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#37 Correspondence between the 217 boundary yods in the 7 enfolded polygons and the 217 points & lines in the 3-d Sri Yantra

The 7 enfolded polygons have 264 yods.

217 yods line the 47 tetractyses in the 7 enfolded polygons.

105 intrinsic, boundary yods are associated with the first 5 polygons (blue) and with the last 2 polygons (red). The 7 green yods are shared with the outer Tree of Life.

The 21 triangles in each half of the 3-d Sri Yantra surrounding the 7 points & lines in the central triangle and bindu comprise 105 points & lines.

The inner form of the Tree of Life consists of two sets of 7 polygons:

triangle, square, pentagon, hexagon, octagon, decagon & dodecagon.

When their 47 sectors are tetractyses, the 7 enfolded polygons contain 264 yods.

Of the 264 yods in the 7 enfolded polygons, 47 hexagonal yods are at the centres of tetractyses. This leaves 217 yods (41 corners & 176 hexagonal yods) lining their sides. One of these is the topmost corner of the hexagon. It coincides with the lowest corner of the hexagon that is part of the inner form of the next higher, overlapping Tree of Life. The tetractyses making up each half of the inner form of successive Trees of Life are shaped by 216 yods that are intrinsic to each Tree. 216 is the number value of Geburah, the sixth Sephirah of Construction, counting from the bottom of the Tree of Life. The 7 enfolded polygons have 36 corners, where 36 is the number value of ELOHA, the Godname of Geburah. This is an example of the conjunction of the numbers of the same Sephirah in the same sacred-geometrical context.

When the 16 triangles in the outer Tree of Life are turned into tetractyses, 7 yods lie on the Pillar of Mercy. They coincide with the 7 green yods on the vertical axis of the hexagon. These are the only yods that the outer Tree shares with the inner Tree in which each yod is the same type in both cases. The 217 yods on the sides of the 47 tetractyses consist of these 7 yods shared with the outer Tree and 210 (=21×10) yods that are intrinsic to the inner Tree. This shows how EHYEH, the Godname of Kether with number value 21, prescribes those yods unshared with the outer Tree that shape the tetractyses making up the inner Tree of Life.

Tabulated below are the numbers of intrinsic yods outside the root edge on the sides of the tetractyses in each polygon:

 Root edge Triangle Square Pentagon Hexagon Octagon Decagon Dodecagon 2+2=4 11 17 22 16 37 46 57 103 103

103 intrinsic yods (coloured blue) line the 25 tetractyses in the first 5 enfolded polygons and 103 intrinsic yods (coloured red) line the 22 tetractyses in the last 2 enfolded polygons. 103 is the number value of SABAOTH ("Hosts"), part of the Godnames of Hod and Netzach (see here). Two blue yods in the root edge can be associated with the former set and two red yods can be associated with the latter. This means that the 210 intrinsic yods divide into two equal sets of 105 yods. It is easily verified that these two sets are unique — there are no other combinations of polygons with 103 yods. The 217 boundary yods comprise 7 shared (green) yods and 210 intrinsic yods (105 blue & 105 red).

The 3-dimensional Sri Yantra consists of four layers of 42 triangles (21 triangles in each half) that surround the central triangle (coloured green) above them. Its three corners represent the three-fold Godhead, or trimurti  (Hindi: त्रिमू्र्ति, translation: "Trinity"), of Shiva, Brahma & Vishnu. A dot, or bindu, representing the Absolute, the source of all manifestation, both physical and superphysical. The central triangle comprises 3 corners & 3 sides. The 42 triangles comprise 84 corners (42 in each half) and 126 sides (63 in each half). Each half of the 3-dimensional Sri Yantra is shaped by (42+63=105) corners & sides. Together with the seven green points & lines making up the bindu and the central triangle, (105+105+7=217) points & lines constitute the Sri Yantra. 216 points & lines surround the central point, 210 (=21×10) belonging to the 42 triangles.

