#22 Correspondence between the five Platonic solids and the 1-tree

Consider the five Platonic solids constructed from triangles. All their faces are regarded as Type A polygons, whilst all their vertices are joined by straight lines to the centres of the polyhedra, creating internal triangles. Except in the case of the tetrahedron, for which no two vertices are diametrically opposite each other, the two straight lines joining opposite vertices of a Platonic solid to its centre are collinear. The table lists the numbers of corners, sides & triangles in the faces and interiors of the Platonic solids that surround the "axis" formed by joining two opposite vertices to the centre of the polyhedron. (90+270+180=540) geometrical elements in their 50 faces surround their axes, which are surrounded by (40+90=130) sides & triangles in their interiors. There are in total (90+270=360=36×10) corners & triangles and 310 (=31×10) sides surrounding their axes. This is how EL, the Godname of Chesed with number value 31, and ELOHA, the Godname of the next Sephirah, Geburah, with number value 36, prescribe the geometrical composition of the five Platonic solids. ADONAI MELEKH, the Godname of Malkuth with number value 155, prescribes their geometry because 155 sides surround their axes in the five half-polyhedra. ADONAI with number value 65 prescribes their interior geometry because their axes are surrounded by (40+90=130) sides & triangles, so that each set of five half-polyhedra has 65 sides & triangles surrounding their half-axes.

The table indicates that the total number of corners, sides & triangles surrounding the axes of the five Platonic solids = 670 = 67×10. 67 is the number value of Binah, the third member of the Supernal Triad. Amazingly, this is the number of yods below Binah in the 1-tree, or lowest Tree of Life, when its 19 triangles become tetractyses (see also here). Binah is the Sephirah that embodies, in a metaphysical sense, the feminine aspect of Creation (one of its titles is "Aima," meaning "Divine Mother"). Fittingly, it determines the geometrical composition of the Platonic solids, whose structure is embodied in the formative, shape-forming, archetype of this Sephirah, namely, its Godname ELOHIM, whose number value 50 specifies the number of their vertices. Each yod below Binah is a potential tetractys, and it represents one of the 67 sets of 10 geometrical elements needed to build the complete form of the five regular polyhedra around straight lines in 3-dimensional space (apart from the tetrahedron) as their axes of construction.

Each axis comprises two vertices, two sides shared by all internal triangles and its central point. The total number of geometrical elements surrounding the centres of the five Platonic solids = 670 + 5×2 + 5×2 = 690. The counterparts in the Tree of Life of these five centres and the 69×10 elements that surround them are Kether — the starting point of its emanation and the 69 yods needed to construct it from tetractyses (see #1). Their counterparts in the 2-dimensional Sri Yantra are the bindu point at its centre and the 69 corners of its 43 triangles.

When the 90 internal triangles of the five Platonic solids are Type A triangles, they have 270 internal triangular sectors. 270 internal sides & 90 corners are added to their 40 internal sides, creating (270+40=310=31×10) internal sides that surround their axes. Listed below are the numbers of geometrical elements in their faces and interiors that surround their axes:

The number value 67 of Binah now determines the number of geometrical elements inside the Platonic solids that surround their axes. In total, 1210 geometrical elements surround them. As 121 = 11²,

where 11 is the tenth integer after 1. The Pythagorean Decad determines arithmetically the number of geometrical elements surrounding the axes of the five Platonic solids. It also determines the total number of elements that surround their centres, because the five axes consist of 10 endpoints (corners of triangles) & 10 sides surrounding their middle points, so that (1210 + 10 + 10 = 1230 = 123×10) elements surround centres, where 123 is the tenth Lucas number L₁₀ (see #20 for the definition of Lucas numbers). The Decad also determines the 190 corners of the 450 triangles surrounding the five centres because 190 = 10×19, where 19 is the tenth odd integer. The average number of geometrical elements in the five Platonic solids surrounding their centres = 1230/5 = 246, which is the number value of Gabriel, the Archangel of Yesod. Including the centres, the average number of geometrical elements making up the five Platonic solids is 247. This is the 31st prime number, showing how EL, the Godname of Chesed with number value 31, prescribes their average geometrical composition.

Fine-structure constant

As indicated in the table in #20, the five Platonic solids have 550 elements in their faces (an average of 110). The average number of elements making up their interiors = 247 – 110 = 137. This is another parameter of holistic systems and is well-known to physicists as the number that approximately determines as its reciprocal the fine-structure constant α = e²/ħc ≈ 1/137. This measures the strength of the coupling of the electron to the electromagnetic field. How fitting it is that the number that can be said to determine the size and structure of atoms should be embodied in the geometry of the Platonic solids that the ancient Greeks believed were the shapes of the particles of the Elements! The average number of yods needed to construct the faces of the tetrahedron, octahedron, cube & icosahedron is 137 (see here). This number is therefore also embodied in the faces of the Platonic solids associated with the four physical Elements.

The number of corners & sides surrounding the axes = 180 + 580 = 760 = 76×10. YAHWEH ELOHIM, the Godname of Tiphareth with number value 76, prescribes the corners & sides making up the five Platonic solids. The number of corners & triangles surrounding the axes = 180 + 450 = 630, which is the number value of Seraphim, the Order of Angels assigned to Geburah. The number of corners & triangles surrounding their centres = 630 + 10 = 640, which is the number value of Shemesh, the Mundane Chakra of Tiphareth. The number of sides & triangles surrounding their centres = 580 + 450 + 10 = 1040. It is amazing that this number is the sum of the number values of the complete Godnames of the ten Sephiroth!:

We have seen that 1230 geometrical elements surround the centres of the five Platonic solids when their faces and internal triangles are Type A. 550 of these are in their faces and 680 are inside them. We saw in #20 that the seven enfolded Type B polygons contain 687 yods. It means that 680 yods surround their seven centres. This is the inner Tree of Life counterpart of the 680 geometrical elements inside the Platonic solids that surround their centres.

The following provides an even clearer indication that the number 680 is, in fact, a Tree of Life parameter. The (7+7) polygons of the inner form of the Tree of Life have 70 corners. The topmost corners of the two enfolded hexagons coincide with the lowest corners of the two hexagons enfolded in the next higher Tree. This means that 68 corners are intrinsic to the 14 polygons enfolded in each overlapping Tree. The number of corners in the 14n polygons enfolded in n Trees = 68n + 2, where "2" denotes the two uppermost corners of the hexagons enfolded in the nth Tree that coincide with the lowest corners of the two hexagons enfolded in the (n+1)th Tree. The number of corners intrinsic to these 14n polygons enfolded in the lowest n Trees = 68n. Therefore, the number of corners intrinsic to the 140 polygons enfolded in the lowest 10 overlapping Trees of Life = 680:

The ancient Greeks associated the tetrahedron with the Element Fire, the octahedron with Air, the cube with Earth and the icosahedron with Water. The table above indicates that 442 geometrical elements (60 corners, 202 sides & 180 triangles) surround their axes. Remarkably, this is the number of hexagonal yods outside the root edge of the 14 polygons in the inner Tree of Life:

The root edge is analogous to the axes of these Platonic solids, with each hexagonal yod outside it symbolizing one of the geometrical elements that surround them. Here is clear and remarkable evidence for the Tree of Life nature of these four solids. It does not, of course, mean that the ancient Greek doctrine of the four Elements is right. What it does imply is that, because they conform to the cosmic blueprint of the Tree of Life, these solids collectively embody numbers of universal significance that should be of interest to physicists. Examples discussed elsewhere are 248, 240, 168 & 137. In that sense, they do embody the basic properties of matter, just as the ancient Greeks intuited.

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