can be created from 10 objects by re-arranging them within each row. Rows 1 & 3 generate the factor 1!×3! = 6, which corresponds to the six yods per sector of a polygon that surround its centre, and rows 2 & 4 generate the factor 2!×4! = 48, which corresponds to the 48 sectors of the seven polygons.

As 172 = 289, this is the number of yods in each half of the combined Trees of Life: 578 = 2×289 = 2×172. As

172 = 1 + 3 + 5 +... + 33,

and 33 is the sum of the number of permutations of the objects in each row:

1! + 2! + 3! + 4! = 33,

we see that we can create 33 arrangements of objects selected from a basic set of four objects and 288 tetractyses when they are selected to form the four rows of a tetractys. The combined outer & inner Trees of Life express the arithmetic identity:

 2 + 242 = 2×172.

The outer Tree has 70 yods (35 yods associated with each half), of which eight black yods are shared with its inner form (see below), leaving 27 green yods in each half. There are two white endpoints of the root edge and 522 yods in the (7+7) enfolded, Type A polygons. Eight yods (the black yods in the diagram shown above) in the 261 yods associated with each half of the inner Tree of Life are shared with its outer form, leaving 253 red or blue intrinsic yods. Associated with each half is one endpoint and (27+253+8=288) yods. The total number of yods in the combined Trees of Life = 2(1+288) = 2 + 576 = 578 = 2 + 242.

The number of yods in the n-gon whose sectors are 2nd-order tetractyses = 72n + 1, where 72 is the number value of Chesed and  "1" denotes its centre. A square (n=4) has 289 yods, where 289 = 172. The two squares present in the inner Tree of Life have (2×289=578) yods when constructed from 2nd-order tetractyses. In the identity highlighted above, the first "2" denotes the two centres of the pair of squares, 242 denotes the number of yods surrounding them and 172 denotes the number of yods in each square. The fact that the yod population of the combined Trees of Life is the yod population of the two squares present in the inner Tree when its sectors are 2nd-order tetractyses is a remarkable illustration of the Tetrad Principle at work (this is discussed in Article 1). It is further illustrated by the fact that

 3 5 7 9 172 − 1 = 288 = 11 13 15 17 19 21 23 25 27 29 31 33

i.e., the number of yods other than the endpoints of the root edge that are s with each half of the combined Trees of Life is a 4×4 array of the first 16 odd integers after 1. Notice that 24 yods surround the centre of a Type A square, so that 242 is the sum of the integers 24 assigned to these 24 yods. Symbolising the Tetrad, the square generates out of itself the yod population of the combined forms of the Tree of Life.

The identity highlighted above is also expressed by the two sets of separate, Type A polygons of the inner Tree of Life when partitioned by the root edge with its two endpoints. 288 yods suround the centres of each set, so that both sets have 242 such yods. Including the endpoints of the root edge separating the two sets, 172 yods are associated  with each set.

Of the 578 yods in the combined Trees of Life, four are corners of triangles in the outer Tree that are unshared with its inner form because they are located at the Sephiroth on the Pillar of Equilibrium, whilst 80 are corners of the (47+47=94) sectors of the (7+7) enfolded polygons. Hence, the (16+47+47=110) tetractyses in the combined Trees have 84 corners, where

84 = 12 + 32 + 52 + 72,

and (578−84=494) hexagonal yods, where

 11 21 31 41 12 22 32 42 494 = 13 23 33 43 14 24 34 44

This is another illustration of the power of the Tetrad to determine properties of holistic systems. It is worth noting that two of the 578 yods are unique in that they behave as both hexagonal yods and corners, i.e., they have a dual character. This is because, when all triangles in the outer Tree of Life are tetractyses, the two hexagonal yods on the Path connecting Chesed and Geburah coincide with the centres of the two Type A triangles in the inner Tree of Life. Whether we regard these shared yods as hexagonal yods of tetractyses in the outer Tree of Life or as corners of tetractyses in the inner Tree is, purely, a matter of context. If we count them as corners, then the total yod population is

578 = 84 corners + 494 hexagonal yods.

If we count them as hexagonal yods, then there are (2+494=496) hexagonal yods and (84−2=82) corners, so that

578 = 82 + 496 = 2 + 80 + 496.

"2" denotes the two endpoints of the root edge, 80 is the number value of Yesod ("Foundation") and 496 ("KIngdom") is the number value of Malkuth. In other words, the yod population of the combined Trees of Life (excluding the two endpoints of the root edge) is the sum of the gematria number values of the last two Sephiroth! Here is the foundation of the mathematical domain or territory that is the cosmic blueprint.

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