#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 E8have (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
