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The graph displays the vertices of the disdyakis triacontahedron projected onto the central XY
plane when the Z axis passes through two diametrically opposite A vertices. The vertices are corners of seven
polygons stacked one above the other. The graph shows the four polygons in the upper half of the polyhedron. The
only sides of the polygons that are not polyhedral edges are the six black sides of the hexagon with A & C
vertices at its corners and two black sides parallel to the Y-axis of an indigo 10-gon that join two pairs of its A
vertices. The table indicates that there are eight geometrical elements unshared with the polyhedron and 240
elements which belong either to its faces or to the internal sectors of the seven polygons.
**80** elements belong to the three polygons above the central, green 12-gon and
**168** elements belong to either faces in the upper half of the polyhedron or to half of the
central 12-gon. Similarly, **80** elements belong to the three polygons below the 12-gon and
**168** elements belong to either faces in the lower half of the polyhedron or to the other half
of the 12-gon. Each half of the disdyakis triacontahedron has **248** geometrical elements. They
are the counterpart of the **248** gauge bosons of E_{8} and E_{8}′ that
transmit the unified forces between E_{8}×E_{8}′ heterotic superstrings.

The disdyakis triacontahedron embodies the number **496** of Malkuth and the
dimension of E_{8}×E_{8}′ as the number of geometrical elements in its faces and seven internal
polygons formed by the 60 vertices that surround an axis joining two opposite A vertices. They include 16 unshared
elements that correspond to the 16 simple roots of E_{8}×E_{8}′ and 480 elements that correspond to
the 480 roots of E_{8} and E_{8}′. They exclude the seven centres of polygons and the two A
vertices.

The total number of geometrical elements that surround the centre of the disdyakis
triacontahedron is **496** + 6 + 2 = 504. This number was encountered in the discussion of the
**Polyhedral Tree of
Life** as:

1. the number of yods lining the edges of the 144 Polyhedron that surround an axis passing through two opposite
vertices;

2. the number of yods surrounding the centre of a Type C dodecagon;

3. the number of yods surrounding the centre of a heptagon with 2nd-order tetractyses as its seven sectors.

It was also described in **Maps of reality/Sri Yantra** as the
504 hexagonal yods that lie on sides of tetractyses in the Sri Yantra with Type A triangles and as the 504 SLs
in CTOL down to the top of the 7-tree that represents the physical plane (physical universe). Just as it was
found in these contexts that the number 504 displays a
**168**:**168**:**168** division, so it does in the disdyakis
triacontahedron, for there are **168** polyhedral edges above and below the central 12-gon,
**168** geometrical elements that are either external triangles (120) or elements of the
12-gon (**48**) and (**80**+**80**+6+2=**168**) elements
that comprise the two A vertices and the elements, including their centres, that make up the six polygons either
above or below the 12-gon. The 84:84 division of **168** is characteristic of holistic
systems. Here, it appears as the 84 polyhedral edges in each half of the disdyakis triacontahedron above or
below the 12-gon, as the 84 elements that are either external triangles (60) in each half or elements in half
the 12-gon (24) and as the 84 elements in each half that are either the A vertex, the centres of the three
polygons or the **80** corners, internal edges, unshared sides & triangles.

The six unshared edges in either half of the polyhedron correspond to the six simple roots of
E_{6}, an exceptional subgroup of E_{8}, and the **72** corners, internal sides
& triangles of the three polygons in its upper or lower halves correspond to the **72** roots
of E_{6}. The 24:24 division of the **48** geometrical elements in the central 12-gon is
characteristic of holistic systems. For example, it was encountered in the 24 lines & broken lines making up
each of the two sets of eight triagrams in the eight diagonal hexagrams in the I Ching table of **64** hexagrams.

In conclusion, the disdyakis triacontahedron embodies the structural parameters 840 & 1680
of the subquark state of the E_{8}×E_{8}′ heterotic superstring and the root composition of the
gauge symmetry group of its unified force. This is one of the reasons why the basic unit of matter described by
Besant & Leadbeater must be identified as this particular type of superstring. All sacred geometries exhibit a
bifurcation into two similar, mirror-image (or in some sense, opposite but complementary) halves, each half
encoding the dimension **248** of E_{8} and embodying the 840 helical turns in the
outer or inner halves of this particle. This duality or split into two sets of geometrical elements that are
opposite to one another is a manifestation of the Taoist principle of the Tao, or cosmic whole, being a cyclic
rhythm of the polar opposite phases of Yang and Yin.

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