Updated: Sep 27, 2020
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. Each coloured hexagonal yod denotes one of the spin-1 gauge bosons, or messenger particles, that transmit the unified force between heterotic superstrings. This enables in principle the quantum state of each messenger particle to be related to its corresponding Sephirah. It means that the 496 gauge bosons consist of seven classes, each corresponding to a Sephirah of Construction.
The number of tetractyses is 80. They have 80 corners surrounding the centre of the octagon.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 with number value 15 because each sector of the octagon has 10 tetractyses with 15 corners. There are 576 yods surrounding the centre of the octagon, where 576 = 242= 12×22×32×42. Thirty-three yods are corners of 1st-order tetractyses that are either on the boundary of the octagon or at its centre, 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 and holistic systems possessing sacred geometry. The division of the 80 corners into 32 on the boundary and 48 inside it matches the division of the 80 yods of the 1-tree into the 48 yods up to Chesed and the 32 yods above it (see here). It also corresponds to the 32 yods that are either corners of the two joined dodecagons or centres of the (7+7) polygons and to the 48 corners of the first (6+6) enfolded polygons outside the root edge (see here). As both the outer and inner Trees of Life display this 32:48 division, this is evidence of the archetypal nature of the octagonal representation of the number 496.