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**#51 The Tetrad determines the numbers of vertices & edges of the 4 _{21} polytope representing the 240 roots of the
exceptional Lie group E_{8}**

The 60 yods surrounding the centre of a Type B square
generate the number 240 when weighted with the Tetrad. This is the number of vertices of the
4_{21} polytope. |
An 8-fold array of 2nd-order tetractyses sharing one corner has 672 yods surrounding its centre.
Weighted with the number 10 (the 4th triangular number), they generate the number 6720, which is
the number of edges of the 4_{21} polytope. Each cross-shaped array of four 2nd-order tetractyses contains 336
yods that represent the 3360 edges of each half of the 4_{21} polytope when
the Decad (10) is assigned to each yod. |

The square is the simplest, geometrical symbol of the Tetrad. The Type B square had 61 yods. Weighted with the number 4, the 60 yods surrounding its centre generates the number 240, where

240 = (1+2+3+4)×1×2×3×4.

This is the number of vertices of the 4_{21} polytope whose 8-dimensional position
vectors are the root vectors of E_{8}, the largest of the five exceptional Lie groups.

The 2nd-order tetractys has 85 yods, where

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

84 yods surround its centre, where

84 = 4^{1} + 4^{2} + 4^{3} = 1^{2} +
3^{2} + 5^{2} + 7^{2}.

There are also 84 yods below its apex. Eight 2nd-order tetractyses arranged around their shared
apices contain (8×84=672) yods. Weighted with the Decad, they generate the number 6720. This is the number of edges
of the 4_{21} polytope. Each cross-shaped array of four red or four blue 2nd-order tetractyses
generates the number (3360) of edges in half this polytope. As the Decad is the *fourth* triangular
number (namely, the 1st-order tetractys), the Tetrad determines the number of vertices and the number of edges of
the 4_{21} polytope. In this way, the number at the heart of the number mysticism of the ancient
Pythagoreans determines the very geometrical object whose vertices represent the symmetry group
E_{8} describing the forces between E_{8}×E_{8} heterotic superstrings. How right
the ancient Pythagoreans were for valuing the Tetrad and tetractys so much!

672 yods are needed to construct the first four Platonic solids (tetrahedron, octahedron, cube
& icosahedron) from tetractyses (see here & here). This means that they embody the number 6720 when the Decad is
assigned to all these yods as their building blocks. *Remarkably, the number 1680
paranormally obtained by C.W. Leadbeater when he counted and checked 135 times the number of turns in each
helical whorl of the UPA is, simply, the number generated by assigning the Decad to the average number of yods
in the four Platonic solids believed by the ancient Greeks to be the shapes of the particles of the four
physical Elements Fire, Air, Earth & Water.* 672 extra yods are needed to transform into
tetractyses all the sectors of the 42 triangles surrounding the centre of the Sri Yantra (see here). It means that they, too, embody the number 6720 when the Decad
is assigned to them (see the home page). We saw in #50 how the inner form of 10 overlapping Trees of Life embodies
this number. *The 4 _{21} polytope conforms to the archetypal pattern of
sacred geometries*. The implication of this is clear: E

The number 6720 is related to the superstring structural parameter 16800 and to the dimension
**248** of E_{8} by:

6720 =
16800×2480/(2^{2}+3^{2}+4^{2}+...+**26**^{2}),

where **26** is both the gematria number value of YAHWEH and the number of dimensions of the
space-time of bosonic strings. As 16800 = **50**×336, where

336 =
(1^{2}+2^{2}+3^{2}+...+**31**^{2})/**31**,

and 2480 = **80**×**31**,

so that

6720 =
4×10×100×(1^{2}+2^{2}+...+**31**^{2})/(2^{2}+3^{2}+4^{2}+...+
**26**^{2}) =
4×(1+2+3+4)(1^{3}+2^{3}+3^{3}+4^{3})(1^{2}+2^{2}+...+**31**
^{2})/(2^{2}+3^{2}+4^{2}+...+**26**^{2}),

where **26** is the sum of the numbers of combinations of 10 objects arranged in the four rows
of a tetractys:

**26** = (2^{1} − 1) + (2^{2} − 1) +
(2^{3} − 1) + (2^{4} − 1) = 1 + 3 + 7 + **15**.*

This shows how the Tetrad expresses the number of edges of the 4_{21} polytope.

* There are ^{n}C_{r} combinations of r objects taken from a set of n objects, where
^{n}C_{r} = n!/r!(n−r)!. The sum of the numbers of combinations of n objects taken one, two,
... n at a time = ^{n}C_{r} = 2^{n} − 1, using the
binomial theorem:

(1 + *x*)^{n} = ^{n}C_{r}*x*^{r} = 1 + ^{n}C_{r}*x*^{r}

and putting *x* = 1.

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