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 Article 40   Part 2 by Stephen M. Phillips Flat 3, 32 Surrey Road South. Bournemouth. Dorset BH4 9BP. England. Website: http://www.smphillips.mysite.com 1

Figure 1

The 34 vertices of the 2-dimensional Sri Yantra and their 34 mirror images correspond to the 34 corners of the seven enfolded polygons outside their root edge and their 34 mirror images in the other set of polygons. The 42 edges and their 42 mirror images in the first three layers of triangles are the counterpart of 42 hexagonal yods on the edges of one set of polygons and their mirror images in the other set. The 42 edges & triangles in the fourth layer and their 42 mirror images are the counterpart of another 42 hexagonal yods on the edges of one set of polygons and their 42 mirror images in the other set.

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 Correspondence between the geometrical composition of the four layers of the 2-d Sri Yantra and the boundary yods of the 14 polygons of the inner Tree of Life. 3

Figure 2

Just as 240 geometrical elements are needed to construct the 2-dimensional Sri Yantra, starting with the central bindu point, so 240 extra yods are needed to construct the 19 triangles of the lowest Tree of Life from tetractyses and 240 extra yods are needed to construct the sectors of the polygons in the inner Tree of Life from tetractyses.

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 The number 240 is a structural parameter of the Sri Yantra, the Tree of Life and its inner, polygonal form. 5

Figure 3

When constructed from tetractyses, the first three Platonic solids have 240 hexagonal yods, as do the icosahedron and the dodecahedron. The number 240 is a structural parameter of holistic systems like the five possible, regular polyhedra.

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Figure 4

There are 13 semi-regular polyhedra (Archimedean solids). They have 13 duals – the Catalan solids – in which each vertex is replaced by a face and vice versa. The two tables list the number of vertices, edges & triangles in their faces. The most complex of the Catalan solids is the disdyakis triacontahedron. 2400 corners, edges & triangles surround an axis through any two opposite vertices when its faces are divided into three triangles and 1680 geometrical elements when its faces are single triangles. This is ten times the corresponding numbers for the triakis tetrahedron, the simplest Catalan solid. The disdyakis triacontahedron is the polyhedral counterpart of the inner Tree of Life, embodying the structural parameter 240. As shown later, 168 and 1680 are also parameters embodied in any manifestation of the Tree of Life blueprint.

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Figure 4

 Case A: triangular face as tetractys

 Case B: triangular face as 3 tetractyses
 C = number of corners E = number of edges F = number of faces N = number of corners, edges & triangles surrounding the axis (case A)  N' = number of corners, edges & triangles surrounding the axis (case B) Tables of properties of the Archimedean and the Catalan solids
 N' F E C Archimedean solid 244 8 18 12 truncated tetrahedron 322 14 24 12 cuboctahedron 490 14 36 24 truncated cube 490 14 36 24 truncated octahedron 646 26 48 24 rhombicuboctahedron 802 38 60 24 snub cube 802 38 60 24 snub cube (chiral partner) 808 32 60 30 icosidodecahedron 882 26 72 48 truncated cuboctahedron 1228 32 90 60 truncated icosahedron 1228 32 90 60 truncated dodecahedron 1618 62 120 60 rhombicosidodecahedron 2008 92 150 60 snub dodecahedron 2008 92 150 60 snub dodecahedron (chiral partner) 2458 62 180 120 truncated icosidodecahedron

 Catalan solid F E C N N' triakis tetrahedron 12 18 8 168 240 rhombic dodecahedron 12 24 14 - 324 triakis octahedron 24 36 14 336 480 tetrakis hexahedron 24 36 14 336 480 deltoidal icositetrahedron 24 48 26 - 648 pentagonal icositetrahedron 24 60 38 - 816 pentagonal icositetrahedron (chiral partner) 24 60 38 - 816 rhombic triacontahedron 30 60 32 - 810 disdyakis dodecahedron 48 72 26 672 960 triakis icosahedron 60 90 32 840 1200 pentakis dodecahedron 60 90 32 840 1200 deltoidal hexacontahedron 60 120 62 - 1620 pentagonal hexacontahedron 60 150 92 - 2040 pentagonal hexacontahedron (chiral partner) 60 150 92 - 2040 disdyakis triacontahedron 120 180 62 1680 2400

2400 geometrical elements surround the axis of the disdyakis triacontahedron constructed from triangles. This is 10 times that for the triakis tetrahedron, the simplest Catalan solid. It illustrates how the number 240 characterizes holistic systems like the disdyakis triacontahedron.

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Figure 5

The Tree of Life parameter 240 appears in superstring physics as the number of non-zero roots of the Lie algebra of E8, the superstring gauge symmetry group. The parameter 168 appears as the number of non-zero roots of E8 that are not non-zero roots of its exceptional subgroup E6.

