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The first four Platonic solids embody the superstring structural parameter 168

The table in Sacred geometry/Platonic solids indicates that, including their centres, the numbers of yods needed to construct from tetractyses the faces and internal triangles of the first four Platonic solids are:

 Tetrahedron:    71
 Octahedron:  135
 Cube:  139
 Icosahedron:  327 
 Total =  672

The average number of yods needed to construct one of these regular polyhedra = 672/4 = 168 (84 yods for each half, the centre being shared by both halves). This shows how they embody the superstring structural parameter 168, as well as the dimension 248 of the exceptional Lie symmetry group E8 as the number of corners & sides of the 120 sectors in their 38 faces. The division: 168 = 84 + 84 is characteristic of holistic systems (see here). It manifests in the E8×E8 heterotic superstring as the 840 turns in an outer or inner half of each whorl of the UPA, i.e., as the 84 turns in an outer or inner quarter-revolution of a whorl. When averaged over the first four Platonic solids, each half of a solid corresponds to a quarter-revolution of an outer or inner half of a whorl: 336/4 = 84. The table indicates that there are 124 yods making up what were called on page 16 the "Type A interiors" of these four Platonic solids. This means that, on average, 31 yods are needed to construct their interiors and (16831=137) yods are needed to create their faces. It shows how the number 137, which is one of the most important numbers in modern physics because its reciprocal is approximately equal to the fine-structure constant α = e2/ħc ≈ 1/137, shapes the first four Platonic solids, which were thought by the ancient Greeks to be the forms of the particles composing the elements Earth, Water, Fire & Air. The fine-structure constant is the electromagnetic coupling constant that measures the strength of the coupling between the negatively charged electron and the electromagnetic field. It is also the ratio of the average velocity of an electron in the hydrogen atom to the velocity of light in vacuo. The number 137 is prescribed by EL because 31 is the number value of this Godname of the Sephirah Chesed. Sixty-eight yods in the 38 faces of the first four Platonic solids are corners of tetractyses, leaving 480 hexagonal yods. This means that their faces on average are made up of 30 tetractyses with (68/4=17) corners and (480/4=120) hexagonal yods. The basic distinction in a tetractys between its corners and its hexagonal yods, which symbolize, respectively, the Supernal Triad and the seven Sephiroth of Construction, generates the division:

137 = 17 + 120.

This will be referred to in the discussion of how, as the outer form of the Polyhedral Tree of Life, the disdyakis triacontahedron embodies structural and dynamical parameters of superstrings (see here).

The significance of the yod population 672 of the first four Platonic solids vis-à-vis the 421 polytope, whose 240 vertices represent the 240 roots of E8, is discussed here.

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