The same number (217) that measures the shape of the sectors of the 7 enfolded polygons quantifies those geometrical elements that shape the Sri Yantra. Moreover, the seven yods in the inner Tree of Life shared with its outer form correspond to the seven points & lines making up the bindu and central triangle. It is remarkable that the set of seven enfolded polygons divide uniquely into two subsets of complete polygons with 105 yods that correspond to the 105 points & lines in the 21 triangles in each half of the Sri Yantra. Both sacred geometries also display the division: 217 = 1 + 216.

These correspondences demonstrates the equivalence of the Sri Yantra and the inner Tree of Life. Their geometrical dissimilarity hides the fact that they embody the same numbers quantifying analogous properties. These numbers are defining parameters of all holistic systems/sacred geometries.

The 217 yods in a triangle with 2nd-order tetractyses as sectors display the pattern 1:6:70:140.

The triangle with its 3 sectors turned into 2nd-order tetractyses contains 217 yods. They comprise its centre (black yod), three green yods at its corners and three green yods at the corners of the 3 1st-order tetractyses whose apices are its centre, 70 red yods in the sector with a horizontal base and 140 blue yods in its two other sectors.

Previously, the 217 yods lining the sides of the 47 tetractyses in the 7 enfolded polygons were divided into sets of 7 and 210. Instead of the former referring to the 7 shared yods that line each side pillar, they may, alternatively, be identified as the 4 yods in the root edge and the 3 yods that coincide with Sephiroth on each side pillar. As before, the topmost corner of the hexagon (black yod) corresponds to the bindu in the Sri Yantra, whilst the 4 green yods in the root edge and the two green yods located at Sephiroth correspond to the 3 corners and 3 sides of the central triangle that symbolises the trimûrti, or triple Godhead of Shiva, Vishnu & Brahma. The 1st 4 polygons have 70 red yods and the last 3 polygons have 140 blue yods, where 140 is the number value of Masloth, the Mundane Chakra of Chokmah. Noting that there are 5 corners & sides per triangle in the Sri Yantra, these two sets of yods correspond to the (5×14=70) red corners & sides of the 14 triangles in the 4th layer of the 3-d Sri Yantra and to the (5×28=140) blue corners & sides in the 28 triangles in its 1st, 2nd & 3rd layers. Note that, if the image of the Sri Yantra is magnified, some corners are half one colour and half the colour of yods in the adjacent layer, or (in the case of adjacent layers having similarly coloured corners) half that colour and half a lighter shade  of it. They denote corners that lie directly over corners of triangles in the adjacent layer.

As well as shaping the inner Tree of Life and the 3-dimensional Sri Yantra, the number 217 has both an arithmetic and a geometric significance vis-à-vis the dimension 248 of E8, the rank 8, exceptional Lie group that plays a pivotal role in one of the two heterotic superstring theories.

Arithmetic
The dimension of SO(32) and E8×E8, the symmetry groups describing the two types of heterotic superstrings, is 496. As the third perfect number,* this number is the sum of its nine divisors:

496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248.

Therefore, 248 is the sum of the eight smallest divisors of 496:

248 = (1+2+4+8+16) + (31+62+124)

= 31 + 217,

where 217 = 31 + 62 + 124 = 7×31. As 31 = 20 + 21 + 22 + 23 + 24, 62 = 2×31 and 124 = 22×31, 217 may be written as the following sum of powers of 2:

217 = 20 + 23 + 24 + 26 + 27,

so that 216 = 23 + 24 + 26 + 27.

Geometric
Constructed from 67 triangles, the 5-tree contains 280 yods when they are tetractyses, where 67 is the number of Binah and 280 is the number value of Sandalphon, the Archangel of Malkuth. There are 248 yods up to Chesed of the fifth Tree of Life (see also here), where 248 is the number of Raziel, the Archangel of Chokmah. This SL is the 31st SL, where 31 is the number value of EL, the Godname of Chesed. Therefore, the 248 yods comprise 31 SLs (shown as black yods in the diagram opposite) and 217 hexagonal yods (shown as red yods). Remarkably, the number 31 determines both the number of corners of tetractyses containing 248 yods and, as 217 = 31×7, the number of separate tetractyses that have 217 hexagonal yods, each having seven such yods. This is a way in which EL prescribes the dimension 248 of E8 at the heart of E8×E8 heterotic supersting theory.

