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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 = 40 + 41 + 42 + 43.
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 E8×E8 heterotic superstring, as remote-viewed by Annie Besant & C.W. Leadbeater over a century ago, manifests in the basic building block of solid geometry.