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#33 Comparison of the hexagonal yod composition of the 1-tree and the 7 enfolded polygons

The number of yods in the n-tree whose triangles are Type A ≡ N(n) = 158n + 93*. Hence, N(1) = 251. The 1-tree has 19 triangles. When they are Type A, they have (19×3=57) sectors. The number of corners of the 57 tetractyses making up the 1-tree with Type A triangles is the sum of the number of corners of the latter and the number of their centres, i.e., 11 + 19 = 30 = 12 + 22 + 32 + 42. The number of hexagonal yods in the 1-tree = 251 − 30 = 221 (see picture opposite). As four hexagonal yods below its apex lie outside the 1-tree on either side of the central Pillar of Equilibrium, there are (221+4+4=229) hexagonal yods below the 1-tree. 229 is the 50th prime number, showing how ELOHIM, the Godname of Binah with number value 50, prescribes the number of hexagonal yods below the top of the 1-tree when it is constructed from Type A triangles.

When their (47+47=94) sectors are tetractyses, the (7+7) enfolded polygons of the inner Tree of Life have 524 yods, of which 80 yods are corners of the 94 tetractyses (see #23 in Sacred geometry/Tree of Life). 80 is the number value of Yesod, the penultimate Sephirah of the Tree of Life. Therefore, they have (524−80=444) hexagonal yods. Two of them lie on the root edge, leaving (444−2=442) hexagonal yods outside it, i.e., each set of polygons has 221 hexagonal yods outside their shared root edge.

We see that the number of hexagonal yods in the 1-tree with Type A triangles is the same as the number of hexagonal yods outside the root edge of the seven enfolded Type A polygons. This is not a coincidence, for the inner Tree of Life encodes the properties of its outer form. The 30 corners of the 19 Type A triangles in the 1-tree correspond to the 30 corners of the seven enfolded polygons outside their shared side that are either not located at Sephiroth (i.e., shared with the 1-tree) or centres of polygons.

* Proof: The n-tree comprises (12n+7) triangles with (6n+5) corners and (16n+9) sides. This is because the Lower Face of the 1-tree is made up of seven triangles with five corners and nine sides, whilst 12 triangles with six corners and 16 sides are added in successive Faces. When each triangle is a Type A triangle, ten yods are inside it. Two hexagonal yods lie on each side of a Type A triangle. The number of yods in the n-tree = 6n + 5 + (16n+9)×2 + (12n+7)×10 = 158n + 93.

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