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**#39 The first four Platonic solids embody the 496 roots of
E _{8}×E_{8}′**

Suppose that a Platonic solid with E edges and F faces, each a regular n-gon with n sectors, is
constructed from tetractyses. There are 2E hexagonal yods on its edges, 2nF hexagonal yods on the internal sides
of the nF tetractyses and nF hexagonal yods at their centres, i.e., 3nF internal hexagonal yods. Suppose also
that its edges are sides of internal triangles, each divided into three tetractys sectors. This means that each
half of the Platonic solid has two red or blue hexagonal yods on the axis drawn through two opposite vertices
and the white centre of the polyhedron (see diagram opposite). The table lists the numbers of hexagonal yods on
and inside the edges of the tetrahedron, octahedron, cube & icosahedron. The number "2" refers to the two
red or blue hexagonal yods on the axis in their upper or lower halves. The first four Platonic solids have eight
hexagonal yods on their upper or lower half-axes and 480 hexagonal yods in their faces (240 black hexagonal yods
in each half). Therefore, (**248**+**248**=**496**) hexagonal yods lie in
their axes and faces. Each of the **248** hexagonal yods in the upper or lower halves of the
first four Platonic solids denotes, respectively, a root of E_{8} or E_{8}', the rank-8,
exceptional Lie group with dimension **248**. The eight red hexagonal yods on their axes in their
upper halves symbolize the eight simple roots of E_{8} and the eight blue hexagonal yods on their
axes in their lower halves denote the eight simple roots of E_{8}′. The 240 black hexagonal yods in each
set of halves denote the 240 roots of E_{8} or E_{8}′:

*The symmetry group of the E _{8}×E_{8}' heterotic
superstring is embodied in the hexagonal yod population of the faces and axes of the first four Platonic
solids*.

The upper/lower halves of the tetrahedron and octahedron (or cube) have
(24+**48**=**72**) hexagonal yods in their faces (see table). The upper/lower halves
of the cube (or octahedron) and the icosahedron have (**48**+120=**168**) hexagonal
yods in their faces. The number **72** is the number of roots of E_{6}, the rank-6 exceptional subgroup of E_{8}. Holistic systems embody
the number 240 and always exhibit this **72**:**168** division, as demonstrated
in many places on this website (see also the diagram above). For example, the 240 yods on the boundary of the
two separate sets of seven enfolded polygons of the inner Tree of Life comprise their
**72** corners and **168** hexagonal yods:

Notice that the faces of the tetrahedron, octahedron & cube have 120 hexagonal yods in each set of halves, as does the icosahedron. The 120:120 division of the number 240 is characteristic of holistic systems embodying this parameter. In the context of the first four Platonic solids, it differentiates not only the hexagonal yod populations of their halves but also the first three Platonic solids from the fourth one.

*We now see that the belief of the ancient Greeks that the shapes of
the first four Platonic solids determined the physics of the physical cosmos is correct, although not for the
reason they supposed, namely, that particles of the Elements Fire, Air, Water & Earth have the shapes of
these regular polyhedra*. Instead, the yod composition of their faces and axes determine the root
structure of E_{8}×E_{8}′, one of the two gauge symmetry groups governing the forces between
heterotic superstrings. Although the second group SO(32) also has the dimension **496**, its roots
do not divide equally into two sets that mirror the two sets of halves of these Platonic solids. This is one of
many reasons for believing that nature now favours E_{8}×E_{8}' over SO(32) as the gauge
symmetry group governing superstrings. The direct product structure arises from the simple geometric fact that
each yod in one half of a Platonic solid has a counterpart in its second half.

It is shown in Article 55 (p. 4) that the tetrahedron, octahedron, cube & icosahedron have
**248** corners & sides of the 120 triangular sectors of the regular polygons in their
faces. The 12 sectors of the four faces of the tetrahedron have eight corners, leaving 240 corners & sides.
They comprise **72** corners & sides (the 60 corners of the octahedron, cube &
icosahedron & 12 internal sides of the sectors of the tetrahedron) and **168** corners
& sides (90 sides of the sectors in the icosahedron & 78 corners & sides in the tetrahedron,
octahedron & cube). See Table 4 in Article 55. The number 78 is the number value of *Cholem* and
the number 90 is the number value of *Yesodoth*, the two words making up the Kabbalistic name
*Cholem* *Yesodoth* for the Mundane Chakra of Malkuth:

The embodiment of the number **248** in the geometrical composition of the
faces of the first four Platonic solids is discussed here. The number of corners and sides of the n sectors in each of the F faces of
a Platonic solid with C vertices = C + F + E + nF. Using the Euler polyhedral formula:

C − E + F = 2,

this simplifies to 2 + 2E + nF. "2" denotes the two vertices that lie on its axis, so that
(2E+nF) corners & sides surround the axis. The first four Platonic solids have 240 corners & sides
surrounding the axes that pass through eight vertices. This compares with the 240 hexagonal yods in the faces in
each set of halves that surround the eight hexagonal yods in each set of their half-axes. So their geometry
embodies the dimension **248** of E_{8} and their hexagonal yod composition
embodies the dimension **496** of E_{8}×E_{8}'. Here is remarkable evidence
for how these Platonic solids, thought by the ancient Greeks to represent the shapes of the particles of the
Elements Earth, Water, Air & Fire, embody the unified force between E_{8}×E_{8}' heterotic
superstrings. How they embody the space-time structure of their subquark states is discussed here.

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