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13. The geometrical composition of the 8 triacontagons
The 8 Type B triacontagons in the E_{8} Coxeter plane of the 4_{21} polytope embody the number (1680) of turns in each helical whorl of the UPA. This and the fact that the interior angle of a triacontagon is 168°, whilst a halfrevolution of a whorl has 168 turns, are remarkable pieces of evidence that the UPA is an E_{8}×E_{8} heterotic superstring. 
Type A triacontagon
The number of geometrical elements in a Type A/1storder ngon* =
4n + 1. This is because each of the n sectors adds one corner, a side of the polygon, an internal side & a
triangle to its centre. A Type A triacontagon (n = 30) has 31 corners of its 30 sectors,
where 31 is the number value of EL, the Godname of Chesed, and 121 geometrical elements,
where 121 = 11^{2} and 11 is the tenth integer after 1. 120 geometrical elements
surround its centre, where
120 = 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21
is the sum of the first ten odd integers after 1. This shows how the Decad determines the geometry of the Petrie polygon of the 4_{21} polytope. It has 30 corners, where
30 = 1^{2} + 2^{2} + 3^{2} + 4^{2}
1 

2  2  
= 
3 
3 
3 

4  4  4  4 
= 2^{1} + 2^{2} + 2^{3} + 2^{4}.
Therefore,
120 = 4×30 = 2^{2}(2^{1} + 2^{2} + 2^{3} + 2^{4}) = 2^{3} + 2^{4} + 2^{5} + 2^{6},
and the number of vertices in the 4_{21} polytope or 8 triacontagons is 240, where
240 = 2×120 = 2^{4} + 2^{5} + 2^{6} + 2^{7}.
The Tetrad expresses these parameters, confirming the holistic nature of the 4_{21} polytope. It also determines the interior angle 168° of a triacontagon because 168 = 13^{2} − 1 = 3 + 5 + 7 +... + 25:
= 
(168 yods surround the centre of a Type C square),
the base angle 84° because
and the sum (24°) of the vertex angles of two adjacent sectors because 24 = 4! = 1×2×3×4.
Type B triacontagon
The number of geometrical elements in a Type B/2ndorder ngon =
10n + 1. A Type B triacontagon has 301 geometrical elements. The number of geometrical elements in 8 concentric
triacontagons = 1 + 8×300 = 2401 = 49^{2}. EL CHAI, the Godname of Yesod with number value
49, prescribes the geometrical composition of the 8 triacontagons when they are Type B. 2400
geometrical elements surround their shared centre, of which 240 are vertices and 2160 (=216×10)
are corners, sides & triangles, where 216 is the number of Geburah. The number of yods
surrounding the centre of the 8 triacontagons = 8×30×15 = 3600 = 36×10×10,
where 36 is the number value of ELOHA, the Godname of Geburah. The number value
21 of EHYEH, the Godname of Kether, determines the number of vertices of the 8 triacontagons
because 240 is the sum of the first 21 binary coefficients other than 1:
YAH, the Godname of Chokmah with number value 15, prescribes the triacontagon with 30 vertices because 30 is the 15th even integer. YAH prescribes all 8 triacontagons with 240 vertices because 240 is the sum of the first 15 even integers:
240 = 2×8×15 = 2×(1+2+3+4+...+15) = 2 + 4 + 6 + 8 +...+ 30.
