Updated: Nov 28, 2020
When their 18 faces are constructed from tetractyses, the tetrahedron, octahedron & cube have 240 hexagonal yods. The 20 faces of the icosahedron likewise have 240 hexagonal yods, as do the 12 faces of the dodecahedron. This 240:240 division of the 480 hexagonal yods shown by the first three Platonic solids and the fourth one is the polyhedral counterpart of the inner Tree of Life, each set of seven regular polygons having 240 hexagonal yods (see here). In both cases, they symbolize the 240 roots of E8 (see here) and the 240 roots of E8′ in the gauge symmetry group E8×E8′ characterizing the E8×E8′ heterotic superstring. The fact that it is the first four Platonic solids that embody the root composition of E8×E8′ illustrates once again the Tetrad Principle formulated in Article 1, just as the 248 corners & sides of the triangles in their faces do, as discussed on page 1, for this number is a defining parameter or signature of holistic systems and the Platonic solids traditionally symbolizing the Elements Fire, Air, Water & Earth constitute such a system. In the case of the 14 polygons of the inner Tree of Life, the direct product character of E8×E8′ corresponds to the mirror symmetry of each set of seven polygons; in the case of the first four Platonic solids, it corresponds to the distinction between one set of their halves and the set of their inverted halves, either set having 19 faces with 240 hexagonal yods.
Just as their faces can be constructed from tetractyses, so, too, the interiors of the Platonic solids are composed of triangles with the centre of a polyhedron as one shared corner that is joined to its vertices. When the faces and interior triangles are Type A, an axis passing through two opposite vertices is composed of two sides that are shared by all the internal triangles. Two hexagonal yods lie on each side, so that the first four Platonic solids have eight hexagonal yods on the shared sides in each half. There are (480+16=496) hexagonal yods either on the axes or in the faces of the first four Platonic solids. 496 is the dimension of E8×E8′. 248 hexagonal yods are on either axis or faces in each half of the first four Platonic solids. The eight hexagonal yods on one half of their axes denote the eight simple roots of E8 and the eight hexagonal yods on the other half denote the eight simple roots of E8′. This is how they embody the 496roots of one of the two symmetry groups governing the unified force between superstrings (see also here). The dimension 248 of E8 or E8′ is the number of points & lines making up the triangles in the faces of the first four Platonic solids, as well as the number of hexagonal yods in the axes and faces of their upper or lower halves. It is not plausible that the presence of these two properties could be a matter of coincidence. Such a remote possibility is rendered even more unlikely by the appearance in the next few pages of the structural parameters 168 and 1680 of the UPA described by Besant & Leadbeater over a century ago and identified by the author as the subquark state of the E8×E8′ heterotic superstring.