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The association between the whorls of the UPA and Sephiroth is expressed arithmetically by the following 2nd-order tetractys representations of the numbers 1680 and 16800:

(a)

1680 =

(b)

16800 =

In (a), the sum of the 70 integers in the ten shaded hexagons is 1680, the sum of the 21 integers in the three hexagons at the corners of the 2nd-order tetractys is 21×24 = 3×168 and the sum of the 49 integers in the seven remaining hexagons is 49×24 = 7×168. In other words, the tetractys factor of 10 splits into the sum of 3 & 7, reflecting the differentiation of the ten Sephiroth into the Supernal Triad and the seven Sephiroth of Construction. In (b), the sum of the 70 integers in the ten shaded hexagons is 16800, the corresponding sums being 21×240 = 3×1680 and 49×240 = 7×1680. The sum of the 21 integers in the tetractyses at corners of the 2nd-order tetractys corresponding to the Supernal Triad is equal to the number of spirillae in the three major whorls, which also correspond to the Supernal Triad, whilst the sum of the 49 integers in the seven tetractyses corresponding to the seven Sephiroth of Construction is equal to the
number of spirillae in the seven minor whorls, which also correspond to these Sephiroth. The integer 240 at the centre of the central tetractys denotes the 240 roots of E8, the largest of the five exceptional Lie groups, which was shown earlier to be embodied in both the lowest Tree of Life as its 240 yods other than Sephiroth when the sectors of its triangles are tetractyses (see here) and the inner Tree of Life as the 240 hexagonal yods of its seven regular polygons (see here). As 16800 = 70×240, assignment of the E8 group parameter 240 to the 70 yods of the outer Tree of Life generates the number of helical turns ("1st-order spirillae") in the ten whorls of the UPA/superstring.

Using the identity:

168 = 132 −  1,

then

16800 = 168×102 = (132 − 1)102 = 1302 − 102 = (2×65)2 – (2×5)2 = 4×(652 − 52).

652 = 1 + 3 + 5 +... + 129

and

52 = 1 + 3 + 5 + 7 + 9.

Therefore,

16800 = 4(11 + 13 + 15 + ... + 129),

i.e., 16800 is the sum of 4(65–5) = 4×60 = 240 odd integers, the largest of which is 129, the number of YAHWEH SABAOTH, the Godname of Netzach, the fourth Sephirah from Malkuth. Not only are the 16800 1st-order spirillae in the UPA the string manifestation of the 240 gauge charges associated with the 240 roots of E8 but the number 16800 is also the sum of 240 odd numbers that terminate in the number of YAHWEH SABAOTH, where

15        15         15        15

15        15         15        15
240 =

15        15         15        15

15        15         15        15 ,

and 15 is the number of YAH, one of the two Godnames of Chokmah. Since 15 = 20 + 21 + 22 + 23, we see the Tetrad Principle at work here (see Article 1), defining the structural parameter 16800 of the UPA/superstring subquark, this number being the sum of 240 odd integers, where

 4! 4! 4! 240 = 4! 4! 4! = 24 + 25 + 26 + 27 (4! = 1×2×3×4) 4! 4! 4! 4!

i.e., 240 is the sum of four consecutive integer powers of 2, starting with 4.

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