Surrounding the centre of four Type B triacontagons are 840 corners & sides and 360 triangles. Surrounding the centre of (4+4) Type B triacontagons are (840+840=1680) corners & sides and (360+360=720) triangles. The division:
2400 = 720 + 1680
is characteristic of holistic systems. For example, it manifests in the disdyakis triacontahedron as the 1680 geometrical elements that surround its axis when its faces are 0thorder triangles and as the 720 extra geometrical elements added when its faces are 1storder triangles (see here). It is yet more evidence that the 4_{21} polytope, whose 240 vertices project onto the corners of 8 triacontagons, constitute a holistic system. This implies, of course, that E_{8}×E_{8} heterotic superstrings exist. The facts that:
constitute persuasive evidence supporting this implication. This is because C.W. Leadbeater counted 1680 turns in each helical whorl of the UPA, its outer and inner halves having 840 turns in 2½ revolutions of a whorl around its axis, 168 turns in a halfrevolution and 84 turns in a quarterrevolution. It is highly implausible that this similar set of numbers could appear by chance in the context of a paranormal account of a remoteviewed subatomic particle! Instead, it demonstrates that the 8 triacontagons, whose corners are the projections of the vertices of the 4_{21} polytope, constitute sacred geometry that embodies both the unified forces between superstrings (namely, the 240 gauge fields associated with the 240 roots of E_{8}) and their structure in 4d spacetime, as described with micropsi by Besant & Leadbeater. The author is not claiming that just this property alone is the explanation for the 1680 turns in each helical whorl of the UPA. Obviously, it is only an aspect of the complete account. What he is asserting is that this number and others related to it, like 84 and 336, with a paranormal provenance do not appear by accident in the geometry of the polytope that represents the symmetry group E_{8} involved in E_{8}×E_{8} heterotic superstring theory. That is just too improbable. On the other hand, its appearance is, precisely, what we could expect if the UPA is, indeed, a certain state of this type of superstring!
As shown in the first diagram, each sector of a Type B polygon adds 7 geometrical elements to the 3 corners & sides lining each sector of the polygon when it is Type A. This 3:7 division of the 10 geometrical elements per sector of a Type B polygon reflects the distinction in the tetractys between the 3 yods at its corners and its 7 hexagonal yods, this representing in the Tree of Life the difference between the Supernal Triad and the 7 Sephiroth of Construction. (7×240=1680) geometrical elements are added in the 8 Type B triacontagons to (3×240=720) geometrical elements in the 8 triacontagons when they are Type A. The division:
240 = 72 + 168
of the 240 roots of E_{8} into the 72 roots of its exceptional subgroup E_{6} and 168 remaining roots is an example of this. It demonstrates the deep parallels between sacred geometry and group theory. It needs to be pointed out that the superstring structural parameter 1680 appears in this calculation NOT because the Petrie polygon of the 4_{21} polytope is the triacontagon but because E_{8} has 240 roots. If its Petrie polygon had been some other regular Ngon, its Coxeter projection would generate n of them, where nN = 240, so that the the number of corners & sides in the n Ngons would have been n×N×7 = 1680. So this parameter depends only on the total number of sectors of all the Petrie polygons, not on what these polygons are. In other words, it depends only on the number (240) of vertices of the 4_{21} polytope. A Type C ngon has 28n corners, sides & triangles surrounding its centre, so that a single triacontagon has (28×30=840) geometrical elements and all 8 triacontagons have 3360 elements. Both of these are structural parameters of the UPA. What are the chances of that occuring if the UPA had been simply an hallucination?!
The regular triacontagon has D_{30} dihedral symmetry, order 60, represented by 30 rotations and 30 lines of reflection. The Tetrad prescribes its order because the centre of a Type B square is surrounded by 60 yods:
D_{30} has 7 dihedral subgroups: D_{15}, (D_{10}, D_{5}), (D_{6}, D_{3}), and (D_{2}, D_{1}). It also has 8 more cyclic symmetries as subgroups: (Z_{30}, Z_{15}), (Z_{10}, Z_{5}), (Z_{6}, Z_{3}), and (Z_{2}, Z_{1}), with Z_{n} representing 360/n degree rotational symmetry. Let us work out how many different rotations are possible that leave the triacontagon in the same orientation. The dihedral group D_{n} is the group of n rotations {mθ_{n}; m = 1−n, θ_{n} = 360°/n)} and n reflections. Below is shown how the number value 65 of ADONAI, the Godname of Malkuth, is the number of different rotations when the single rotation through 360° is included):
The Tetrad and the Pythagorean integers 1, 2, 3 & 4 express the total rotation angle because 32×360 = 4×8×10×36 = 4×10×288 = 4(1+2+3+4)(1!×2!×3!×4!). They express the 33 circles when the rotation through 360° is included because 33 = 1! + 2! + 3! + 4!. This number can be interpreted as the number of permutations, one row at a time, of 10 objects arranged in the 4 rows of a tetractys.
* See Power of the polygons/Collective properties for a discussion of ngons of any order.
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