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Powersof2 representation of some parameters of the inner Tree of Life and its interpretation in terms of the 4_{21} polytope
The (7+7) enfolded Type B polygons contain 1370 yods (for proof, see Table 3 here). This parameter has a representation in terms of powers of 2. To derive it, start with the arithmetic identity: 84 = 2^{2} + 4^{2} + 8^{2} = 2^{2} + 2^{4} + 2^{6}. ∴ 168 = 2×84 = 2^{3} + 2^{5} + 2^{7}, 336 = 2×168 = 2^{4} + 2^{6} + 2^{8}, and 672 = 2×336 = 2^{5} + 2^{7} + 2^{9}. ∴1370 = 26 + 1344 = 26 + 672×2, where 26 = 2 + 8 + 16 = 2^{1} + 2^{3} + 2^{4}. ∴ 1370 = 2^{1} + 2^{3} + 2^{4} + 2^{6} + 2^{8} + 2^{10}. 

Geometrical interpretation
1368 = 2^{3} + 2^{4} + 2^{6} + 2^{8} + 2^{10}.
1360 = 2^{4} + 2^{6} + 2^{8} + 2^{10} = 4^{2} + 4^{3} + 4^{4} + 4^{5}.


1344 = 2^{6} + 2^{8} + 2^{10} = 4^{3} + 4^{4} + 4^{5}.
There are 13 other yods in each set of 7 enfolded polygons (12 intrinsic) that are either shared with the outer Tree of Life (6) or centres of polygons (7), where 13 = 1 + 4 + 8 = 2^{0} + 2^{2} + 2^{3}. 

Six of the 1370 yods in the (7+7) enfolded Type B polygons are located at the positions of the 6 Sephiroth on the side pillars of the outer Tree of Life. They are shown above as black dots. There are (1370−6=1364) other yods, i.e., (1364−4=1360) yods outside the root edge, where 1360 = 4^{2} + 4^{3} + 4^{4} + 4^{5} = 2^{4} + 2^{6} + 2^{8} + 2^{10}. ∴ 1364 = 4^{1} + 4^{2} + 4^{3} + 4^{4} + 4^{5} = 2^{2} + 2^{4} + 2^{6} + 2^{8} + 2^{10}. This demonstrates in a beautiful way the power of the Tetrad to express parameters of the inner Tree of Life. As 1370 = 2^{1} + 2^{3} + 2^{4} + 2^{6} + 2^{8} + 2^{10}, the number of yods associated with each set of 7 enfolded polygons = 685 = 2^{0} + 2^{2} + 2^{3} + 2^{5} + 2^{7} + 2^{9}, the number of yods in each set = 685 + 2 = 687 = 2^{0} + 2^{1} + 2^{2} + 2^{3} + 2^{5} + 2^{7} + 2^{9}, and the number of intrinsic yods outside the root edge in each set = 687 − 4 − 1 = 682 = 2^{1} + 2^{3} + 2^{5} + 2^{7} + 2^{9}. This provides an alternative interpretation of the number 1364 (=2×682): it is the number of yods outside the root edge in the inner form of successive Trees of Life. The latter is designed in such a way that 1364 = 4^{1} + 4^{2} + 4^{3} + 4^{4} + 4^{5}. Very beautiful, don't you think? And not a coincidence ...... 
There are 1360 yods outside the root edge not coinciding with the positions of the 6 Sephiroth on the two side pillars of the outer Tree of Life. It is remarkable that this number is the sum of the gematria numbers of Malkuth (496), its Godname ADONAI (65), its Archangel Sandalphon (280), its Angelic Order Ashim (351) and its Mundane Chakra Cholem Yesodeth (168): 1360 = 496 + 65 + 280 + 351 + 168. The total yod population of the (7+7) enfolded Type B polygons (1370) can be represented as a pentagonal array of these numbers enclosed in a square with the Pythagorean integers 1, 2, 3 & 4 assigned to its corners. The number 4 denotes the 4 yods in the root edge and the sum (6) of the integers 1, 2 & 3 denotes the 6 yods at the positions of the 6 Sephiroth on the two side pillars. 

