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Five-fold division of the hexagonal yod population of the dodecahedron

__Inner Tree of Life__The seven separate polygons with their

__Dodecahedron__When its 12 pentagonal faces are each divided into their five sectors,
the dodecahedron has 240 hexagonal yods. On its 30 edges are 60 blue hexagonal yods. Each of the five vertices of a
given face can be associated with one edge of that face and with five edges of other faces, i.e., with six edges,
so that the 30 edges can be grouped into five sets of six edges. As each edge has two blue hexagonal yods, the 60
blue hexagonal yods can be grouped into five sets of 12 hexagonal yods. In each pentagonal face are five pairs of
green hexagonal yods on sides of tetractyses, so that there are 12 sets of five pairs of these yods, or five sets
of 12 pairs of green hexagonal yods, i.e., five sets of 24 green hexagonal yods. At the centre of each tetractys is
a red hexagonal yod, so that there are 12 sets of five red hexagonal yods, which can be regarded as five sets of 12
hexagonal yods. Hence, the 60 tetractyses contain five sets of (12+24+12=24+24=

Comparing the distribution of the 240 hexagonal yods in the inner Tree with that of the 240
hexagonal yods in the faces of the dodecahedron, we see that they are analogous, with the same number of hexagonal
yods of the same colour. Their 240 hexagonal divide naturally into five sets of **48**, i.e, in either
case a given type of yod appears **48** times. This is evidence that the dodecahedron is the
single polyhedral counterpart of the seven polygons constituting half of the inner Tree of Life. The division:
**48** = 24 + 24 found in both the seven polygons and their polyhedral counterpart is
characteristic of holistic systems (see **The holistic pattern**). So is the
further division: 24 = 12 + 12. The five sets of **48** hexagonal yods manifest these two
divisions in the dodecahedron, for each set consists of 24 red & blue hexagonal yods and 24 green hexagonal
yods, the former consisting of 12 red hexagonal yods and 12 blue hexagonal yods and the latter comprising 12
pairs of green hexagonal yods. Alternatively, each of the five types of hexagonal yod appears
**48** times, 24 in one half of the polyhedron and 24 times in its other half.

As the last of the five Platonic solids, the dodecahedron was regarded by the ancient Greeks as
associated with Aether, the fifth Element. The basic particle of the matter making up human biofields is discussed
here. It was identified as the superstring of shadow matter, whose forces have
the symmetry of E_{8}′. By correlating the sacred geometry of the inner Tree of Life with its five
whorls, it was deduced that **48** of the 240 gauge charges of E_{8}′ are spread along
each whorl, as opposed to the UPA, for which 24 of the 240 gauge charges of E_{8} are distributed
around each whorl. This is the superstring manifestation of the 5-fold pattern of
**48** hexagonal yods in the faces of the dodecahedron. Superstring theory predicts that
shadow matter is invisible, interacting only gravitationally with superstrings of ordinary matter, each type of
matter being confined to its own 10-dimensional space-time sheet, the two parallel sheets being separated by a
narrow gap extending along the tenth dimension of space predicted by M-theory. If ordinary matter is what the
ancient Greeks meant by the Elements Earth, Water, Air & Fire, then *shadow matter must be the fifth
Element*. If so, the reason why the dodecahedron should be associated with it turns out to be,
mathematically speaking, far more soundly based than what historians and philosophers of science have attributed
to the Platonic and Aristotlean view of the physical universe. Instead of the association between the
dodecahedron and Aether being due to its resemblance to a sphere and, therefore, to the Celestial Sphere
encompassing all space (not very convincing!), the *real* reason is more rigorous, mathematically
speaking. In a Pythagorean context, the 5-foldness of the superstring of shadow matter arises from the fact that
five hexagonal yods are intrinsic to each tetractys sector of a polygon, each one being associated with either
Malkuth, Yesod, Hod, Netzach & Tiphareth, to which the respective Elements Earth, Water, Air, Fire &
Aether are assigned.

Some particle physicists have interpreted Aether as so-called "quintessence," a scalar field
that is a form of dark energy hypothesized to be responsible for the acceleration of the expanding universe (see
here). This is incorrect. Dark energy is only a particular component of all that makes up
shadow matter — both its fermions and the gauge fields of E_{8}′ that transmit its forces. As a category
of matter, Aether must be interpreted as shadow matter, which is equally as fundamental as ordinary matter
described by E_{8}. It is not merely an energy field pervading interstellar space that causes the
universe to accelerate. Cosmologists have made an association between the four physical Elements and the four
types of components of the mass/energy of the universe: baryonic matter, hot dark matter, cold dark matter and
gravitational self-energy, thereby providing themselves with a reason to associate the fifth factor governing
the universe's expansion with the fifth Element. But this comparison is totally specious, for cold dark matter
(WIMPs, MACHOs & axions) is purely hypothetical and may not even exist, as well as being conceptually confused,
for MACHOs (Massive, Astronomical, Compact Halo objects) are not different species of matter but, simply,
astronomical *bodies* that do not emit light, like black holes, neutron stars and brown dwarfs,
i.e., they are *objects* made up of normal, baryonic matter, so they cannot be regarded as a
category of Element that is distinct from the other three.

Analysis in Article 57 of the first four enfolded polygons making up the inner Tree of Life
shows that the rows of their 17 tetractys sectors have **496** permutations. The first three
polygons with 10 sides have **248** permutations, 240 of them being those of the four yods on
each side. The remaining rows have **248** permutations, 240 of them forming five sets of
**48**. This is analogous to the 10-fold pattern in the 240 hexagonal yods in the faces of the
icosahedron (see next page) and the 5-fold pattern in the 240 hexagonal yods in the faces of the
dodecahedron.

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