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#52 The outer and inner Tree of Life basis of E_{8} and E_{8}×E_{8}′

The outer & inner Tree of Life basis of the heterotic
superstring symmetry group E8
Outer Tree of Life
A Type B triangle is composed of 9 triangles with 3 external corners and 4 corners that are inside it. The 16 Type B triangles of the outer Tree of Life have 10 corners on their 22 sides and (16×4=64) internal corners of their (16×9=144) triangles with (16×12=192) sides. The outer Tree of Life comprises its root, its trunk and its branches. In terms of the 74 corners of its 144 triangles, the root consists of two corners whose projection onto the plane containing the (7+7) enfolded polygons of the inner Tree of Life are the two endpoints of their shared root edge. One corner is located at Daath, which is the centre of the triangle formed by Chokmah, Binah & Tiphareth. The other corner coincides with Tiphareth, which is the centre of the Tree of Life in both a geometrical and a metaphysical sense. It is where the root joins the trunk of the Tree of Life, which is defined as the sequence:
pointlinetriangletetrahedron
symbolising the integers 1, 2, 3 & 4, Tiphareth being located
at the lowest corner of the triangle in this sequence. The corners making up the trunk are
composed of the 9 red corners of all 16 primary triangles, the 5 green centres of
its 5 primary triangles and the
(5×3=15) blue corners of their (5×9=45) internal triangles. The branches
are the remainder of the outer Tree of Life outside the root &
trunk. They comprise 11 primary triangles with 10
green centres and (3×11=33) blue corners.
The 74 corners comprise the two black corners forming the root
and the 72 corners of the trunk (29 corners) and branches
(43 corners). The 16 primary triangles have
(10+16=26) corners & centres, where
and
so that
This compares with the 26 dimensions predicted by
quantum mechanics for spinless strings being made up of the time dimension, the longitudinal dimension and 24 transverse dimensions
comprising 9 dimensions of the 11dimensional spacetime predicted by
Mtheory and the 15 additional, bosonic string dimensions. The root
of the outer Tree of Life is analogous to the time and longitudinal dimensions of such a string
in 26dimensional spacetime. Neither of these participates in creating shape.
Rather, in each case they are its rootlike source. It means that the 9 dimensions are the trunk of spacetime and
the 15 bosonic string dimensions are its branches.
Inner Tree of Life
The 7 enfolded, Type B polygons have 47 sectors with 41 corners. Their (47×3=141) triangles have (41+47=88) corners (86 outside the root edge). The (7+7) enfolded, Type B polygons have (2×141=282) triangles with (2 + 2×86 = 174) corners. 282 is the number value of Aralim, the Order of Angels assigned to Binah. The centre, top and bottom of each hexagon coincide with corners of triangles belonging to the outer Tree of Life. These six white corners are shared and (174−6=168) corners are unshared. (168/2=84) corners (turquoise or pink) are associated with each set of 7 enfolded polygons, which have 85 unshared corners, where
84 = 1^{2} + 3^{2} + 5^{2} +
7^{2}
and
85 = 4^{0} + 4^{1} + 4^{2} +
4^{3}.
The (47×2=94) sectors of the (7+7) enfolded polygons have 80 corners. 87 corners of (47×3=141)
triangles are associated with each set of 7 enfolded polygons when they are Type B. 80 is the number value of
Yesod and 87 is the number value of Levanah, its Mundane
Chakra. The separate outer and inner Trees of Life
have (74+174=248) corners of (144+282=426) triangles. Excluding the
root, there are 246 corners,
where 246 is the number value of Gabriel, the Archangel of
Yesod. 248 is the dimension of the rank8, exceptional Lie group E_{8} appearing in E_{8}×E_{8}′
heterotic superstring theory. The 8 white or black corners symbolise the 8 simple roots of
E_{8}; the (72+168=240) other corners symbolise its 240 roots.
The 72 corners in the trunk and branches symbolise the 72 roots of
E6, the rank6, exceptional Lie
group that is a subgroup of E8.
The 168 unshared corners in the inner Tree of Life denote
the 168 roots of E8 that do not belong to E6.
The 84:84 division of the superstring structural parameter 168 is characteristic
of sacred geometries (see Article 64).
The (282+144=426) triangles in the combined
outer & inner Trees of Life have 240 corners other than those forming the root.
Alternatively, they have 240 corners other than the two endpoints of the root edge.
This embodiment of the number 240 is characteristic of sacred geometries (see #1) because it quantifies the 240 roots of
E_{8}, which this website demonstrates is part of the divine paradigm.


