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**4. The triacontagon has the holistic
pattern of the Type B dodecagon & the disdyakis triacontahedron**

The 12:84:84 pattern in the Type B dodecagon and the disdyakis triacontahedron. |

The dodecagon is the last of the seven types of regular polygons making up the inner form
of the Tree of Life. 180 yods surround its centre when it is Type B. The 360 yods surrounding the centres of the
two Type B dodecagons in the (7+7) regular polygons symbolise the 360 degrees in a circle, so that each dodecagon
represents a half-revolution of 180°. Its 12 corners denote the 12 degrees in each sector of the triacontagon. The
**168** other yods surrounding its centre denote the **168** degrees in the two base angles of a sector. The
6 corners & 84 yods in the pair of 6 sectors of the Type B dodecagon denote the angles of 6° & 84° in the
pair of right-angled triangles making up a sector of the triacontagon. Of the **168** yods, 60 belong to the
Type A dodecagon as yods other than its corners, leaving 108 yods. The interior angles of an equilateral triangle
and a pentagon are, respectively, 60° and 108°. The significance of these polygons is that the triacontagon is the
largest regular polygon whose interior angle is the sum of the interior angles of smaller polygons, namely, the
triangle and the pentagon.

The disdyakis triacontahedron has **62** vertices: 30 A, 12 B & 20 C
vertices. When orientated so that its vertical axis passes through two diametrically opposite A vertices, a 12-gon
whose corners coincide with 4 A, 4 B & 4 C vertices lies in its equatorial plane. All its sides are edges of
the polyhedron. The 180 edges of the disdyakis triacontahedron comprise the 12 edges forming the central 12-gon and
**168** edges, arranged as 84 edges either above or below it. This 84:12:84 division of edges
corresponds to the three angles of a sector of the triacontagon, the Petrie polygon of the 4_{21} polytope.
It is additional evidence for the holistic character of the 4_{21} polytope and therefore the
*reality* of E_{8}×E_{8} heterotic superstrings.

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