Interpretation in terms of the 4_{21} polytope The inner form of the ntree consists of n sets of (7+7) enfolded polygons, the topmost corners of the two hexagons in the mth set coinciding with the lowest corners of the pair of hexagons in the (m+1)th set. Each set has 1368 yods that are intrinsic to that set. The number of yods in the n sets ≡ N(n) = 1368n + 2. But 1368 = 1370 − 2 = 26 − 2 + 1344 = 24 + 1344. ∴ N(n) − 2 = 24n + 1344n. The number of yods intrinsic to the 10tree = 240 + 13440 = 13680. The (70+70=140) polygons have 240 yods that are either shared with the 10 Trees of Life or centres of polygons that are not also corners. Surrounding the latter are 13440 yods that consist of 6720 pairs of yods. One yod in a pair belongs to a polygon in one set of 7 polygons and its partner belongs to the corresponding polygon in its mirrorimage set. Compare these properties of the inner form of 10 Trees of Life with the 4_{21} polytope when it is constructed from tetractyses. It has 240 vertices that represent the 240 roots of the rank8, exceptional Lie group E_{8}. Its 60480 triangular faces have 6720 edges. When each face is turned into a tetractys, two hexagonal yods line each edge. The number of yods lining the sides of the 60480 tetractyses = 240 + 6720×2 = 13680. This is the number of yods that are intrinsic to the inner form of 10 Trees of Life. The correspondences:
have only one implication: the 4_{21} polytope is the 8dimensional realisation of the inner form of 10 Trees of Life. Because the 4_{21} polytope possesses sacred geometry (see p. 2 et seq in the section 4d sacred geometries), the symmetry group E_{8} must be realised in Nature, so that E_{8}×E_{8} heterotic superstrings must exist. 

E_{8} Coxeter plane projection of the 4_{21} polytope. 
The number of vertices in the 4_{21} polytope = 240 = 2^{4} + 2^{5} + 2^{6} + 2^{7}. As 240 = 16×15, where 16 = 4^{2} and 15 = 4^{2} − 1, 240 = 4^{2}(4^{2}−1) = 4^{4} − 4^{2} = 2^{8} − 2^{4}. The number of edges in the 4_{21} polytope = 6720 = 10×672. As 10 = 2^{1} + 2^{3} and 672 = 2^{5} + 2^{7} + 2^{9}, 6720 = 2^{6} + 2^{9} + 2^{11} + 2^{12}. The number of vertices & edges = 240 + 6720 = 6960 = 2^{4} + 2^{5} + 2^{8} + 2^{9} + 2^{11} + 2^{12}. The number of hexagonal yods lining the sides of the 60480 tetractyses in the 4_{21} polytope = 2×6720 = 13440 = 2^{7} + 2^{10} + 2^{12} + 2^{13}. The number of yods lining these sides = 240 + 13440 = 13680 = 2^{4} + 2^{5} + 2^{6} + 2^{8} + 2^{10} + 2^{12} + 2^{13}. This is the number of yods intriunsic to the inner form of 10 Trees of Life, namely, the (70+70) Type B polygons (see discussion of previous diagram). Its powerof2 expression is the sum of (4+4) powers of 2:
For more discussion of the yod composition of the 4_{21} polytope, see here. 
Sceptics, please note: the author is, of course, aware that any positive integer can be expressed as the sum of different powers of 2. The points that his analysis makes here are:
It is unreasonable to dismiss both these two nontrivial findings as the product of chance because, selfevidently, the possibility is far too improbable. Instead, they confirm the Kabbalistic belief that the Tree of Life is, indeed, designed by a transcendental Intelligence that is the reason for the beautiful harmony between number and sacred geometries. Far too many examples of such design exist for the alternative explanation of coincidence to be plausible. These findings also provide evidence that E_{8}×E_{8} heterotic superstrings exist because — if they did not — the undeniable correlation between the geometries of the inner Tree of Life and the 4_{21} polytope (arithmetically confirmed, as shown above) would have to be due, simply, to coincidence, an explanation which even sceptics would find hard to believe!
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