The inner Tree of Life basis of the heterotic superstring
symmetry group E8×E8′
Each of the n sectors of a Type C ngon is divided into three
smaller sectors that are further divided into three triangles. The
9n triangles have (5n+1) corners. The 7 separate
Type C polygons with 48 corners have 247
corners. Consider the two sets of 7 such polygons
separated by the root edge. The two endpoints of the latter can be regarded as two corners
of a triangle in the limit that its third corner
collapses into its base. 248 corners are
associated with each set of 7 polygons separated by the root edge. The first 6 separate polygons with 36 corners have 324
triangles with 186 corners. Hence, 187 corners are associated with the root edge and
each set of the first 6 polygons. 248 is the number value of Raziel, the
Archangel of Chokmah, and 187 is the number value of Auphanim, the Order of
Angels assigned to this Sephirah.
The 248 corners associated
with each set of 7 polygons and the separate root edge correspond to
the 248 roots of the rank8, exceptional Lie group E8.
Their 7 centres and an endpoint of the root edge correspond to its 8 simple roots; the 240
other corners correspond to its 240 roots. The
direct product E8×E8′
(one of the two possible superstring symmetry groups) is the
consequence of the inner form of the Tree of Life
having two halves, one of which is the mirror
image of the other. The (7+7) Type C
polygons and the root edge
have 496 corners. 496 is the number value
of Malkuth (material universe) and the dimension that a
superstring symmetry group must have to make
all its quantum
anomalies disappear.


The outer and inner Trees of Life embody E_{8}×E_{8} heterotic superstring theory 
Outer Tree of Life Inner Tree of Life
Combined outer & inner Trees of
Life Collecting together the various results discussed above: 496 = 52 + 444 = (16+36) + (70+2) + 24 + 22 + 24 + 22 + 24 + 256. The 256 hexagonal yods lining tetractyses in the first (6+6) enfolded polygons comprise eight black hexagonal yods on the four sides of the pair of triangles outside the root edge and 248 khaki hexagonal yods (let us write this set as 248′). Therefore, 496 = (16+36) + (70+2) + 24 + 22 + 24 + 22 + 24 + 8 + 248′ = 248 + 248′, where 248 = (16+36) + (70+2) + 24 + 22 + 24 + 22 + 24 + 8 = 8 + (70+2) + (36+24+24) + (16+24+22+22)
We find that the hexagonal yod population 496 splits into 248 and 248′. This is analogous to how the dimension 496 of E_{8}×E_{8}′ is the sum of the dimension 248 of E_{8} and the dimension 248′ of E_{8}′. That the geometry of the combined Trees allows such a natural splitting of this number into two equal numbers is evidence that it is not by chance that they have 496 hexagonal yods. This conclusion is confirmed by the fact that the number 248 further divides naturally into the number 8 denoting the eight simple roots of E_{8}, the number 72 denoting the 72 roots of E_{6} (an exceptional subgroup of E_{8}) and the number 168 denoting the 168 remaining roots of E_{8}. 168 further divides into two equal numbers 84 (the first diagram shown above indicates that the 168 8tuples group into (56+28=84) 8tuples and (3×28=84) 8tuples). This bifurcation is displayed in all the sacred geometries/holistic systems discussed in this website (see here under "168 = 84 + 84"), as is the 72:168 division (see here). Amazing though it may seem, what we are encountering here is not the highly improbable result of miraculous chance but the manifestation of the group mathematics of heterotic E_{8}×E_{8}′ superstring theory in the sublime sacred geometry of the outer & inner Trees of Life. This has enormous implications, for how can sacred geometries central to the mystical teachings of certain religions have properties exactly analogous to certain mathematical discoveries in theoretical physics unless a "divine design" underpins the universe that superstring theory and Mtheory have begun to reveal? 


