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**#47 Inner form of 10 Trees of Life with Type C polygons is
the geometrical counterpart of the two 4 _{21} polytopes associated with
E_{8}×E_{8}**

We
found in #46 that the inner form of 10 overlapping Trees of Life, namely, 70 Type B polygons, requires 6720 yods to
transform their 480 sectors, each yod corresponding to one of the 6720 edges in the 4_{21} polytope.
We now show that, when transformed into Type C polygons, these polygons have (6720+6720) geometrical elements
surrounding their centres that correspond to the (6720+6720) edges of the pair of 4_{21} polytopes
associated with the symmetry group E_{8}×E_{8}. In other words, the geometrical composition of the
inner form of 10 Trees of Life reproduces the number of edges belonging to the two polytopes representing the 480
roots of E_{8}×E_{8}.

The Type B triangle:

Type B triangle |

has
seven corners and **15** sides of 9 triangles, i.e., **31** geometrical
elements, where **31** is the number value of EL, the Godname of Chesed. The Type C n-gon has n
sectors that are Type B triangles, each sharing one side and two corners with an adjacent sector. This means that
the number of geometrical elements in a Type C n-gon = 28n + 1, where "1" denotes its centre. The seven separate
Type C polygons making up the inner form of the Tree of Life have **48** sectors with
(**48**×28=1344) geometrical elements surrounding their centres. Curiously, the number 28 also shows
up in the representation of 1344 as the sum of squares of four integers, starting with the integer 4:

1344 = 4^{2} + 12^{2} + 20^{2} + 28^{2}.

The
number 1344 is the arithmetic mean of the squares of the first **31** even integers:

1344 =
(2^{2}+4^{2}+6^{2}+...+**62**^{2})/**31**.

We saw in #45 that the (7+7) enfolded Type B polygons have (672+672=1344) yods surrounding their centres that are unshared with the outer Tree of Life. That the same number appears again in the next higher-order polygons belonging to both sets of seven polygons cannot, plausibly, be dismissed as chance. The division:

**48** = 24 + 24,

which
is characteristic of holistic systems, manifests in the **48** corners of the seven separate
polygons as the 24 corners of the triangle, square, pentagon & dodecagon and the 24 corners of the hexagon,
octagon & decagon. The former set of polygons (coloured red in the diagram below) has (24×28=672) geometrical
elements surrounding their centres, as do the latter set (coloured blue).

Coxeter plane projection of the 4 |
The 10 sets of Type C triangles, squares, pentagons & dodecagons that are part of the inner form of 10 overlapping Trees of Life have 6720 geometrical elements surrounding their centres, as do the 10 sets of Type C hexagons, octagons & decagons. |

This
division, therefore, generates the direct product E_{8}×E_{8} because the 10 sets of red
polygons have 6720 geometrical elements surrounding their centres, as do the 10 sets of blue polygons, so that they
correspond to the 6720 edges of the 4_{21} polytope representing each group
E_{8} in the direct product. This is the Tree of Life basis
of E_{8}×E_{8} heterotic superstring theory.

The 28
geometrical elements per Type B sector of the Type C n-gon comprise five corners, 14 sides & nine triangles,
i.e., 14 corners & triangles and 14 sides. Hence, the **48** sectors of the seven Type C
polygons consist of 672 corners & triangles and 672 sides surrounding their centres, so that the 480 sectors of
the 70 Type C polygons in the inner form of 10 Trees of Life comprise 6720 corners & sides and 6720 sides
surrounding their centres. We may, alternatively, see the division 13440 = 6720 + 6720 (and, therefore, the direct
product E_{8}×E_{8}) as arising from the distinction between the sides and corners & triangles
in the polygons. To attribute this, too, to chance is, clearly, implausible in the extreme.

As the last of the regular polygons making up the inner Tree of Life, the Type C dodecagon has 336 geometrical
elements (**168** corners & triangles, **168** sides) surrounding its centre:

Each half of the 4 |
Constructed from Type B triangles, each of the two Type C dodecagons has 336 corners, sides & triangles surrounding its centre, i.e., 3360 for the 10 dodecagons belonging to the 70 polygons in the inner form of 10 Trees of Life. |

(see also here). The triangle, square & pentagon, too, have 336 geometrical
elements surrounding their centres. The 40 triangles, squares, pentagons & dodecagons in the complete set of
70 Type C polygons have 6720 geometrical elements (3360 corners & triangles, 3360 sides) surrounding their
centres, as do the 30 hexagons, octagons & decagons. They include the 10 dodecagons with 1680 corners &
triangles and 1680 sides. *Embodied in the dodecagon is the superstring structural
parameter 1680 paranormally obtained by C.W. Leadbeater over a century ago when he applied his micro-psi vision
to the UPA. Its identification by the author as the subquark state of the
E _{8}×E_{8} heterotic superstring is confirmed here in the inner Tree of Life basis of the
4_{21} polytope representing the exceptional Lie group E_{8}.*

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