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In the discussion in **Superstrings as sacred
geometry** of the Tree of Life basis of superstrings, it was shown that the 240 yods
lining the sides of the two separate sets of seven enfolded polygons comprise **72** corners
and **168** hexagonal yods (see here). This **72**:**168** division is reproduced
in the first three Platonic solids as the **72** hexagonal yods in half of the tetrahedron and
half of the octahedron and as the **168** hexagonal yods in their other halves and in the cube
(see here). It will be now be shown how this division is embodied in the disdyakis
triacontahedron.

The red and green diagonals of the rhombic triacontahedron are the edges of, respectively, a
dodecahedron and an icosahedron. Their lengths are in the proportion of the Golden Ratio Φ = 1.618... . Sticking
pyramids on each of the 30 Golden Rhombic bases of the rhombic triacontahedron generates the disdyakis
triacontahedron with **62** vertices, 120 triangular faces & 180 edges. When these faces are
turned into tetractyses, (180×2=360) red hexagonal yods line their sides and 120 black hexagonal yods are at their
centres. Hence, their faces have (120+360=480) hexagonal yods. The 240 hexagonal yods in each half of the
polyhedron consist of 60 black hexagonal yods at centres of tetractyses and 180 red hexagonal yods on their 90
sides. The equatorial plane perpendicular to an axis passing through two diametrically opposite A vertices is
bounded by a 12-gon with four A, four B & four C vertices and with four AB sides, four BC sides & four CA
sides. Each half of the 12-gon has two AB, two BC & two CA sides with 12 red hexagonal yods on them. Therefore,
there are (180−12=**168**) red hexagonal yods on the 84 edges above the equatorial plane and
**168** red hexagonal yods on the 84 edges below it. In each half of the polyhedron are
(60+12=**72**) hexagonal yods either lining half of the central 12-gon or at centres of 60
tetractyses. We see that there are (**72**+**168**) hexagonal yods in the 60 faces in
each half of the disdyakis triacontahedron. This **72**:**168** division is the
single, polyhedral counterpart of the similar division found earlier in the two separate sets of seven enfolded
polygons and in the first three Platonic solids. It confirms the archetypal character of this polyhedron as the
*single*, polyhedral version of the universal pattern embodied in holistic systems. The 240 hexagonal yods
in each half symbolize the 240 roots of E_{8}, so that the disdyakis triacontahedron embodies the root
composition of E_{8}×E_{8}. This is further confirmation of the identification of the UPA as the
subquark state of the E_{8}×E_{8} heterotic superstring because, just as each whorl has 1680
circularly polarized oscillations, so, too, the disdyakis triacontahedron has 1680 geometrical elements surrounding
any one of its **31** possible axes (see here). The **72** hexagonal yods in each half
of the polyhedron denote the **72** roots of E_{6}, the rank-6 exceptional subgroup of
E_{8}, and the **168** hexagonal yods on edges in each half above or below its
equatorial plane denote its remaining roots. The two halves, one the mirror image of the other, correspond to
the two similar groups E_{8} in the direct product E_{8}×E_{8}.

Each of the 30 pyramids in the disdyakis triacontahedron with a Golden Rhombus as its base has
four inclined faces that consist of two similar ones outlined in yellow and two similar ones outlined in violet
(see picture). The latter are the mirror image of the former. The 60 faces in each half of this polyhedron comprise
two sets of 30 triangles. The 60 black hexagonal yods at their centres divide into two sets of 30. The 12 red
hexagonal yods on the six sides of each half of the 12-gon consist of two sets of six hexagonal yods because the
six sides comprise three different sides that are repeated. Therefore, the **72** hexagonal yods
naturally divide into two sets of **36**. This is analogous to the **36** corners in
each of the two separate sets of seven enfolded polygons making up the inner Tree of Life (see here). The **168** red hexagonal yods on the 84 edges either
both or below the equator of the disdyakis triacontahedron are arranged as 84 pairs. This corresponds to the 84
hexagonal yods in each set of seven enfolded polygons. The (**36**+84=120) yods lining the 42 sides
in each set have their counterpart in the disdyakis triacontahedron as the (**36**+84=120)
hexagonal yods in each half of the polyhedron that get duplicated in the second set of 30 faces and in the
second set of edges in half the 12-gon.

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