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#55 Correspondence between the 64 hexagrams and the combined outer & inner Trees of Life
Combined Trees of Life
The (7+7) enfolded polygons have 444 hexagonal yods, of which 94 are at the centres of their 94
sectors/tetractyses. Therefore, (444−94=350) hexagonal yods line their 175 sides. Eight hexagonal yods lying on
four vertical, internal sides of sectors of the two hexagons coincide with hexagonal yods on the two side pillars
of the outer Tree of Life constructed from tetractyses and superposed on its inner form. Hence, (350−8=342)
hexagonal yods in the latter are unshared with the former, which has 44 hexagonal yods on the 22 sides of its 16
triangles when they are tetractyses. When the outer and the inner Trees combine, the two hexagonal yods on the Path
connecting Chesed and Geburah coincide with the centres of the two triangles belonging to the set of 14 polygons.
They are not hexagonal yods from the perspective of both Trees. In the outer Tree, 42 pure hexagonal
yods line tetractyses. They are coloured white in the adjacent diagram. The combined Trees have (342+42=384) yods
lining tetractyses that are hexagonal yods from the point of view of both of them. Of these, two white hexagonal
yods are on the root edge and two white hexagonal yods are on the lower side of each triangle belonging to the
inner Tree of Life. This leaves (342−6=336) hexagonal yods in the (7+7) enfolded polygons outside the root edge
that are unshared with the outer Tree. 168 red hexagonal yods belong to one set of seven
enfolded polygons and 168 blue hexagonal yods belong to the other set. The 384 hexagonal yods
consist of (42+6=48) white hexagonal yods and 336 red & blue hexagonal yods.
64 hexagrams
Compare this 48:336 division with the 384 lines & broken lines making up the
64 hexagrams used in I Ching, the ancient Chinese system of divination (see here). The eight diagonal hexagrams in the 8×8 square array of hexagrams consist
of 48 white lines & broken lines. 168 red lines & broken
lines make up the 28 hexagrams above them; 168 blue lines & broken lines make up the
28 hexagrams below the diagonal.
The following correspondences are established:
Three unbroken (Yang) lines compose the Heaven trigram that starts the sequence of the eight trigrams and the table of their 64 possible pairings. They correspond to the upper, white hexagonal yod in the root edge and to the two white hexagonal yods on a side of one of the two triangles belonging to the (7+7) polygons. The counterpart of the three white broken (Yin) lines composing an Earth trigram are the mirror images of these three yods, namely, the lower, white hexagonal yod in the root edge and the pair of white hexagonal yods on the lower side of the triangle in the second set of polygons (the white circles surrounding each pair of these yods denote the white hexagonal yods that lie directly behind them on the Paths of the outer Tree joining Tiphareth to Chesed and Geburah. Interchanging the trigrams A & B in a hexagram (A/B) generates its inversion (B/A) reflected across the diagonal of the 8×8 array of hexagrams. This inversion corresponds to reflection of hexagonal yods across the axis of the central Pillar of Equilibrium as a mirror. The two diagonal halves of the table of hexagrams correspond to the two mirrorimage halves of the outer & inner Trees of Life.
An ngon has 4n hexagonal yods lining the 2n sides of its n tetractys sectors. Tabulated below are the numbers of hexagonal yods in the seven enfolded polygons outside their root edge that are unshared with the outer Tree of Life:
triangle 
square 
pentagon 
hexagon 
octagon 
decagon 
dodecagon 

8 
14 
18 
14 
30 
38 
46 

Total = 
84 
84 
The triangle has 12 such yods (10 outside the root edge). It has eight hexagonal yods other than the two white hexagonal yods on its lower side that are grouped with the hexagonal yods lining tetractyses in the outer Tree. We see that 84 hexagonal yods line tetractyses in the first five enfolded polygons and that 84 hexagonal yods line tetractyses in the decagon & dodecagon. The 168 hexagonal yods in each set of seven enfolded polygons naturally divide into two sets of 84 hexagonal yods.
Compare this 84:84 division with the fact that the 168 red lines & broken lines in the 28 offdiagonal hexagrams consist of 84 lines & 84 broken lines, as do the 168 blue lines & broken lines. Their counterpart in the 168 red hexagonal yods and the 168 blue hexagonal yods unshared with the outer Tree of Life are the 84 hexagonal yods lining tetractyses in the first five enfolded polygons and the 84 hexagonal yods in the decagon & dodecagon:
The 84:84 division is characteristic of sacred geometries (see The holistic pattern). An example
is the first (6+6) enfolded polygons shown above. 84 yods line the sides of each set of six polygons outside
their root edge. Another example is found in the complete inner Tree of Life. Its (7+7) enfolded polygons have
94 sectors with 80 corners and 94 centres. As explained above, six corners (shown as white
yods in the diagram below) coincide with Sephiroth of the outer Tree. This leaves
(80+94−6=168) unshared corners & centres, 84 of which are associated with
each set of seven enfolded polygons:
The significance for superstrings of this 84:84 division displayed by sacred geometries is that each of the 10 whorls of the UPA described with micropsi by Annie Besant & C.W. Leadbeater consists of 1680 1storder spirillae that wind fives times around its axis of spin. There are 840 spirillae in the outer section of 2½ revolutions of a whorl and 840 spirillae in its inner section of 2½ revolutions. Their inner Tree of Life counterparts are the 840 unshared corners & centres of the 470 sectors of the 70 polygons enfolded on each side of the central pillar of 10 Trees of Life (see here). The number 168 represents the number of spirillae in a halfrevolution of a whorl and the number 84 represents the number of spirillae in a quarterrevolution (see here). The two mirrorimage halves of the inner form of the 10 Trees manifest in the subatomic world as the outer and inner halves of each stringlike whorl of the UPA. This is represented in the 64 hexagrams by the 84 unbroken Yang lines and the 84 broken Yin lines making up the 28 offdiagonal hexagrams in each diagonal half of the 8×8 square array. One may visualize each halfrevolution of a whorl as comprising 168 complete, oscillatory cycles, each having a positive (Yang) phase and a negative (Yin) phase. The 168 unbroken lines in the 56 offdiagonal hexagrams symbolize the positive phases of these cycles; their 168 broken lines represent the negative phases. The 84 lines and the 84 broken lines in each set of 28 offdiagonal hexagrams represent the 84 positive and 84 negative phases of the 84 circularly polarized oscillations, or 1storder spirillae, that occur in a whorl when it turns 90° around the axis of spin of the UPA/subquark.
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