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#54 E_{8}×E_{8} conforms to the holistic division 384 = 192 + 192 embodied in sacred geometries such as the inner form of the Tree of Life

The 240 vertices of the 8dimensional 4_{21} polytope represent the 240 nonzero roots of E_{8}.128 of these roots are 8tuples with halfintegral coordinates (±½) and the remaining 112 are 8tuples with integral coordinates (either 0 or ±1). This means that (8×240=1920=192×10) coordinates specify the positions of the 240 vertices. In the case of the symmetry group E_{8}×E_{8} appearing in E_{8}×E_{8} heterotic superstring theory, (1920+1920=3840=384×10) coordinates specify the (240+240=480) vertices of a pair of 4_{21} polytopes. We see that, apart from the factor of 10, the number of coordinates needed to specify these vertices is the number 384. It is the global parameter of holistic systems (see here), whose division: 384 = 192 + 192 is studied in many examples of sacred geometries discussed in this website (e.g., see here, here, here, here & here). This property of the 480 roots of E_{8}×E_{8} amounts to powerful evidence of the archetypal status of the E_{8}×E_{8} heterotic superstring, the subquark state of which manifests as the yettodiscovered UPA, the subatomic particle remoteviewed over a century ago by the Theosophists Annie Besant and C.W. Leadbeater (see Occult Chemistry).
It is shown in #35 of Miscellaneous wonders that the number 3840 is the number of sides of triangles outside the root edge of the inner Tree of Life whose (7+7) enfolded polygons are 4thorder polygons (Type D polygons with Type C triangles as their sectors). The seven 4thorder polygons in each half have 1269 triangles with 1920 sides and 650 (=65×10) corners outside their root edge. This is how ADONAI, the Godname of Malkuth with number value 65, prescribes the shape of the inner Tree of Life defined by its corners. Outside the root edge of the (7+7) enfolded 4thorder polygons are 1300 corners, where
1^{4} 

2^{4}  2^{4}  
1300 = 1^{5} + 2^{5} + 3^{5} + 4^{5} = 
3^{4} 
3^{4} 
3^{4} 

4^{4}  4^{4}  4^{4}  4^{4} . 
1300 = 1^{5} + 2^{5} + 3^{5} + 4^{5} = 


Such Pythagorean character in the number of shapedetermining corners of triangles making up the inner Tree of Life with 4thorder polygons is further evidence of the numbers 384 and 3840 being parameters of holistic systems.
Including the root edge, the inner Tree of Life with 4thorder polygons contains 2538 triangles with 3841 sides. This number can be represented by a 10fold array of the squares of the integers 110 because
1^{2} 

2^{2} 
3^{2} 
^{ }  
385 = 
4^{2} 
5^{2} 
6^{2} 

_{ } 
7^{2} 
8^{2} 
9^{2} 
10^{2} , 
so that
3840 = 10×384 = 10×(2^{2}+3^{2}+4^{2}+5^{2}+6^{2}+7^{2}+8^{2}+9^{2}+10^{2})
and
3841 = 1^{2} + 10×(2^{2}+3^{2}+4^{2}+5^{2}+6^{2}+7^{2}+8^{2}+9^{2}+10^{2}).
The representation comprises 91 squares, where
91 = 1^{2} + 2^{2} + 3^{2} + 4^{2} + 5^{2} + 6^{2}.
The central square 1^{2} corresponds to the root edge. The sum (1920) of the 45 squares of 210 in each pentagram is the number of sides of the 1269 triangles in each half of the inner Tree of Life outside the root edge shared by the two sets of seven polygons. This is the arithmetic counterpart of a geometrical object that has a holistic character because it embodies the same patterns and parameters as other sacred geometries. The total number of sides of triangles in the seven enfolded, 4thorder polygons is 1921, which is the sum of the 46 squares of 110 in either pentagram. The number 46 is the number of yods in a Type B triangle (see here). It is also the human diploid number, being the number of types of chromosomes in the human cell.
The Tetrad (4) and the integers 1, 2, 3 & 4 symbolised by the four rows of dots in the tetractys express the number 3640 because
240 = 1×2×3×4(1+2+3+4),
so that
3840 = 1920 + 1920 = 8×240 + 8×240 = 16×240 = 1×2×3×4(1+2+3+4)4^{2}.
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