The number 217 is a defining parameter of the inner Tree of Life, as now explained. Its seven enfolded Type A polygons have 223 hexagonal yods, of which two lie on the root edge and four lie on a side pillar of the outer Tree, so that they are shared with it. They are coloured black in the diagram below:

This means that (223−2−4=217) hexagonal yods outside the root edge are intrinsic to the inner Tree of Life. Symbolising the seven Sephiroth of Construction, they quantify the number of bits of information that are needed to define the sacred geometry of an object embodying a holistic number like 248.

Sacred geometrical counterpart of the zodiacal age
217 yods line the sides of the 47 tetractyses making up each half of the inner Tree of Life. As the topmost corner of the hexagon in the inner Tree of Life coincides with the lowest corner of the hexagon in the inner form of the next higher, overlapping Tree of Life, 216 boundary yod are intrinsic to the set of seven polygons enfolded in successive Trees. 216 is the number value of Geburah. The number of yods on the boundaries of the 47n tetractyses in the 7n polygons enfolded in the n-tree ≡ B(n) = 216n + 1, where "1" denotes the shared corner of the hexagon. The 470 tetractyses in the 70 polygons enfolded in the 10-tree are shaped by B(10) = 2160 yods that are intrinsic to them. In fact, in every 10 Trees, there are 70 polygons whose 470 tetractyses are lined by 2160 yods. According to the discussion of CTOL in Maps of reality, 91 Trees of Life map the seven cosmic planes. These Trees were emanated in 10 stages, each of 10 Trees, the last Tree in each stage being the first Tree of the next stage. The number 2160 represents a measure of each cycle, being the number of yods needed to shape the inner form of the 10 Trees emanated in each stage. Astronomers have calculated that the axis of the Earth rotates in a circle once every 25,920 years approximately (the so-called 'Platonic Year'). In this precession of the equinox, the axis rotates through 30° about every 2160 years, creating an astrological age that is dominated by one of the 12 zodiacal signs.** It makes the stars move west to east about one degree about every 72 years. These numbers are only approximate because the lengths of the ages are decreasing with time as the rate of precession is increasing. Therefore, no two ages are of equal length. With this qualification, the astronomical number 2160 may be seen as a counterpart of the parameter 2160 associated with the 10-fold emanation of CTOL. It may be regarded as an example of the hermetic principle "As above, so below." Remarkably, the first four enfolded polygons have 72 intrinsic yods on the sides of their 17 tetractyses, the last three polygons having 144 such yods lining their 22 tetractyses, so that the 216 boundary yods intrinsic to each set of seven polygons naturally divide into a set of 72 yods and a set of 144 yods. The number value of Chesed is 72 and the number value of Geburah, the next Sephirah, is 216, where 216 = 23 + 24 + 26 + 27. Notice that this can be re-arranged as (23+26) + (24+27) = (8+64) + (16+128) = 72 + 144, which are the numbers of intrinsic yods lining the tetractyses in, respectively, the first four and the last three enfolded polygons of the inner Tree of Life. When geometry is sacred, it always exhibits a beautiful harmony with the properties of integers. The division 216 = 72 + 144 is found also in the 144 Polyhedron, part of the Polyhedral Tree of Life (see here). Formed from the disdyakis dodecahedron with 26 vertices, 72 edges & 48 faces by attaching tetrahedra to its faces, it has (26+48=74) vertices, (72 + 3×48 = 216) edges and (3×48=144) triangular faces. Its 216 edges comprise the 72 edges of the disdyakis dodecahedron and the extra 144 edges of the added 48 tetrahedra. It has 432 geometrical elements surrounding an axis passing through two opposite vertices. In terms of years, this number, which plays a part in Hindu chronology concerning the four yugas, corresponds to the period of 1/5 of an astrological age, or a 6° (1/60th circle) rotation of Earth's axis. Notice that 144 (= 122) is the 12th Fibonacci number after 0 and the only non-trivial, square Fibonacci number.

