#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:
48 lines & broken lines in the 8 diagonal hexagrams ↔
48 hexagonal yods either lining tetractyses in the outer Tree or on the lower
side of each triangle in the inner Tree;
168 lines & broken lines in the 28 upper, off-diagonal hexagrams ↔
168 hexagonal yods outside root edge in one set of 7 enfolded polygons that are
unshared with the outer Tree;
168 lines & broken lines in the 28 lower, off-diagonal hexagrams ↔
168 hexagonal yods outside root edge in the other set of 7 enfolded polygons that
are unshared with the outer Tree.
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 mirror-image halves of the outer & inner Trees of
Life.
An n-gon 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 off-diagonal 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 micro-psi by Annie Besant & C.W. Leadbeater consists of 1680
1st-order 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 half-revolution of a whorl and the number 84 represents the number of spirillae in a
quarter-revolution (see here). The two mirror-image halves of the inner form of the 10 Trees manifest in
the subatomic world as the outer and inner halves of each string-like 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 off-diagonal hexagrams in each diagonal half of the 8×8 square array. One may visualize each half-revolution
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 off-diagonal 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 off-diagonal hexagrams represent the 84
positive and 84 negative phases of the 84 circularly polarized oscillations, or 1st-order spirillae, that occur
in a whorl when it turns 90° around the axis of spin of the UPA/subquark.
Combined Trees of Life
