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The direct product nature of the heterotic superstring gauge symmetry group E8×E8 arises geometrically because an octagon with 2nd-order tetractyses as sectors has 496 coloured hexagonal yods, which is twice the 248 similar yods in a square for the simple reason that 8 = 2×4.

The number of tetractyses is 80. This is the number value of Yesod, the Sephirah immediately above Malkuth in the Tree of Life. This archetypal pattern is prescribed by the Godnames. For example, it is prescribed by the Godname YAH (Hebrew: יה, number value 15) because each sector of the octagon has ten tetractyses with 15 corners. There are 576 yods surrounding the centre of the octagon, where 576 = 242= 12×22×32×42. Of these, 33 yods are corners of 1st-order tetractyses, where 33 = 1! + 2! + 3! + 4!. These are examples of how the integers 1, 2, 3 & 4 symbolized by the four rows of the Pythagorean tetractys express the properties of archetypal patterns of numbers and holistic systems possessing sacred geometry.

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