Ten-fold division of the hexagonal yod population of the first four Platonic solids

Constructed from tetractyses, the Platonic solids have three types of hexagonal yods: 1. hexagonal yods at centres of tetractyses; 2. hexagonal yods on edges of solids, and 3. hexagonal yods on sides of tetractyses inside the boundaries of faces. Their numbers are calculated and tabulated below:
 

(numbers are further divided into their two halves to indicate the numbers of hexagonal yods in each half of the solid). The first three Platonic solids have 240 hexagonal yods comprising 120 hexagonal yods either at centres of tetractyses or on edges and 120 hexagonal yods on interior sides. This 120:120 division in the 240 hexagonal yods in the 18 faces of the first three Platonic solids re-occurs in the icosahedron and in the dodecahedron, each half having 120 hexagonal yods. Its inner Tree of Life counterpart was discussed on page 30 at Superstrings as sacred geometry/Tree of Life. Each of the two separate sets of seven enfolded polygons has 120 yods on their boundaries. Both sets have 240 yods on their boundaries that comprise 72 corners and 168 hexagonal yods. We saw there that these 240 boundary yods symbolize the 240 roots of the superstring gauge symmetry group E₈, the 72 corners denoting the 72 roots of its exceptional subgroup E_6, leaving 168 roots represented by the 168 hexagonal yods. The table indicates that their counterparts in the first three Platonic solids are the 72 hexagonal yods in one half of the tetrahedron and the octahedron and the 168 hexagonal yods in their other half and in the cube. The 72:168 division in the holistic parameter 240 manifests in the first three Platonic solids. Moreover, just as the 72 corners were found to be composed of three sets of 24 corners of a unique combination of polygons, so the 72 hexagonal yods consist of 24 hexagonal yods in one half of the tetrahedron, 24 hexagonal yods that either line edges of the octahedron or are centres of tetractyses and 24 hexagonal yods on sides of tetractyses inside the faces of the octahedron. Finally, just as the 168 hexagonal yods on the sides of the two sets of polygons were found on page 30 at Superstrings as sacred geometry/Tree of Life to be distributed amongst the polygons in a way that naturally generates seven sets of 24, so, too, the table shows that the 168 hexagonal yods in the other half of the tetrahedron & octahedron and in the cube group naturally into seven sets of 24:

1. 24 (6+6+12) hexagonal yods that are in that half of the tetrahedron;
2. 24 (12+12) hexagonal yods that are either centres of tetractyses or line edges of the other half of the octahedron;
3. 24 hexagonal yods that are sides of tetractyses in that half;
4. 24 (12+12) hexagonal yods at centres of tetractyses in the cube;
5. 24 (12+12) hexagonal yods that line its edges;
6. 24 hexagonal yods lining sides of tetractyses in one half of the cube, and
7. 24 hexagonal that line sides in the other half of the cube.

(the red numbers are those appearing in the table). The mathematical analogy between the two systems of sacred geometries exists because one encodes in the yods shaping the two-dimensional, regular polygons what exists in the faces of the three-dimensional, regular polyhedra. As with the two sets of seven enfolded polygons, the 240 hexagonal yods in the 18 faces of the tetrahedron, octahedron & cube can be grouped into ten sets of 24. Here, therefore, are two sacred geometries that encode in analogous ways the 10-fold structure of the UPA as ten whorls, each carrying 24 gauge charges of E₈. The importance of this discovery for the future development of M-theory cannot be emphasized too strongly. This 10-fold nature is the manifestation in the subatomic world of the 10-fold nature of God, as represented by the Kabbalistic Tree of Life and Pythagoras' tetractys. The predicted 10-foldness in the 3-dimensional form of the superstring constituents of quarks will be a sign of the correctness of any future candidate for M-theory.

The 240 hexagonal yods repeated in the fourth Platonic solid symbolize the 240 roots of the second, similar Lie group E₈′ that is part of the gauge symmetry group E₈×E₈′ governing the unified force between one of the two types of heterotic superstrings. The first four Platonic solids embody not only the dimension 248 of E₈ as the 248 corners & sides of the sectors of their faces but also the numbers of so-called "non-zero roots" of E₈ and E₈×E₈′. We saw on page 5 that 1680 geometrical elements surround their axes when their faces and internal triangles are divided into their sectors. Here is irrefutable evidence that the UPA remote-viewed by Besant & Leadbeater is an E₈×E₈′ heterotic superstring, for both the basic structural parameter of the UPA and the root composition of this symmetry group (as well as E₈) are embodied in the tetrahedron, octahedron, cube & icosahedron — three truly remarkable features of these regular polyhedra that cannot, plausibly, be dismissed as all coincidences. Once again, the fact that they manifest in the first four Platonic solids is consistent with the Tetrad Principle formulated in Article1.

It was pointed out in Superstrings as sacred geometry/Platonic solids that the 48 corners outside the root edge of the two separate sets of the first (6+6) enfolded polygons denote the 48 roots of F₄, the rank-4 exceptional subgroup of E₆ whose 72 roots are denoted by the 72 corners of the two separate sets of the (7+7) enfolded polygons. What is their counterpart in the first three Platonic solids? The table above indicates that the 72 hexagonal yods in the six faces making up half the tetrahedron and half the octahedron are:

The 48 hexagonal yods in the four faces of half the octahedron denote the 48 roots of F₄.

In the section Superstrings as sacred geometry/Disdyakis triacontahedron, we shall discuss the properties of the disdyakis triacontahedron, the Cartan solid with 120 triangular faces faces which emerged in Sacred geometry/Polyhedral Tree of Life as the outer form of the "Polyhedral Tree of Life." We shall find that it embodies both these properties in the same way that the first four Platonic solids do collectively, namely, 1680 geometrical elements surround an axis of the disdyakis triacontahedron, whilst 240 hexagonal yods are in the 60 faces making up each half of it when they are turned into tetractyses. Like these Platonic solids, the disdyakis triacontahedron also embodies the number 248 (see here). It cannot be a coincidence that the number 1680 (the paranormally-obtained structural parameter of the 10 vortex rings of the subquark state of the E₈×E₈ superstring) re-appears in a polyhedron that was identified independently here as the outer form of the Polyhedral Tree of Life. Instead, its appearance confirms the archetypal nature of this number, whose significance for string theory awaits discovery by theoretical physicists when they start to realize that subquarks, not quarks, are superstrings.

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