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#78 The 56:56:56 & 84:84 divisions in six sacred geometries and the seven diatonic musical scales and their counterparts in the exceptional Lie group E_{8}
Demonstrated below is how six sacred geometries, the 64 hexagrams of I Ching and the seven diatonic musical scales embody a 56:56:56 division analogous to that shown by the 168 roots of the rank8, exceptional Lie group E_{8} that are not also roots of its exceptional subgroup E_{6}. They are evidence that not only these different representations/expressions of the cosmic blueprint encode the group mathematics of superstrings but also the Theosophist C.W. Leadbeater possessed genuine micropsi ability because the E_{8} grouptheoretical parameter 168 is at the heart of his description of the UPA, identified by the author as the subquark state of the E_{8}×E_{8} heterotic superstring.
168 = 3×56 168 = 2×84
and their inversions:
The 168 roots comprise 84 roots and their 84 inversions. 

The 240 roots of the rank8, exceptional Lie group E_{8} are represented by 240 vectors in 8dimensional, Euclidean space. These vectors all have the same length (by convention, it is taken to be √2). These vectors are the position vectors of the 240 vertices of the semiregular polytope called the "Gosset polytope," or 4_{21} polytope (this is discussed here). The 28 permutations of (−1, −1, 0, 0, 0, 0, 0, 0) are the mirror images of the 28 permutations of (1, 1, 0, 0, 0, 0, 0, 0). The 56 permutations of (1, −1, 0, 0, 0, 0, 0, 0) contain their mirror images, so that it contains 28 permutations and their 28 mirror images, in which +1→−1 & −1→+1. The 28 permutations of (−½, −½, −½, −½, −½, −½, ½, ½) are the mirror images of the 28 permutations of (−½, −½, ½, ½, ½, ½, ½, ½). There are 168 permutations that consist of three sets of 28 permutations and three sets of their mirror images. The remaining 72 roots consist of 70 permutations of (−½, −½, −½, −½, ½, ½, ½, ½), that is, 35 permutations and their mirror images, and two roots, namely, (½, ½, ½, ½, ½, ½, ½, ½) and its mirror image (−½, −½, −½, −½, −½, −½, −½, −½). The 72 roots comprise (35+1=36) roots and their 36 mirror images.
When their 47 sectors are tetractyses, the seven enfolded polygons making up each half of the inner Tree of Life have 120 yods lining their boundaries. They comprise 36 corners and 84 hexagonal yods. The topmost corner of the hexagon coincides with the lowest corner of the hexagon enfolded in the next higher Tree of Life. It is the only corner of any of the seven polygons that is not intrinsic to them. It is coloured black in the diagram above, its counterpart in the mirror image set of polygons being coloured white. The 36 corners of each half of the inner Tree of Life consist of 35 intrinsic corners (coloured grey) and one corner shared with the inner form of the next higher Tree. Both separate halves contain 72 corners that comprise 70 intrinsic corners and two shared corners. 2×84 hexagonal yods line the 84 sides. They form three sets of 28 hexagonal yods lining each set of seven enfolded polygons, i.e., 3×56 hexagonal yods for both sets.
If the two hexagonal yods in the shared root edge are associated with the dodecagon, this polygon can be paired only with the triangle to create 14 sides with 28 hexagonal yods. This in turn means that the decagon can be combined only with the hexagon to create 14 sides, which means that the octagon can be combined only with the square and pentagon to create 14 sides with 28 hexagonal yods. If, instead. the two hexagonal yods are associated with one of the six other polygons, there are six possible combination schemes that can generate three sets of 28 hexagonal yods:
Such ambiguity in correspondence between hexagonal yods and roots of E_{8} is unattractive because, unless there is a plausible reason why it should exist (and there is none here), a unique, onetoone correspondence should exist between hexagonal yods and these roots. Associating the root edge with the dodecagon is the most likely possibility because it leads to a unique, triple combination of polygons with 28 hexagonal yods in their 14 sides.
Comparing the six sets of 28 permutations of the 8tuples with the sets of yods lining sides of the polygons in the two halves of the inner Tree of Life, the 72 corners correspond to the 72 8tuples of the last three types listed in the diagram and the 168 hexagonal yods correspond to the 168 8tuples of the first five types. The two chiral halves of the inner Tree of Life with 120 boundary yods correspond to the 120 8tuples and their 120 mirror images. The two shared corners of the hexagons correspond to the root at (½, ½, ½, ½, ½, ½, ½, ½,) and its mirror image at (−½, −½, −½, −½, −½, −½, −½, −½). 