The 1tree is the lowest Tree of Life of any set of overlapping Trees of Life. When its 19 triangles are Type A, it contains 251 yods.* 11 of these are corners of these triangles to which the 10 Sephiroth and Daath are assigned. There are (251−11=240) white yods that are either hexagonal yods or internal corners of the the Type A triangles. Outside the 1tree and below its apex are four red hexagonal yods on either side of the central Pillar of Equilibrium. Therefore, (240+8=248) yods lie below the top of the 1tree that are not Sephirothic corners of triangles. The number 248 is both the gematria number value of Raziel, the Archangel of Chokmah, and the dimension of E_{8}. The eight red yods outside the 1tree symbolise the eight simple roots of E_{8} and the 240 yods that are not Sephirothic corners denote its 240 roots. See also #1 & #2. * Proof: The 19 triangles in the 1tree have 25 sides and 11 corners. (25×2 + 11 = 61) yods line their sides. There are 10 yods inside each Type A triangle. Number of yods in the 1tree = 61 + 19×10 = 251.


Correspondence between the 496 roots of
E_{8}×E_{8} and the The outer Tree of Life is composed of 16 triangles. When they are Type A, they have (16×3=48) sectors with (10+16=26) corners and (22 + 16×3 = 70) sides. These 144 geometrical elements comprise those distributed on either side of the central Pillar of Equilibrium as pairs of elements that are mirror images of each other and those either located on this Pillar or spanning both sides of it in a symmetrical way; the latter can be divided into two equal sets that can be formally associated with either half of the outer Tree of Life. This means that 72 geometrical elements are associated with each half if it. Those associated with the lefthand set of 7 polygons are written as 72′. The 7 enfolded polygons making up each half of the inner Tree of Life are composed of 47 triangles with 41 corners and 88 sides. These 176 geometrical elements comprise the three geometrical elements (two corners & one side) making up their shared root edge, the top, centre & bottom corner of the hexagon, the two vertical, internal sides of its sectors and 168 other geometrical elements. The hexagon in each set of polygons has 5 geometrical elements that coincide with their counterparts (coloured green in the diagram opposite) in the outer Tree when it combines with its inner form. The set of 5 shared elements in the lefthand set of polygons is written as 5′ and the 3 elements making up their root edge are written as 3′; their unshared elements outside the root edge are written as 168′. Hence, 144 = 72′ + 72, The total number of geometrical elements in the separate outer and inner Trees with Type A polygons = 176′ + 144 + 176 = 496 = (3′+5′) + 72′ + 168′ + (3+5) + 72 + 168 = 8′ + 72′ + 168′ + 8 + 72 + 168 = 248′ + 248, where 8′ ≡ 3′ + 5′ and 8 = 3 + 5. The 8 geometrical elements in the righthand set of polygons that either belong to the root edge or are shared with the outer Tree of Life correspond to the 8 simple roots of E_{8}; their 8 counterparts in the lefthand set (written as 8′) correspond to the 8 simple roots of an identical group (written as E_{8}′). The 72 geometrical elements associated with either half of the outer Tree of Life correspond to the 72 roots of E_{6}, an exceptional subgroup of E_{8}. The remaining unshared 168 elements in each set of polygons correspond to the 168 roots of E_{8} that are not roots of E_{6}. The direct product E_{8}×E_{8} appearing in string theory as the symmetry group describing the unified interactions of one of the two types of heterotic superstrings arises from the leftright mirror symmetry of both the outer and inner forms of the Tree of Life. This natural isomorphism between the root composition of E_{8}×E_{8} and the geometrical composition of the Tree of Life can have no other plausible explanation than that E_{8}×E_{8} heterotic superstrings exist as the subatomic manifestation of the universal blueprint known as the "Tree of Life." 