#40 The geometrical composition of the combined outer & inner Trees of Life reflects the root composition of E₈×E₈'

Outer Tree of Life
When they are Type A triangles, the 16 triangles making up the outer Tree of Life have (16×3=48) sectors with 16 internal corners and 48 internal sides, 10 external corners and 22 external sides. The outer Tree of Life is composed of 48 triangles with (10+16=26) corners and (22+48=70) sides, i.e., 144 geometrical elements, where 144 (=12²) is the 12th Fibonacci number.

Inner Tree of Life
When they are Type A, the (7+7) enfolded polygons making up the inner Tree of Life have (47+47=94) sectors with 80 corners. Two of these are endpoints of their shared root edge, 72 corners are unshared with the triangles of the outer Tree and six corners coincide with Sephirothic corners of triangles in the outer Tree. The 94 sectors have 175 sides, of which one is the root edge and four sides are sides of triangles in the outer Tree. The inner Tree of Life has 349 geometrical elements, of which 10 corners & sides are shared, leaving (349−3−10=336) intrinsic geometrical elements outside the root edge. Each half of the inner Tree of Life comprises 168 geometrical elements that are unshared with its outer form. The number value of the Mundane Chakra of Malkuth measures the points, lines & triangles that are intrinsic to each half of the inner Tree of Life.

Combined Trees of Life
When combined, the outer & inner Trees of Life have (48+94=142) triangles with (26+80−6=100=10²) corners & (70+175−4=241) sides. Outside the root edge, they have (241−1=240) sides and (100−2+142=240) corners & triangles. This 240:240 division of the 480 geometrical elements in the outer & inner Trees of Life outside the root edge is characteristic of sacred geometries. It manifests in superstring physics as the 240 roots of each rank-8 Lie group E₈ in the symmetry group E₈×E₈ describing the forces between one of the two types of heterotic superstrings. Notice that these 480 geometrical elements contain 336 elements that are intrinsic to the inner Tree of Life. We see that the "dynamical" parameter 480 contains the superstring structural parameter 336 — the number of turns in a helical whorl of the UPA in each one of its five revolutions around the spin-axis. The division: 480 = 144 + 336 shown between the outer & inner Trees of Life reflects the division: 240 = 72 + 168 between the 72 roots of E₆, the rank-6 exceptional subgroup of E₈, and the remaining

168 roots of E₈. This division can be made explicit by suitably dividing the outer Tree of Life into a left-hand half and a right-hand half, each with 72 geometrical elements (48 corners & sides, 24 triangles) and associating each half with the left-hand or right-hand set of seven enfolded polygons, each with 168 unshared geometrical elements outside their root edge. The metaphysical distinction between the outer & inner Trees finds expression in the special significance of E₆, which string theorists have long considered as an intermediate stage in the symmetry breakdown of E₈ that leads to the symmetries U(1)×SU(2)×SU(3) of the Standard Model. The inner Tree encodes the oscillatory form of the UPA/subquark state of the E₈×E₈ heterotic superstring, whilst the outer Tree encodes the E₆ subgroups of each E₈. In #37, it was shown that the root and trunk of the outer Tree have three sets of 24 yods, whilst its branches have seven sets of 24 yods. This is the analogue of the three major whorls carrying the 72 gauge charges of E₆, 24 per whorl, and the analogue of the seven minor whorls carrying 168 gauge charges of E₈, 24 per whorl. Whereas the geometrical composition of the outer Tree of Life picks out E₆ from other subgroups of E₈, its yod composition differentiates between the major and minor whorls of the UPA. They correspond, respectively, to the basic distinction between the root and trunk of the Tree of Life with three sets of 24 yods and its branches with seven sets of 24 yods.

This 240:240 division in the geometrical composition of the outer & inner Trees is also discussed in #29.

We find that the combined Trees of Life can be divided into two halves, one the mirror image of the other. The 240 geometrical elements in each half outside the root edge comprise 49 corners, 120 sides & 71 triangles, that is, (49+71=120) corners & triangles and 120 sides. The 120:120 division in the number 240 is characteristic of holistic systems embodying this parameter. It manifests in 4-dimensional sacred geometries as the 120 vertices, edges, faces & octahedral cells in each half of the 24-cell (see here) and as the 120 vertices of each of the two 600-cells whose compound is the 4-dimensional projection of the 4₂₁ polytope (see discussion under "The 4₂₁ polytope" here). See here for its appearance in other sacred geometries.

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