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Figure 6

There are 260 vertices, edges & triangles in the 3-dimensional Sri Yantra. 168 geometrical elements are in the first three layers of triangles.

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 Geometrical composition of the 3-d Sri Yantra. 13

Figure 7

As confirmation that the 2-dimensional Sri Yantra is the counterpart of seven overlapping Trees of Life, compare their geometrical compositions. The seven overlapping Trees of Life are composed of 260 vertices, edges, triangles & tetrahedra. The 3-dimensional Sri Yantra has 260 vertices, edges & triangles. They are both composed of 260 geometrical elements. The seven overlapping Trees of Life are a map of the seven planes of consciousness. The number value 26 of YAHWEH, Godname of Chokmah, prescribes these two maps of the seven planes.

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Figure 8

Constructed from tetractyses, the seven enfolded polygons have 260 yods outside their shared root edge. Each yod symbolizes one of the geometrical elements composing the 3-dimensional Sri Yantra. This demonstrates that the inner Tree of Life and the Sri Yantra are equivalent representations of holistic systems.

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 The inner Tree of Life is composed of 260 yods outside its root edge. This demonstrates its identity to the 3-d Sri Yantra, which comprises 260 geometrical elements. Each yod symbolizes an element. 17

Figure 9

The 260 geometrical elements of the Sri Yantra comprise the eight elements making up the central bindu point and innermost triangle and the 252 elements of the eight layers of triangles. Their counterparts in the inner Tree of Life are the eight yods that are either centres of polygons or Sephiroth of the Tree of Life and the 252 other yods outside the shared root edge of the polygons.

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 3-d Sri YantraBindu + innermost triangle: 4 vertices + 3 edges + 1 triangle = 8 geometrical elements 4 layers of triangles: 84 vertices + 126 edges + 42 triangles = 252 geometrical elements Total = 260 Inner Tree of Life 8 yods () are either centres of polygons or locations of Sephiroth of the outer Tree of Life; has 252 coloured yods Total = 260 The correspondence between the Sri Yantra and the inner Tree of Life. 19

Figure 10

When their triangles are turned into tetractyses, there are 384 yods up to the top of the seventh, overlapping Tree of Life. The 42 triangles of the Sri Yantra have 378 yods, whilst the central triangle has six hexagonal yods on its edges, a total of 384 yods. This numerical correlation is not an accident but indicates that the Sri Yantra and the seven Trees of Life are isomorphic representations.

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 The Sri Yantra is equivalent to the lowest 7 Trees of Life because it comprises as many yods as there are yods up to the top of these trees. 21

Figure 11

According to Plato, the celestial sphere was designed in the proportions of the squares of the numbers 1, 2 & 3. Arranged in the shape of the Greek letter lambda, this set of seven integers (the so-called “Lambda”) is but two sides of a tetractys array of 10 integers (let us call it the “Lambda Tetractys”) that add up to 90. The four integers 1, 3, 9 & 27 on one side of the array add up to 40. The sum of the remaining integers is 50. The Sri Yantra is formed from five downward-pointing triangles expressing the Shakti (feminine) aspect of creation and four upward-pointing triangles expressing its Shiva (masculine) aspect. If each triangle is considered a tetractys, the number of yods in the nine tetractyses is 90, which is the sum of the integers in the tetractys extension of Plato’s Lambda. The four Shiva triangles/tetractyses have 40 yods and the five Shakti triangles/tetractyses have 50 yods. The source of the Sri Yantra therefore conforms to the Lambda Tetractys pattern, confirming its archetypal quality.

The four integers adding to 40 are all odd integers. The six integers adding to 50 are all even. The Pythagoreans regarded even integers as female and odd integers as male. This is consistent with the five triangles that embody the Shakti creative energy (female principle) having 50 yods and the four triangles that embody the Shiva energy (male principle of creation) having 40 yods. The division of the Lambda Tetractys into even and odd integers matches precisely its counterpart as five Shakti triangles and four Shiva triangles. The Lambda Tetractys is an arithmetic expression of the paradigm underlying different sacred geometries.

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Figure 12

As tetractyses, the nine primary triangles generating the Sri Yantra have 27 yods at corners and nine central yods, i.e., 36 yods. The sum of the integers at the corners of the Lambda Tetractys is 36 (the largest of its integers is 27). The nine tetractyses have 54 hexagonal yods. The sum of the seven integers in the Lambda Tetractys arranged at the corners and centre of a hexagon is 54. This further demonstrates that the Lambda Tetractys arithmetically expresses the geometrical origin of the Sri Yantra. The integers 1 and 8 at two corners of the Lambda Tetractys denote, respectively, the central yod of the unpaired Shakti tetractys (the smallest, downward pointing, blue tetractys and the eight central yods of the four pairs of Shiva & Shakti tetractyses forming Stars of David. The integer 27 at the third corner of the Lambda Tetractys is the number of yods at the corners of the nine tetractyses.

