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#63 Embodiment of the finestructure constant number 137 in both the inner Tree of Life and the five Platonic solids
It was pointed out on the previous page in the discussion of the five Platonic solids that, when their faces are Type A polygons and their internal triangles are Type A triangles, 1820 yods surround their centres, of which 450 hexagonal yods are at the centres of tetractyses. It means that (1820−450=1370) yods line sides of tetractyses. This is the number of yods in 137 tetractyses. It is also the yod population of the (7+7) enfolded Type B polygons of the inner Tree of Life (see here). This correspondence is amazing. It is powerful evidence for the five Platonic solids being the regular polyhedral realisation of the inner form of the Tree of Life. The two sets of halves of the five Platonic solids correspond to the two mirrorimage sets of seven enfolded polygons. Two pairs of hexagonal yods lie on an axis of each Platonic solid that passes through its centre and two opposite vertices. The 20 hexagonal yods on the axes of the five Platonic solids and their 50 vertices constitute the basic "trunk" of this polyhedral version of the Tree of Life. Their counterparts in the (7+7) enfolded polygons are the 20 corners of the two dodecagons outside the root edge and the 50 corners of the first (6+6) enfolded polygons. Surrounding the axes of the five Platonic solids are (1370−70=1300) yods lining sides of tetractyses other than vertices, where
1^{4} 

2^{4}  2^{4}  
1300 = 1^{5} + 2^{5} + 3^{5} + 4^{5} = 
3^{4} 
3^{4} 
3^{4} 

4^{4}  4^{4}  4^{4}  4^{4} . 
This demonstrates how the Tetrad expresses this beautiful property. On average, there are (1300/5=260=26×10) such yods. This demonstrates the power of YAHWEH, the Godname of Chokmah with number value 26, to quantify how many extra yods, given their vertices, are needed to create the shapes of the five Platonic solids when they are constructed from tetractyses. 650 yods other than vertices (i.e., the yods in 65 tetractyses) in the five upper or five lower halves of the Platonic solids surround their axes. This shows how ADONAI, the Godname of Malkuth with number value 65, prescribes their forms:
The number 650 is the sum of the squares of the first 12 integers:
650 = 1^{2} + 2^{2} + 3^{2} + 4^{2} + 5^{2} + 6^{2} + 7^{2} + 8^{2} + 9^{2} + 10^{2} + 11^{2} + 12^{2}.
As the five Platonic solids have 1820 yods surrounding their centres, the total number of yods in each set of five halves is 910 = 91×10. Therefore, (910−650=260=26×10) yods make up the axes, vertices and the hexagonal yods at centres of tetractyses in each set. The division: 91 = 65 + 26, which expresses the number value of ADONAI TETRAGRAMMATON, manifests in the five Platonic solids!
As 1370 yods other than vertices line the sides of tetractyses in the 10 halves of the five Platonic solids, the average number of such yods in one half = 137. This is a way in which the five Platonic solids embody the number whose reciprocal is approximately the dimensionless finestructure constant α = e^{2}/ħc ≈ 1/137, which measures the strength of the coupling of the electric charge of an electron to the electromagnetic field. This number is the average number of yods (including vertices) in the faces of the first four Platonic solids constructed from tetractyses (see here). It is also the sum of the first, second & third powers of 2, 3 & 4 located on the inclined edges of the Tetrahedral Lambda:
137 = 2^{1} + 2^{2} + 2^{3} + 3^{1} + 3^{2} + 3^{3} + 4^{1} + 4^{2} + 4^{3}.
(see here).
As
91 = 1^{2} + 2^{2} + 3^{2} + 4^{2} + 5^{2} + 6^{2}
and
1820 = 91×20 = 91×4×5 = 5×2^{2}×(1^{2} + 2^{2} + 3^{2} + 4^{2} + 5^{2} + 6^{2}) = 5×(2^{2} + 4^{2} + 6^{2} + 8^{2} + 10^{2} + 12^{2}) = 5×364,
the average number of yods surrounding the centre of a Platonic solid = 364 = 2^{2} + 4^{2} + 6^{2} + 8^{2} + 10^{2} + 12^{2}. The first term: 2^{2} = 4 denotes the four hexagonal yods on the axis. The average number of yods other than these axial yods = 360 = 36×10 = 4^{2} + 6^{2} + 8^{2} + 10^{2} + 12^{2}. It is prescribed by ELOHA, the Godname of Geburah, because its gematria number value is 36. This is how the Godname determines the average number of yods needed to construct the form of the five Platonic solids from tetractyses. ELOHIM, the Godname of the Sephirah above Geburah on the Pillar of Judgement of the Tree of Life, prescribes their form because its number value 50 is the number of their shapedetermining vertices.