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The Type C triangle contains 127 yods, where 127 is the 31st prime number and 31 is the number value of EL, the Godname of Chesed. 123 yods are outside the root edge. Two enfolded Type C triangles have (127+123=250) yods. The Pillars of Mercy and Judgement of the Tree of Life are in the plane of the two sets of seven enfolded polygons making up its inner form. This means that the left-hand and right-hand corners of the triangles coincide with, respectively, Geburah and Chesed. (250–2=248) yods are unshared with the Tree of Life, where 248 is the number value of Raziel, the Archangel of Chokmah. These intrinsic yods symbolize the 248 roots of the rank-8 Lie group E_{8}, which is present in the symmetry group E_{8}×E_{8} describing the forces between one of the two types of heterotic superstrings.
246 intrinsic yods surround the centres of the two triangles. This is the number value of Gabriel, the Archangel of Yesod. The two triangles consist of 54 tetractyses, each with a hexagonal yod at its centre. Hence, (250–54=196) intrinsic yods line their sides, of which two coincide with Geburah and Chesed, leaving 194 yods. 194 is the number value of Tzadekh, the Mundane Chakra of Chesed. 95 intrinsic yods outside the root edge line sides of the 27 tetractyses in each triangle, where 95 is the number value of Madim, the Mundane Chakra of Geburah.
The eight purple yods at the centres of each triangle and its sectors symbolize the eight simple roots of E_{8} and the 240 black and white yods denote its 240 roots. The 72 black yods that belong to the pair of triangles when regarded as Type B triangles denote the 72 roots of E_{6}, the rank-6 exceptional subgroup of E_{8}. The six purple yods at the centres of the sectors of the two triangles symbolize the six simple roots of E_{6}. The 168 white yods denote the 168 roots of E_{8} that are not also roots of E_{6}. They comprise 84 white yods in one triangle and 84 white yods in its mirror image. This 84:84 division of the number 168 is characteristic of sacred geometries and is discussed in many articles listed in the section Articles. It was encountered on page 19 of this section as the 84 yods that line the sides of each set of the first six enfolded polygons outside their root edge.
If the two triangles were Type A triangles, they would have 32 intrinsic yods. The number of components of the Dirac wave function of a D-dimensional fermion is 2^{D/2}. Hence, spin-½ fermions in 10-dimensional superstring space-time are described by Dirac wave functions with (2^{5}=32) components. We find that, when the triangles are Type A, the pair of joined triangles generate the number of components of the wave function of a 10-dimensional fermion and, when the triangles are Type C, they generate the number of gauge bosons transmitting the E_{8}-symmetric force between 10-dimensional E_{8}×E_{8} heterotic superstrings.
The numbers of corners, lines & triangles in the two enfolded Type C triangles that are unshared with the Tree of Life are shown below:
corners |
lines |
triangles |
30 |
83 |
54 |
They comprise 167 geometrical elements, that is, 165 elements surround their centres, where
165 = 1^{2} + 3^{2} + 5^{2} + 7^{2} + 9^{2}.
There are 137 lines & triangles, showing how the number 137 determining the fine-structure constant in physics characterizes the pair of Type C triangles that embodies the dimension 248 of the superstring gauge symmetry group E_{8}. Of the 84 corners & triangles, where
84 = 1^{2} + 3^{2} + 5^{2} + 7^{2},
two corners are on the root edge, whilst the left-hand and right-hand corners coincide with Sephiroth. Hence, 80 corners & triangles outside the root edge generate the 248 intrinsic yods, where 80 is the number of Yesod. They comprise 26 corners and 54 triangles. This shows how YAHWEH, the Godname of Chokmah with number value 26, prescribes the two enfolded Type C triangles embodying the 248 gauge bosons that mediate the forces between E_{8}×E_{8} heterotic superstrings of ordinary matter.
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