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**#34 Correspondence between (248+248) yods in
the 10-tree, the 248 yods in a tiled 3-torus and the 248 yods in its form turned
inside-out**

The 3-torus tiled with 56 tetractyses has **248** yods. It turns inside
out into another 3-torus with **248** yods.

(Animated image provided by Greg Egan at http://www.gregegan.net/SCIENCE/KleinQuartic/KleinQuartic.html).

When the triangles of overlapping Trees of Life are turned into tetractyses, there are
**248** yods up to Chesed of the fifth Tree. They comprise the **80** black
yods in the lowest Tree and a further **168** red yods up to this Chesed. Above it are
**248** blue yods up to (but not including) Chesed of the tenth Tree. As ten overlapping Trees of
Life are a complete representation of the ten Sephiroth of the Tree, the division of the
**496** yods into two sets of **248** yods reflects the basic division of the
Tree of Life into its Lower Face spanned by the five lowest Sephiroth (Tiphareth-Malkuth) and the rest of it
spanned by the five highest Sephiroth (Kether-Geburah). It is the Kabbalistic basis for the direct product nature
of the gauge symmetry group E_{8}×E_{8} associated with one of the two types of heterotic
superstrings. As Chesed of the fifth Tree is the **31**st SL and Chesed of the tenth Tree is the 61st
SL, where 61 is the **31**st odd integer, the Godname EL of Chesed with number value
**31** prescribes both the dimension **496** of the two possible, anomaly-free
gauge symmetry groups for 10-dimensional superstrings (SO(32) & E_{8}×E_{8}) and the dimension
**248** of E_{8}. Another way in which EL prescribes this number at the heart of
superstring theory is that **496** is the **31**st triangular number:

1 + 2 + 3 + ... + **31** = **496**.

As discussed in #25 of **Wonders of
Superstrings**, when the 3-torus is assembled from four triangular prisms and six square
antiprisms and their 56 triangular faces are then turned into tetractyses, there are
**80** black yods at either the corners (24) or centres (56) of these tetractyses and
**168** red hexagonal yods on their 84 sides. We see that a remarkable correspondence exists
between the 3-torus, its version turned inside-out and the ten overlapping Trees of Life. The lowest five Trees
with **248** yods up to the **31**st SL are the counterpart of the tiled 3-torus
whose 56 tetractyses have **248** yods. The **80** black yods in the lowest
Tree are the counterpart of the **80** black yods at corners and centres of these tetractyses
and the **168** red yods above them up to Chesed of the fifth Tree are the counterpart of the
**168** red hexagonal yods that line their 84 sides. The **248** blue yods
beyond them up to (but not including) Chesed of the tenth Tree are the counterpart of all
**248** blue yods in the 3-torus when it is turned inside out. The 5:5 division of the Tree of
Life has its counterpart in the 3-torus, which maps the **168** automorphisms of the Klein
quartic, and in its turned inside-out version, which maps the **168** anti-automorphisms
(rotations plus reflections) of this famous equation in mathematics.

More amazing parallels between the tiled 3-torus and its Tree of Life counterpart are analyzed in Article 43.

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