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#29 The counterpart of the root composition of E8×E8 in the geometry of the combined outer & inner Trees of Life

Outer Tree of Life
The 16 triangles making up the outer Tree of Life have 10 corners and 22 sides. Regarded as Type A triangles, they have (16×3=48) sectors, where 48 is the number value of Kokab, the Mundane Chakra of Hod, (10+16=26) corners and (22+48=70) sides. The outer Tree of Life is composed of 144 geometrical elements, where 144 (=122) is the twelfth Fibonacci number. Its trunk is the geometrical sequence: point, line, triangle & tetrahedron. Its composition is shown below:

 Corners Sides Triangles Total Point 1 0 0 1 Line 2 1 0 3 Triangle 4 6 3 13 Tetrahedron 8 18 12 38 Total 15 25 15 55

The trunk of the outer Tree of Life is composed of 55 geometrical elements, so that its branches (the remainder of the outer Tree) has 89 elements. 55 is the tenth Fibonacci number and 89 is the eleventh Fibonacci number. The geometrical composition of the Tree of Life is determined by three successive Fibonacci numbers!

Starting from a point (Kether), 95 more points & lines are needed to construct the outer Tree of Life from triangles. 95 is the number value of Madim, the Mundane Chakra of Geburah.

Inner Tree of Life
The seven enfolded polygons in each half of the inner Tree of Life have 47 sectors with 41 corners & 88 sides (39 corners & 87 sides outside their shared, white root edge, where 87 is the number value of Levanah, the Mundane Chakra of Yesod). They are composed of 176 geometrical elements. Both sets of enfolded polygons have 94 sectors with 80 corners & 175 sides, totalling 349 geometrical elements.

Combined outer & inner Trees of Life
When the outer Tree of Life is superposed on the (7+7) enfolded polygons so that they are contained in the plane formed by its two side pillars, the latter are aligned with the four white, vertical sides of sectors of the two hexagons. For each set of polygons, three corners and two white sides of triangles in the outer Tree are shared with the hexagon. Therefore, ten of the 349 geometrical elements in the inner Tree of Life are shared with the outer Tree, leaving (349−10=339) unshared, geometrical elements. 336 geometrical elements outside the root edge (168 elements in each set of seven enfolded polygons) are unshared. These 168 unshared elements comprise (39−3=36) corners, (87−2=85) sides & 47 triangles. The geometrical composition of the outer & inner Trees of Life is tabulated below:

Geometrical composition of the combined outer & inner Trees of Life

 7 enfolded polygons Tree of Life Root edge 7 enfolded polygons Total Geometrical element Total Unshared Total Unshared Total Unshared Corners 39 36 26 20 2 39 36 2 + 26 + 72 = 2 + 98 = 100 Sides 87 85 70 66 1 87 85 1 + 70 + 170 = 1 + 240 = 241 Triangles 47 47 48 48 0 47 47 48 + 94 = 142 Total 173 168 144 134 3 173 168 3 + 240 + 240 = 3 + 480 = 483

The combined Trees of Life are composed of 483 geometrical elements. The 480 elements outside the root edge consist of 240 sides and (98+142=240) corners & triangles. The latter comprise (36+36=72) corners of the (7+7) enfolded polygons that are unshared with the outer Tree of Life, 26 corners & 142 triangles, i.e., 168 more corners & triangles. This 72:168 division is characteristic of the number 240 as a parameter of holistic systems (see here). It was discussed in #23 for five sacred geometries. Both 240 and 480 are determined by the Tetrad because

240 = 24 + 25 + 26 + 27,

i.e., 240 is the sum of four successive powers of 2, starting with the power of 4, and

480 = 42 + 82 + 122 + 162,

the terms starting with the number 4 and increasing by four units. The number 480 is, too, a parameter of sacred geometries and its 240:240 division is characteristic of holistic systems. For example:

• each set of seven separate polygons with tetractyses as their sectors contains 240 hexagonal yods (see here);
• each set of seven separate Type B polygons has 144 triangles with 240 sides (see here and comment (14) under heading "Type B polygons" here);
• the faces of the first three Platonic solids have 240 hexagonal yods when constructed from tetractyses, just as the faces of the icosahedron have (see here);
• each of the two sets of the first six polygons enfolded in the 10-tree has 240 corners associated with it (see here);
• the disdyakis triacontahedron and the seven internal polygons formed by its vertices that are perpendicular to an axis passing through two opposite A vertices have 240 geometrical elements in each half of them other than the 16 sides of polygons that are not polyhedral edges (see here).

