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The association between the whorls of the UPA and Sephiroth is expressed arithmetically by the following 2ndorder 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 2ndorder 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 2ndorder 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
E_{8}, 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 E_{8} group parameter
240 to the 70 yods of the outer Tree of Life generates the number of helical turns ("1storder spirillae") in
the ten whorls of the UPA/superstring.
Using the identity:
168 = 13^{2} − 1,
then
16800 = 168×10^{2} = (13^{2} − 1)10^{2} = 130^{2} − 10^{2} = (2×65)^{2} – (2×5)^{2} = 4×(65^{2} − 5^{2}).
65^{2} = 1 + 3 + 5 +... + 129
and
5^{2} = 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 1storder spirillae in the UPA the string manifestation of the 240 gauge charges associated with the 240 roots of E_{8} 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 = 2^{0} + 2^{1} + 2^{2} + 2^{3}, 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! 
= 2^{4} + 2^{5} + 2^{6} + 2^{7} 
(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.