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**1. The eight triacontagons in the
E _{8} Coxeter plane projection of the 4_{21} polytope.**

The superstring structural parameters 168 & 84
are, respectively, the interior angle of the Petrie polygon of the 4_{21} polytope with
E_{8} symmetry and the base angle of its sectors. |

When projected into 4-dimensional space, the 8-dimensional 4_{21} polytope has a
particular projection in which its vertices appear arranged at the 240 corners of eight concentric,
similar regular polygons called triacontagons, each with 30 corners. The triacontagon is known to be the
so-called "Petrie polygon" for the E_{8} Coxeter plane projection of the 4_{21} polytope.
The 120 red vertices in four triacontagons are the projected vertices of a 600-cell; the remaining 120 blue
vertices of four other triacontagons are the projected vertices of another, smaller 600-cell. This polychoron is
discussed in #1 of Polychorons & Gosset polytope.

A triacontagon has an interior angle of **168**°. This is remarkable because the
number **168** is the gematria number value of *Cholem Yesodoth*, the Mundane Chakra of
Malkuth. Each of the 30 sectors of the triacontagon is an isosceles triangle with a base angle of 84° and an apex
angle of 12°. This display of the 84:84 division of **168** is characteristic of sacred geometries, being
part of the archetypal pattern that they represent (see here). It manifests in the UPA, the subquark state of the
E_{8}×E_{8} heterotic superstring remote-viewed over a century ago by Annie Besant &
C.W. Leadbeater, as the **168** circular turns in every half-revolution of each of its 10 helical
whorls. For a whorl with 1680 turns spirals 5 times around the axis of the UPA, making 2½ revolutions with 840
turns in its outer half and 2½ revolutions with 840 turns in its inner half. Every ¼ revolution of a whorl
comprises 84 turns, where

84 = 1^{2} + 3^{2} + 5^{2} + 7^{2} =
4^{1} + 4^{2} + 4^{3}.

*The fact that the Petrie polygon of the E _{8} Coxeter
plane projection of the 4_{21} polytope has an interior angle equal to the basic structural parameter
168 of the UPA cannot, plausibly, be dismissed as coincidence because the odds are extremely
small that a number characterising the geometry of such a complex mathematical object would just happen by chance
to be identical to that reported 109 years ago as a result of alleged remote-viewing of the basic constituents of
matter! Instead, here is smoking gun evidence that the UPA described by Annie Besant & C.W. Leadbeater is
a state of the E_{8}×E_{8} heterotic superstring. There is no alternative explanation for the
appearance*

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