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**#24 Correspondence between the tetrahedron and the
Tree of Life**

When the four triangular faces of the tetrahedron are simple tetractyses, the four tetractyses contain 20 yods (four corners & 16 hexagonal yods. When the six internal triangles created by joining the four vertices to the centre of the tetrahedron are tetractyses, they contain (4×2 + 6 = 14) hexagonal yods and one corner, namely, the centre of the polyhedron. Hence, the 10 external & internal tetractyses making up the tetrahedron have 35 yods (30 hexagonal yods & five corners).

Construction of the Platonic solids from tetractyses requires their faces to be divided into
their sectors. For its construction to be consistent with that of the other Platonic solids, the faces of the
tetrahedron must be treated as Type A triangles, otherwise comparison between the Platonic solids is,
mathematically speaking, illegitimate. A Type A triangle contains 19 yods (see here), i.e., nine more yods than a tetractys, so that this adds nine yods per face. The
tetrahedron now has (35 + 4×9 = 71) yods (nine corners & **62** hexagonal yods,
**31** hexagonal yods being in each half). **31** is the number of EL, the
Godname of Chesed and **62** is the number value of *Tzadkiel*, the Archangel of this
Sephirah. 70 yods in 18 tetractyses surround the centre. **64** yods surround the seven yods
spaced along an axis consisting of the straight lines joining two vertices to the centre.
**64** is the number value of *Nogah*, the Mundane Chakra of Netzach. (71–18=53) yods
line the 22 sides of the 18 tetractyses with nine corners. Starting from the centre,
**26** such yods shape the nine tetractyses in each half of the tetrahedron. This shows how
YAHWEH, the Godname of Chokmah with number **26**, prescribes the simplest Platonic solid. It has
nine corners and 22 sides of 18 triangles, i.e., **31** corners & sides and
(**31**+18=**49**) corners, sides & triangles, showing how EL CHAI, the Godname of
Yesod with number value (**31**+18=**49**) prescribes the tetrahedron. The number
**31** of EL is the number of corners & sides and the number 18 of CHAI is the number of
triangles. **48** geometrical elements surround its centre, where **48** is
the number value of *Kokab*, the Mundane Chakra of Hod.

The counterpart in the outer Tree of Life of the 70 yods present in the
**48** geometrical elements surrounding the centre of the tetrahedron is the 70 yods making up
the **48** geometrical elements (16 triangles with 10 corners & 22 sides) that compose the
Tree. Moreover, the counterparts in the Tree of Life of the eight vertices, 22 sides & 18 triangles, i.e.,
the **26** vertices & triangles and their 22 sides in the tetrahedron, are its
**26** corners & triangles and 22 sides. Here is clearcut evidence of the correspondence
between the two geometrical objects: they have the *same* yod population and the
*same* number of geometrical elements! The Platonic solids really are sacred in the sense that they
embody Divine Ideas about the mathematical nature of reality, and it should come as no surprise that, as the
simplest regular polyhedron, the tetrahedron should display analogies with the Tree of Life and other examples
of sacred geometry that embody the same archetypes.

Further evidence for the analogous properties possessed by the tetrahedron is provided by considering its faces as 2nd-order tetractyses (for their definition, see here). As the next higher-order differentiation of the Pythagorean tetractys (1st-order tetractys), the 2nd-order tetractys has 85 yods, where

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

Inside its boundary lined by **36** yods are **49** yods, where
**36** is the number value of ELOHA, the Godname of Geburah, and **49** is the
number value of EL CHAI, the Godname of Yesod. 11 yods on each side of the 2nd-order tetractys are between each
pair of corners, so that, when the faces of the tetrahedron are transformed into 2nd-order tetractyses, the number
of yods lining its six edges = 6×11 + 4 = 70. They comprise 24 red hexagonal yods of coloured tetractyses
symbolizing the Sephiroth of Construction, 24 white hexagonal yods belonging to tetractyses at corners of each
2nd-order tetractys that symbolize the Supernal Triad and 22 black yods at corners of tetractyses that symbolize
members of the Supernal Triad.

The counterparts of these numbers in the inner Tree of Life are the 24 red corners of one set of the first six enfolded polygons outside the root edge, the 24 white corners of the other set of the first six polygons outside the root edge and the 22 black corners of the two joined dodecagons. The two endpoints of the root edge correspond to the two "poles" of the axis of the tetrahedron. Alternatively, they correspond to the 22 corners of the first (4+4) enfolded polygons and to the 24 corners of each set of enfolded octagons, decagons & dodecagons.

The number of yods in the faces of the tetrahedron when they are 2nd-order tetractyses = 70 +
4×**49** = 266. Therefore, 264 yods in the four faces surround the bent axis passing through
two vertices. They comprise 68 blue yods lining its edges and 196 orange yods inside them. Compare this with the
inner Tree of Life: the seven enfolded polygons have 264 yods, of which 196 orange yods belong to the first six
enfolded polygons or include the centre of the dodecagon, which is surrounded by 68 blue yods. Just as 68 more
yods delineate the tetrahedron, starting with one vertex and ending with the vertex at the other end of the
axis, so, starting with its centre, 68 yods outside the root edge shape the dodecagon, which always embodies
numbers that express the Malkuth (outer form) aspect of the Tree of Life. In terms of the outer Tree of Life,
the 68 yods extend between Kether and Malkuth; in terms of the inner Tree of Life, the (7+7) enfolded polygons
are shaped by 68 corners outside the root edge, whose two endpoints represent the beginning and the completion
of the 14 polygons.

Another property of the tetrahedron with 2nd-order tetractyses as its faces that is analogous to the inner Tree of Life is the following: the 40 tetractyses in the four faces have 34 corners and 232 hexagonal yods. The latter is the number of yods outside the root edge that line the sides of the (7+7) enfolded polygons (see #6). The four vertices of the tetrahedron are analogous to the four yods in the root edge and the 232 hexagonal yods are analogous to the 232 yods on sides of the (7+7) polygons outside their shared root edge. Alternatively, the 34 corners are analogous to the 34 corners of the seven enfolded polygons that are outside their root edge, whilst the 232 hexagonal yods correspond to the 232 yods in the seven enfolded polygons other than these corners. Whichever analogy we favour, the message is unmistakable, namely, the seven enfolded polygons is the "flat" form of the simplest building block in 3-dimensional space — the tetrahedron.

Each 2nd-order tetractys has seven coloured tetractyses representing the seven Sephiroth of
Construction. The (4×7=28) colored tetractyses in the four faces have 172 hexagonal yods arranged in 28
hexagons. Hence, (172–4=**168**) hexagonal yods surround the centres of the four faces. As the
latter are the centres of the central tetractyses representing Malkuth, they symbolize Malkuth
*par* *excellence*. The number value **168** of its Mundane Chakra emerges
as the number of hexagonal yods that symbolize differentiations of Sephiroth of Construction other than the most
Malkuth-like ones, namely, the central yod of the central tetractys in each 2nd-order tetractys that symbolizes
this Sephirah. We found in #20 of **Wonders of
Superstrings** that **168** corners, sides & triangles surround the
axis of the tetrahedron when its faces and internal triangles formed by its edges and by the sides of sectors in
its faces are Type A triangles. Here, therefore, are two independent ways in which the structural parameter
**168** of the E_{8}×E_{8} heterotic superstring, as remote-viewed by
Annie Besant & C.W. Leadbeater over a century ago, manifests in the basic building block of solid
geometry.