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**The 480 hexagonal yods in the faces of the first 4 Platonic
solids denote the 480 roots of E _{8}×E_{8}′**

When their 18 faces are constructed from tetractyses, the tetrahedron, octahedron & cube have
240 hexagonal yods. The 20 faces of the icosahedron likewise have 240 hexagonal yods, as do the 12 faces of
the dodecahedron. This 240:240 division of the 480 hexagonal yods shown by the first three Platonic solids and
the fourth one is the polyhedral counterpart of the inner Tree of Life, each set of seven regular polygons
having 240 hexagonal yods (see here). In both cases, they symbolize the 240 roots of
E_{8} (see here) and the 240 roots of E_{8}′ in the gauge symmetry group
E_{8}×E_{8}′ characterizing the E_{8}×E_{8}′ heterotic superstring. The fact
that it is the first *four* Platonic solids that embody the root composition of
E_{8}×E_{8}′ illustrates once again the Tetrad Principle formulated in Article 1, just as the **248** corners & sides of the triangles in
their faces do, as discussed on page 1, for this number is a defining parameter or signature of holistic systems
and the Platonic solids traditionally symbolizing the Elements Fire, Air, Water & Earth constitute such a
system. In the case of the 14 polygons of the inner Tree of Life, the direct product character of
E_{8}×E_{8}′ corresponds to the mirror symmetry of each set of seven polygons; in the case of
the first four Platonic solids, it corresponds to the distinction between one set of their halves and the set of
their inverted halves, either set having 19 faces with 240 hexagonal yods.

Just as their faces can be constructed from tetractyses, so, too, the interiors of the Platonic
solids are composed of triangles with the centre of a polyhedron as one shared corner that is joined to its
vertices. When the faces and interior triangles are Type A, an axis passing through two opposite vertices is
composed of two sides that are shared by all the internal triangles. Two hexagonal yods lie on each side, so that
the first four Platonic solids have eight hexagonal yods on the shared sides in each half. There are
(480+16=**496**) hexagonal yods either on the axes or in the faces of the first four Platonic solids.
**496** is the dimension of E_{8}×E_{8}′. **248** hexagonal yods are on
either axis or faces in each half of the first four Platonic solids. The eight hexagonal yods on one half of their
axes denote the eight simple roots of E_{8} and the eight hexagonal yods on the other half denote the eight
simple roots of E_{8}′. This is how they embody the **496** roots of one of the two
symmetry groups governing the unified force between superstrings (see also here). *The dimension 248 of
E_{8} or E_{8}′ is the number of points & lines making up the triangles in the faces of the
first four Platonic solids, as well as the number of hexagonal yods in the axes and faces of their upper or
lower halves*. It is not plausible that the presence of these two properties could be a matter
of coincidence. Such a remote possibility is rendered even more unlikely by the appearance in the next few pages
of the structural parameters

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