#16 The pair of joined, Type C hexagons embody the (248+248) roots of the heterotic superstring gauge symmetry group E₈×E₈′

The Tetrad Principle discussed in Article 1 states that the fourth member of a class of mathematical objects embodies numbers of universal (and therefore scientific) significance. As the hexagon is the fourth type of regular polygon and the fourth of the seven regular polygons making up the inner form of the Tree of Life, it should not be surprising that the pair of joined hexagons embodies parameters of the Tree of Life. For example, when their 12 sectors are tetractyses, they contain 70 yods (10 corners & 60 other yods). They correspond to the 10 Sephiroth of the Tree of Life and its 60 hexagonal yods when its 16 triangles become tetractyses. The Type C hexagon is the fourth class of hexagon:

hexagon → Type A → Type B → Type C → etc.

A Type C n-gon has (42n+1) yods. A Type C hexagon (n=6) has 253 yods. Two joined Type C hexagons have 502 yods. 500 (=50×10) yods surround their centres, showing how ELOHIM, the Godname of Binah with number value 50, prescribes this particular pair of hexagons. Outside the four black yods in their shared edge are 496 yods. This is the gematria number of Malkuth, the last Sephirah of the Tree of Life, signifying on a cosmic scale the physical universe (the space-time continuum):

248 such yods belong to each hexagon, where 248 is the number value of Raziel, the Archangel of Chokmah. They symbolize the sets of 248 roots of E₈ and E₈′ in the heterotic gauge symmetry group E₈×E₈′.

The following correspondences exist:

• two white yods → two of the eight simple roots of E₈ (E₈′) that are not roots of E₆ (E₆′), its rank-6, exceptional subgroup.
• six yellow yods → six simple roots of E₆ (E₆′).
• 72 red yods → 72 roots of E₆ (E₆′).
• 168 blue yods → 168 roots of E₈ (E₈′) that are not roots of E₆ (E₆′).

An alternative interpretation
As the topmost corner of each hexagon coincides with the bottom corner of its counterpart enfolded in the next higher Tree of Life, there are 500 (=50×10) yods that are intrinsic to each pair of hexagons enfolded in any Tree. This shows how ELOHIM, the Godname of Binah with number 50, prescribes these hexagons. Their top & bottom corners and their centres are located at the positions of the Sephiroth on the Pillars of Mercy and Judgement. The number of yods intrinsic to the two hexagons = 502 – 6 = 496. 248 such yods are associated with each hexagon. For the hexagon on the right of the root edge, they comprise two black yods on the root edge, six yellow centres of sectors, 72 red yods that are either hexagonal yods at centres of tetractyses or their corners and 168 blue yods that are either hexagon yods on sides of tetractyses or the two corners on the right-hand side of the hexagon. For the hexagon on the left, the 248 yods comprise a similar set of yods. The 496 yods intrinsic to the two hexagons symbolize the two sets of 248 roots of E₈ in the heterotic gauge symmetry group E₈×E₈′.

The hexagon can be regarded as Nature's primary polygon, manifesting, for example, in the molecule of benzene, the basic building block of the aromatic hydrocarbon compounds, graphite and the honeycombs of bees. Perhaps its most remarkable and fundamental property is that discussed here, namely, its embodiment of the dimension 248 of the gauge symmetry group E₈ governing the forces between one of the two types of heterotic superstring.

Article 60 and #39 at Superstrings as sacred geometry/Tree of Life discuss how the 496 roots of E₈×E₈′ manifest in the inner Tree of Life when its polygons are Type C.

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