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#54 E8×E8 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 8-d 421 polytope representing the 240 roots of E8 have (8×240=1920) coordinates. They correspond to the 1920 sides outside the root edge of the 7 enfolded, 4th-order polygons making up each half of the inner Tree of Life. The 3840 sides of the complete inner form correspond to the (1920+1920=3840) coordinates of the (240+240=480) vertices of two 421 polytopes representing E8×E8. The Decad determines the number 3840 because it is the sum of the 90 squares of 2-10 arranged in two 5-fold arrays.

The 240 vertices of the 8-dimensional 421 polytope represent the 240 non-zero roots of E8.128 of these roots are 8-tuples with half-integral coordinates (±½) and the remaining 112 are 8-tuples 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 E8×E8 appearing in E8×E8 heterotic superstring theory, (1920+1920=3840=384×10) coordinates specify the (240+240=480) vertices of a pair of 421 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 E8×E8 amounts to powerful evidence of the archetypal status of the E8×E8 heterotic superstring, the subquark state of which manifests as the yet-to-discovered UPA, the subatomic particle remote-viewed 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 4th-order polygons (Type D polygons with Type C triangles as their sectors). The seven 4th-order 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 4th-order polygons are 1300 corners, where

 14 24 24 1300 = 15 + 25 + 35 + 45 = 34 34 34 44 44 44 44  .

 1300 = 15 + 25 + 35 + 45 = Outside the root edge of the (7+7) enfolded 4th-order polygons with Type C triangular sectors are 1300 corners of triangles.

Such Pythagorean character in the number of shape-determining corners of triangles making up the inner Tree of Life with 4th-order 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 4th-order polygons contains 2538 triangles with 3841 sides. This number can be represented by a 10-fold array of the squares of the integers 1-10 because

 12 22 32 385 = 42 52 62 72 82 92 102 ,

so that

3840 = 10×384 = 10×(22+32+42+52+62+72+82+92+10 2)

and

3841 = 12 + 10×(22+32+42+52+62+72+82+92+10 2).

The representation comprises 91 squares, where

91 = 12 + 22 + 32 + 42 + 52 + 62.

The central square 12 corresponds to the root edge. The sum (1920) of the 45 squares of 2-10 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, 4th-order polygons is 1921, which is the sum of the 46 squares of 1-10 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)42.

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