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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.

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 = 24^{2}= 1^{2}×2^{2}×3^{2}×4^{2}.
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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