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**Representation of CTOL parameter 550 by two joined
squares**

550 yods outside their root edge surround the centres of two joined squares with 2nd-order tetractyses as their sectors. |

A 2nd-order tetractys contains 85 yods, where

85 = 4^{0} + 4^{1} + 4^{2} + 4^{3}.

13 yods line each side, so that (85−13=**72**) yods are added by each sector of an
n-gon when it is a 2nd-order tetractys. The number of yods in an n-gon constructed from 2nd-order tetractyses ≡
N(n) = **72**n + 1, where "1" denotes its centre. The number of corners of 1st-order tetractyses = 10n
+ 1 and the number of hexagonal yods = **62**n. Notice that **72** is the number of
Chesed and that **62** is the number of *Tzadkiel*, the Archangel of
*this* Sephirah. For the square (n=4), the number of yods surrounding its centre is

N(4) = 4×**72** = 288 = 1^{1} + 2^{2} +
3^{3} + 4^{4} = 1!×2!×3!×4!.

They comprise 40 corners (35 outside one side) and **248** hexagonal yods (240
outside one side), where **248** is the number of *Raziel*, the Archangel of Chokmah. For
two joined squares, (288−13=275) yods outside their shared root edge surround the centre of each square. Symbol of
the Tetrad, the square determines the number of SLs in CTOL because (275×2=550) yods outside their root edge
surround the centres of two joined squares. They comprise (35+35=70) black corners of (40+40=**80**)
1st-order tetractyses and (240+240=480) coloured hexagonal yods, each square having 240 hexagonal yods outside the
shared side. Hence, the two separate squares contain
(**248**+**248**=**496**) hexagonal yods. This is the square representation
of the (**248**+**248**=**496**) roots of E_{8}×E_{8}, one
of the two symmetry groups known to describe superstring forces that are free of quantum anomalies. The two sets of
eight hexagonal yods that line two sides of the separate squares that become the root edge of the joined squares
symbolise the two sets of eight simple roots of E_{8}×E_{8}. We discover that the square provides a
*natural* connection between the number (550) of Sephirothic emanations in CTOL and the root
composition of E_{8}×E_{8}. The Tetrad determines all properties of the pair of squares. For
example, it determines their 70 corners outside the root edge because 70 is the *fourth*, 4-dimensional,
tetrahedral number* after 1, whilst 35 (number of corners of 1st-order tetractyses in each square outside the root
edge) is the *fourth* tetrahedral number after 1. It determines their
**80** 2nd-order tetractyses because **80** = 10×8, where 10 = 1 + 2 + 3 + 4
and 8 = 4th even integer. It determines their 480 hexagonal yods because

480 = 16×30 = 4^{2}×(1^{2}+2^{2}+3^{2}+4^{2}) =
4^{2} + 8^{2} + 12^{2} + 16^{2}.

It determines their 550 yods outside the root edge that surround their centres because 550 =
10×55, where 10 = 1 + 2 + 3 + 4 and 55 is the *fourth*, square pyramidal number** after 1. Including its
centre, each square has **36** corners of 40 1st-order tetractyses outside the root edge, where
**36** = sum of the first *four* odd integers and the first *four* even
integers:

**36** = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = (1+3+5+7) + (2+4+6+8).

* A tetrahedral number is the number of points in a tetrahedral stack of triangular arrays of
points; a 4-dimensional, tetrahedral number is the number of points needed to construct the 4-dimensional version
of a tetrahedron, called the "pentachoron," "4-simplex" or "5-cell" (see here).

** A square pyramidal number is the number of points in a stack of square arrays of points that form a pyramid.

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