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**#11 The octagon embodies the dimension 496 of the symmetry
group E _{8}×E_{8} governing interactions between E_{8}×E_{8} heterotic
superstrings**

The direct product nature of the heterotic superstring gauge symmetry group
E_{8}×E_{8} 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 = 24^{2}= 1^{2}×2^{2}×3^{2}×4^{2}.
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**.

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