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**#38 10-fold geometrical composition of the disdyakis
triacontahedron**

Article 27 showed that, when both the internal
triangles and the faces of the disdyakis triacontahedron are Type A triangles, 2400 corners, sides &
triangles surround any axis passing through two diametrically opposite vertices (see Table 1, p. 4). It will now
be proved that the 10-fold division of vertices & edges found above for this polyhedron extends to its
*total geometrical composition*. In other words, these 2400
geometrical elements divide *naturally* into ten sets of 240 elements
that divide further into three sets of 240 and seven sets of 240. The meaning
of this 10-fold division is that, associated with the 240 roots of E_{8} are 240 vector gauge fields, each with ten
components, so that 240 field components are defined by each of the 10 dimensions of superstring
space-time. Each geometrical element making up the disdyakis triacontahedron that surrounds its axis bears a
formal correspondence to a component of the 240 spin-1 gauge fields of E_{8}.

The adjacent diagram shows one of the 120 faces with centre O. Sides a, b & c are the internal sides of its three sectors that terminate, respectively, on A, B & C vertices. It also depicts an internal triangle with polyhedral vertices at two corners and the centre of the polyhedron at its third corner. The three sides meeting at O′ are of two types: one (X) that terminates on a vertex and one (Y) that ends at the centre. The three sectors of an internal triangle are of two types: one (p) that has a polyhedral edge as one side and two (q) that do not have an edge as a side. The edges are of three types: AB, BC & AC. The disdyakis triacontahedron has 180 edges, there being 60 edges of each type. Each one generates in an internal triangle two X sides & one Y side, one triangle of class p & two triangles of class q, as well as the centre O′. The table below shows the numbers of geometrical elements in the faces and interior of this polyhedron that surround its axis:

Numbers of corners, sides & triangles surrounding the axis of the disdyakis triacontahedron.

In the surface of the
polyhedron are 120 triangular faces, each with three sectors with a, b & c sides, i.e., three sets of 120
sides of sectors and three sets of 120 sectors, i.e., three sets of 240 sides & triangles, or 720
(=**72**×10) geometrical elements. There are also 120 corners that are centres of faces and 60 vertices, as well as 60 sides joining the latter to the
centre of the polyhedron. These form a set of 240 corners & sides. There are three sets of 120 X sides
& 120 q triangles (indicated in orange, yellow & green), i.e., three sets of 240 geometrical elements,
and there are three sets of 60 edges, 60 internal corners, 60 internal sides & 60 internal triangles
(indicated in blue, indigo & violet), i.e., three sets of 240 elements. Hence, there are seven more sets of
240 geometrical elements. All these numbers, of course, refer to the geometrical elements that surround the
axis.

We see that the 2400 geometrical elements
comprise ten sets of 240 that are made up of three sets of 240 sides & triangles in the faces and seven sets
that comprise two triplets of sets and a single set. This 3:3:3:1 pattern reproduces the pattern of Sephiroth
of the Tree of Life, which group into three triads and a final, single one:

Kether-Chokmah-Binah

Chesed-Geburah-Tiphareth

Netzach-Hod-Yesod

Malkuth

There are 720 elements in the faces that are unshared with the
remaining 1680 elements. This pattern is the counterpart of the **72**
roots of E_{6} and the **168** other roots of E_{8}. The 1680
elements comprise 60 vertices & 180 edges, i.e., 240 elements in the faces, leaving 1440 elements, 720
elements belonging to each half of the disdyakis triacontahedron. The division:

1680 = 720 + 240 + 720

Is embodied in the 240 yods
other than SLs in the 1-tree and in the 720 yods that surround the centres of each of the two sets of seven
Type B polygons making up the inner Tree of Life (see diagram here). Every yod or geometrical element corresponds to a circular
turn of each helical whorl of the UPA/subquark superstring. In other words, they denote circularly polarized
oscillations of each whorl. The same division manifests in the compound of two 600-cells whose 240 vertices are
the 4-dimensional projection of the 240 vertices of the 4_{21} polytope (see here). This amounts to convincing evidence that the UPA is an
E_{8}×E_{8} heterotic superstring, for these 240 vertices define vectors that are identical to
the 240 root vectors of the rank-8, exceptional Lie group E_{8}. In other words, the 240 gauge
charges of this symmetry group are spread along the 16800 turns of the 10 whorls of the UPA.