Outer Tree of Life Inner Tree of Life Outer & inner Trees of Life 

The 1st (4+4) enfolded Type B polygons have
(28+28= 56) corners of 102 triangles; the last (3+3) enfolded Type B polygons have
(56+56=112) corners of 180 triangles. 
168 = 3×56 168 = 2×84 

168 = 3×56 85 = 4^{0} + 4^{1} + 4^{2} + 4^{3}, of which 84 geometrical elements surround its centre, where 84 = 1^{2} + 3^{2} + 5^{2} + 7^{2}. There are 28 such elements per sector. There are 56 elements per sector and its counterpart in the mirror image triangle making up the (7+7) polygons of the inner Tree of Life. The two Type C triangles have (3×56=168) geometrical elements surrounding their centres. 168 = 2×84


84 black/white yods 
168 = 3×56 168 = 2×84 

56 black & white yods in either red, green or blue triangles 
168 = 3×56 168 = 2×84 

The 64 hexagrams 
168 = 3×56 168 = 2×84
The three rows of a trigram correspond to the three types of 8tuple:
Interchanging lines & broken lines in a trigram corresponds to inverting the 8d root vector, i.e., (+) signs → (−) signs and vice versa. Each offdiagonal line and broken line denotes one of the 168 8tuples. The fact that they comprise 84 lines and 84 broken lines reflects the fact that these 8tuples consist of 84 8tuples and their 84 inversions, in which all (+) signs → (−) signs and vice versa. The 168 lines & broken lines in the 56 offdiagonal trigrams in the lower half of the 8×8 array of hexagrams denote the 168 8tuples expressing those roots in the second E_{8}′ group belonging to E_{8}×E_{8}′ that are not roots of its exceptional subgroup E_{6}. 

The 3dimensional Sri Yantra 
168 = 3×56 168 = 2×84 

168 = 3×56 If each triangular face is considered a tetractys, the 60 tetractyses either above or below the equatorial plane are lined by (84×2=168) hexagonal yods. Assigning blue hexagonal yods to AB edges, green hexagonal yods to BC edges and red hexagonal yods to AC edges, each set of 168 hexagonal yods comprises (28×2=56) blue yods, 56 green yods and 56 red yods. 168 = 2×84 

168 = 3×56 X^{3}Y + Y^{3}Z + Z^{3}X = 0 have a {3,7} tiling on the Poincaré disc. It consists of 56 triangles with 168 vertices. As the Klein quartic has genus 3, its 168 symmetries can be mapped onto the hyperbolic surface of a 3torus. Regarded as Type A triangles (albeit distorted because they lie not on a plane but on a hyperbolic surface), the 56 triangles have 168 sectors. 168 = 2×84 Including the 56 hexagonal yods at the centres of tetractyses and the 24 corners, (168+56+24=248) yods make up the 56 tetractyses. This is the dimension of E_{8}. This is another indication that the {3,7} tessellation of the 3torus constitutes sacred geometry. Together with its version turned inside it, 496 yods make up their 112 tetractyses. This is the dimension of E_{8}×E_{8}, one of the two symmetries that govern heterotic superstrings. (For details, see Table 3 on p. 11 of Article 43). 

168 = 3×56 168 = 2×84 There are (28+11=39) repeated D & E intervals and (23+17+5=45) repeated F, G & A intervals. Hence, the seven diatonic scales have (39+39=78) repeated, rising and falling D & E intervals and (45+45=90) repeated, rising and falling F, G & A intervals. The seven diatonic scales embody the gematria number values of Cholem (78) and Yesodoth (90), the two words in Cholem Yesodoth, the Kabbalistic name of the Mundane Chakra of Malkuth. (For more details, see Article 16 and here). The 84 repeated, rising and 84 repeated, falling, Pythagorean intervals are the counterparts of the 84 roots in E_{8} and their 84 inversions that are not roots of E_{6}. Amazingly, in each case they form three sets of 28. This is because the seven diatonic scales constitute a holistic system that conforms to the archetypal pattern embodied in sacred geometries. 

The 84:84 division in a helical whorl of the UPA 
When he observed the UPA with micropsi, Charles W. Leadbeater counted 1680 circular turns in each of its 10 helical whorls. As a whorl twists five times around the spin axis of the UPA, one revolution of a whorl consists of 336 turns. A halfrevolution consists of 168 turns and a quarterrevolution consists of 84 turns, where 84 = 1^{2} + 3^{2} + 5^{2} + 7^{2} = 4^{1} + 4^{2} + 4^{3}. This 84:84 division (or its 840:840 counterpart in the outer and inner halves of each whorl) manifests in sacred geometries (see Article 64) and in the seven diatonic scales (see also here). The vibrational ground state of the E_{8}×E_{8} heterotic superstring is the microscopic realisation of the archetypal pattern embodied in sacred geometries, whilst the seven diatonic scales are its macroscopic manifestation in music. 
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