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#29 The counterpart of the root composition of E_{8}×E_{8} 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 (=12^{2}) is the twelth 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 = 2^{4} + 2^{5} + 2^{6} + 2^{7},
i.e., 240 is the sum of four successive powers of 2, starting with the power of 4, and
480 = 4^{2} + 8^{2} + 12^{2} + 16^{2},
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:
As
11^{2} − 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 E_{8}×E_{8} heterotic superstring is one of the five types of superstrings (see here). The Lie group E_{8} has eight simple roots and 240 roots (see here & here under "Superstring gauge symmetry group"). E_{8}×E_{8} 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, E_{8} has E_{6} 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 E_{8}×E_{8} 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 E_{8}×E_{8} 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:
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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