As

112 − 1 = 120 = 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21,

120 is the sum of the first 10 odd integers after 1 and

240 = 2×120 = 6 + 10 + 14 + 18 + 22 + 26 + 30 + 34 + 38 + 42

is the sum of the first 10 even integers that start with 6 and are spaced by 4 units. Also, 240 = 1×2×3×4(1+2+3+4), showing how the integers 1, 2, 3 & 4 denoting the numbers of points in the four rows of the tetractys express this holistic parameter. It shows how the Decad symbolised by the tetractys arithmetically determines the geometrical composition of the combined Trees of Life.

The E8×E8 heterotic superstring is one of the five types of superstrings (see here). The Lie group E8 has eight simple roots and 240 roots (see here & here under "Superstring gauge symmetry group"). E8×E8 has (240+240=480) roots. The geometrical composition of the combined Trees — a set of 240 sides and a mixed set of 240 corners & triangles — is their remarkable counterpart. Furthermore, E8 has E6 as an exceptional subgroup with 72 roots. The 72 corners of the 94 sectors of the (7+7) enfolded polygons outside the root edge that are not part of the outer Tree are their counterpart. This analogy between the root composition of E8×E8 and the geometrical make-up of the combined Trees of Life cannot be a mere coincidence. Instead, it amounts to powerful evidence that the heterotic E8×E8 superstring conforms to the cosmic blueprint of the outer and inner Trees of Life.

The table indicates that the outer Tree of Life has 134 geometrical elements that are unshared with its inner form. The root edge is unshared with the outer Tree, so that both consist of (134+3=137) unshared geometrical elements (22 corners, 67 sides & 48 triangles). Embodied in the geometry of the combined Trees is the number 137 whose reciprocal is approximately equal to the fine-structure constant. This should not be surprising in view of the fact that this number is a characteristic parameter of all sacred geometries (see the discussion under the heading "137" here). The number of geometrical elements in the combined Trees is:

483 = 10 shared elements + 137 unshared elements in the outer Tree of Life & root edge + 336 unshared elements in the inner Tree outside its root edge.

This displays the remarkable conjunction in a number describing the geometrical composition of the combined Trees of Life of the number 137 that determines the fine-structure constant and the superstring structural parameter 336, namely, the number of turns in a complete revolution of each helical whorl of the UPA/subquark superstring around its spin axis.

The 240 corners & triangles consist of the 48 triangles in the outer Tree and 192 corners & triangles. The 240 sides comprise the 48 sides of sectors inside the 16 original triangles of the Tree of Life and 192 sides. This 48:192 division is characteristic of the parameter 240 of holistic systems. For example:

• surrounding the centres of the seven separate polygons of the inner Tree of Life are 240 yods that line the sides of their 48 tetractyses. They comprise 48 corners and 192 hexagonal yods.
• the 144 triangles in the seven separate Type B polygons have 240 sides (48 external & 192 internal);
• the faces of the tetrahedron have 48 hexagonal yods and the faces of the octahedron & cube have 192 hexagonal yods when their sectors are tetractyses (see here);
• the 240 geometrical elements surrounding the centre of the 2-d Sri Yantra consist of the 48 elements in the central triangle & first group of eight triangles and the 192 elements in the second, third & fourth groups of triangles (see table here);
• the 2400 (=240×10) geometrical elements surrounding the axis of the disdyakis triacontahedron (see here) consist of 480 (=48×10) sectors & centres of its 120 faces and 1920 (=192×10) other geometrical elements.

Article 54 analyzes the properties of the combined outer & inner Trees of Life with either triangles, tetractyses, Type A or Type B triangles in the former and either Type A or Type B polygons in the latter.

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