Embodiment of the number of Tiphareth
According to the table shown above, (46+57=103) yods outside the root edge line the 22 tetractyses in the decagon & dodecagon. This number does not include the centre of the former, which the table counted as a corner of the pentagon. Let us, instead, count it with the decagon. Including the four yods in the root edge, there are 108 boundary yods in the last two polygons and 109 boundary yods in the first five polygons. The latter includes the corner of the hexagon shared with the hexagon belonging to the polygons enfolded in the next higher Tree. The number of boundary yods outside the root edge in the 5n polygons of the first five types enfolded in the n-tree ≡ b(n) = 108n + 1. There are (b(10)=1081) such yods shaping the 50 polygons of these types enfolded on either side of the central pillar of the 10-tree. This is the gematria number of Tiphareth, the Sephirah at the centre of the Tree of Life! Although remarkable, the appearance of this number in the inner form of the 10-tree is not unexpected in view of the holistic character of this set of Trees, as demonstrated in Superstrings as sacred geometry in how it is prescribed by Godnames and embodies superstring structural and dynamical parameters. Using the formula given above for B(n), there are also (B(5)=1081) yods lining the 235 tetractyses in the 35 polygons of all seven types enfolded in the 5-tree. This is discussed here. Therefore, the number of the Sephirah dividing the 10 Sephiroth into two sets of five Sephiroth also divides 10 Trees into two sets of five Trees, there being 1081 boundary yods in the inner form of the lowest five Trees and 1081 yods (one yod being shared with the former set) in the inner form of the next five Trees.

The number of boundary yods outside the root edge in the first (5+5) polygons enfolded in the n-tree = 2b(n) = 2×(108n+1) = 216n + 2, where "2" denotes the top corners of the two hexagons enfolded in the nth Tree that coincide with the lowest corners of the hexagons enfolded in the (n+1)th Tree. The first (5+5) enfolded polygons have 50 sectors, showing how they are prescribed by ELOHIM, the Godname of Binah with number value 50. The (50+50) polygons of the first five types enfolded in the 10-tree have (50×10=500) tetractyses with 2160 boundary yods outside their root edges that are intrinsic to them. Compare this with the fact that the 70 polygons enfolded on either side of the 10-tree contain 2160 intrinsic boundary yods. It demonstrates the geometrical connection between the number values of Chesed, Geburah and Tiphareth, which are successive Sephiroth in the Tree of Life. It also shows how the same holistic parameters re-appear in different holistic subsets of a holistic system, defining their properties.

We prove in #20 at Wonders of Superstrings that, when the vertices and face centres of the five Platonic solids are joined to their centres and their faces and all internal triangles are divided into their sectors, there are 2160 sides & triangles that surround their axes passing through two opposite vertices and their centres. This means that, on average, 216 lines and triangles in each half of a Platonic solid surround its axis. Moreover, on average, 72 corners surround their axes. The presence in the regular polyhedra of the same parameters as were found above in the inner form of the 10-tree is evidence of their analogous, holistic character.

* A perfect number is an integer that is the sum of its divisors. For example, the number 6 is the first perfect number because 6 = 1 + 2 + 3, where 1, 2 & 3 are its divisors, and 28 is the second perfect number because 28 = 1 + 2 + 4 + 7 + 14, where 1, 2, 4, 7 & 14 are its divisors.

** The zodiacal signs are the the twelve 30° sectors of the ecliptic and need to be distinguished from the 12 constellations of the zodiac, which take up varying amount of ecliptic longitude, e.g., the constellation of Virgo takes up five time as much longitude as the constellation of Scorpius. This means that the Sun is not in each constellation for the same amount of time.

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