Updated: Nov 28, 2020
The seven separate polygons with their48 sectors as tetractyses have 240 hexagonal yods. There are five hexagonal yods per sector. The first three polygons have 12 tetractys sectors with five blue hexagonal yods per sector, i.e., 60 blue hexagonal yods. The next three polygons have 24 tetractyses with 120 green hexagonal yods, i.e., five sets of 24 green hexagonal yods. The dodecagon has 60 red hexagonal yods, i.e., five sets of 12 such yods. The seven separate polygons have 240 hexagonal yods. They consist of five sets of 12 blue hexagonal yods, five sets of 24 green hexagonal yods and five sets of 12 red hexagonal yods, i.e., five sets of (12+24+12=24+24=48) hexagonal yods.
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=48) hexagonal yods 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 E8′. 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 E8′ are spread along each whorl, as opposed to the UPA, for which 24 of the 240 gauge charges of E8 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 E8′ 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 E8. 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.