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#14 Equivalence of the 10-pointed star and the outer & inner Trees of Life

A 2nd-order tetractys is part of a triangular array of 91 yods with 13 yods along each side. With their bases joined, two such arrays form a parallelogram made up of (91+91−13=169) yods. Of these, 25 are corners of 32 1st-order tetractyses. Suppose that 10 parallelograms are linked at a corner shared between them. They form a 10-pointed star composed of 1680 yods surrounding its centre. 240 of these are corners of 320 tetractyses. This leaves 720 red hexagonal yods in a red pentagram and 720 blue hexagonal yods in an inverted, blue pentagram.

Compare this 720:240:720 pattern with the yods in the 1-tree and the (7+7) separate polygons of its inner form. When its 19 triangles are Type A, the 1-tree has 251 yods, of which 11 coincide with its SLs. It has 240 new yods created by the transformation of each triangle. The number of yods in a Type B n-gon = 15n + 1, where "1" denotes its centre. The number of yods in the seven separate Type B polygons = Σ(15n+1) = 15Σn + 7 = 15×48 + 7 = 727. Therefore, 720 yods surround their centres. The number of yods in both sets of polygons and in the 1-tree other than SLs = 720 + 240 + 720 = 1680. This is the same as the number of yods surrounding the centre of the 10-pointed star! Moreover, the following correspondences exist:

 10-pointed star 1-tree & (7+7) separate Type B polygons 240 corners of 320 tetractyses ↔ 240 new yods in 1-tree; 720 red hexagonal yods in red pentagram ↔ 720 yods surrounding centre of 7 separate Type B polygons; 720 blue hexagonal yods in blue pentagram ↔ 720 yods surrounding centre of mirror image set of 7 separate Type B polygons.

They demonstrate the uniqueness of this star as a counterpart of the 1-tree and its inner form. Each "point" of the star contains 168 yods (78 yods in its outer half and 90 yods in its inner half). This reproduces the respective number values of Cholem and Yesodoth in the Kabbalistic name of the Mundane Chakra of Malkuth:

As each parallelogram has 12 yods at centres of the "gaps" between the 20 1st-order tetractyses that make up its pair of 2nd-order tetractyses, there are (1680−120=1560) yods belonging to the 200 1st-order tetractyses in the 10-pointed star. This is the number of yods in 156 1st-order tetractyses, where 156 is the 155th integer after 1. This shows how ADONAI MELEKH, the complete Godname of Malkuth with number value 155, prescribes the 10-pointed star representation of the superstring structural parameter 1680. This Godname, too, has a representation by a 10-pointed star (see #11).

The 720:240:720 pattern is characteristic of sacred geometries. For a discussion of how it manifests in the first four Platonic solids and in the disdyakis triacontahedron, see, respectively, #20 in Wonders of Superstrings and #39 in Wonders of correspondences.

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