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Figure 13

The nine triangles with 27 vertices overlap to form the 3-dimensional Sri Yantra whose 43 triangles have 87 vertices. 60 new vertices are generated. This is comparable with the Tree of Life when its 16 triangles with 10 vertices are turned into tetractyses made up of 70 yods: 60 new yods appear.

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 60 new vertices are generated when 9 triangles with 27 vertices form the 3-d Sri Yantra with 87 vertices in its 43 triangles. Likewise, 60 yods are created by turning the 16 triangles of the Tree of Life into tetractyses. 27

Figure 14

The four layers of triangles in the 3-dimensional Sri Yantra have 84 vertices. This is the number of yods surrounding the centre of the 2nd-order tetractys. The 24 triangles of the outer two groups have 48 vertices. This is the number of hexagonal (brown) yods in the seven tetractyses arranged in a hexagon that surround the centre. The two inner groups of triangles have 36 vertices. 36 yods in the 2nd-order tetractys do not belong to these tetractyses. These correlations show how the Sri Yantra is equivalent to the 2nd-order tetractys – a higher differentiation of the tetractys symbolizing the 10-fold nature of Divine Unity.

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 Correspondence between the 84 vertices in the 4 layers of triangles of the 3-d Sri Yantra and the 84 yods surrounding the centre of the 2nd-order tetractys. 29

Figure 15

When the lowest Tree of Life is constructed from tetractyses, there are 84 yods up to the top of the lowest Tree of Life. They correspond to the 84 corners of the 42 triangles of the Sri Yantra. The third and fourth layers of triangles have 48 corners corresponding to the 48 yods up to Chesed, the first Sephirah of Construction, and the first and second layers have 36 vertices corresponding to the 36 yods between Chesed and the top of the lowest tree.

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 Correspondence between the Tree of Life and the 3-d Sri Yantra. 31

Figure 16

336 yods lie on the edges of the 42 triangles of the Sri Yantra. 168 yods form the edges of each half. 168 yods also lie on the edges of the 14 triangles in the lowest layer.

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 336 yods lie on the 126 edges of the 42 triangles of the 3-d Sri Yantra. 168 yods lie on the edges of one half and 168 yods lie on the edges of the other half. Remarkably, the same number of yods also lie on the edges of the 14 triangles in the lowest layer of the Sri Yantra, whilst 168 hexagonal yods lie on the edges of the 28 triangles in its first 3 layers. 33

Figure 17

When the 42 triangles of the Sri Yantra are converted into tetractyses, 168 yods lie on the edges of the 28 triangles in the first three layers. 168 yods lie on the edges of the 14 triangles in the fourth layer.

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Figure 18

336 yods lie on the edges of the 42 triangles of the 3-dimensional Sri Yantra. 168 yods lie on the edges of the 21 triangles in one half and 168 yods lie on the edges of the 21 triangles in the other half.

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Figure 19

The first (6+6) enfolded polygons of the inner Tree of Life have 42 corners that do not coincide with Sephiroth of the outer Tree of Life. The eight corners that do coincide are shown as white yods (although not a Sephirah, Daath can be formally treated here as a Sephirah because it is Yesod of the next higher, overlapping Tree of Life). Each set of six polygons have 168 yods (red or blue) that are not corners. Compare this with the 42 centres of triangles and the 168 yods on the edges of each half of the Sri Yantra. The correspondence is complete if we include the seven hexagonal yods of the central triangle (corresponding to the seven Sephiroth whose positions coincide with corners) and the bindu, which corresponds to Daath.

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 Correspondence between the inner Tree of Life and the 3-d Sri Yantra 39

Figure 20

When the triangles of the Sri Yantra are each constructed from three tetractyses, its outermost 14 triangles have 168 hexagonal yods on the edges of their 42 tetractyses.

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 The 14 outermost triangles of the Sri Yantra embody the superstring structural parameter 168 as the number of hexagonal yods on the edges of their 42 tetractyses. 41

Figure 21

The dodecagon is the seventh and last of the regular polygons in the two sets that constitute the inner Tree of Life.

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Figure 22

The dodecagon requires 168 extra yods when its sectors are divided into three tetractyses. 14 yods are added per sector. Therefore, six sectors have 84 extra yods. As the dodecagon is two hexagons rotated through 30°, the number 168 embodied in the dodecagon divides naturally into 84 and 84. This is the counterpart of the 84 vertices of the 42 triangles of the Sri Yantra and the 84 hexagonal yods on the edges of its outermost 14 triangles.

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 168 extra yods are needed to construct each of the sectors of the dodecagon from three tetractyses. 45

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