
The tetractys 
The Tree of Life 





Whorl
Each whorl is a closed helix with 1680 circular turns, or 1storder spirillae. 
= 
The gematria number value of Cholem Yesodoth, the Mundane Chakra of Malkuth, is 168: This is the number of points, lines & triangles below the top of the 1tree constructed from 19 Type A triangles with 25 sides, i.e., 168 geometrical elements are needed to construct the 1tree, starting from the point at its apex:
Below the apex of the 1tree are:
(10+19=29) corners of triangles; (25+ 3×19 = 82) sides of triangles; (3×19=57) triangles. Total = 168.







The 240 hexagonal yods in the 48 tetractyses of the 7 separate polygons making up half of the inner Tree of Life denote, in the context of superstrings, the 240 roots/gauge charges of the exceptional Lie group E_{8} whose associated spin1 gauge fields determine the forces between the E_{8}′singlet states of E_{8}×E_{8}^{′} heterotic superstrings. 


Interpretation 1 Interpretation
2 Which is the correct interpretation? The fact that each 600cell has 1200 faces that, taken separately, have 8400 corners, sides & triangles corresponding to the 8400 turns in the inner or the outer half of the UPA supports Interpretation 2 as the more natural one because it explains not only why the UPA has two halves but why each half comprises five revolutions, each 600cell being a compound of five 24cells. In the case of interpretation 1, five whole whorls would have to correspond to each 600cell, so that a whorl would have to correspond to a 24cell, which leaves unexplained why it has an inner and an outer half and why each half has five halfrevolutions. As shown on #3 of 4d sacred geometries, sacred geometries comprise 240 structural components (yods or geometrical elements) that can be grouped naturally into a pair of five sets of 24. Each "half" of these sacred geometries has its 4dimensional counterpart in the 600cell, so that we can feel sure that the latter does, indeed, correspond to an inner or outer half of the UPA rather than to five complete whorls. It suggests, therefore, that a 24cell defines a halfrevolution of all 10 whorls of the UPA rather than one complete whorl. 


The Godname of Malkuth — the physical manifestation of the Tree of Life blueprint — is ADONAI. Its number value is 65, which is the number of Sephirothic levels (white, red & blue corners of triangles) in the 10tree. This is equivalent to a tetractysdivided decagon that is enclosed in a square. ADONAI prescribes the 10 dimensions of spacetime predicted by superstring theory and mapped by 10 Trees of Life. EL ("God"), the Godname of Chesed with number value 31, also prescribes them because the 10tree has 127 triangles, where 127 is the 31st prime number. EHYEH ("I am"), the Godname of Kether wiith number value 21, prescribes the 10tree because each side pillar of it has 21 Sephirothic levels.



Each of the 10 whorls spirals five times around the axis of the UPA. Each revolution of the 10 whorls comprises 3360 helical turns (1storder spirillae), 336 per whorl. An outer or inner halfrevolution of a whorl comprises 168 turns and a quarterrevolution comprises 84 turns.


Divided into their sectors, the (70+70) polygons enfolded in 10 overlapping Trees of Life are composed of 3360 points, lines & triangles that are unshared with the outer Trees (shared geometrical elements are coloured green). Each set of (7+7) enfolded polygons has (168+168=336) geometrical elements that are unshared with its outer Tree of Life.



16800 yods surround the centre of the 7pointed star, where 16800 = 7×2400. Every point of the star is a
parallelogram constructed from tetractyses with




There are 248 hexagonal yods in a square with 2ndorder tetractyses as its sectors. Each yod symbolizes a root of E_{8}, the rank8, exceptional Lie group. The square also provides an arithmetic representation of the dimension 496 of the two possible superstring gauge symmetry groups SO(32) & E_{8}×E_{8}:

There are 248 yods below the top of the 1tree with its triangles turned into Type A triangles. The eight red yods outside the 1tree denote the eight simple roots of E_{8} and the 240 white yods other than Sephiroth denote its 240 roots.



The outer & inner Tree of Life basis
of the heterotic superstring symmetry group E8
Outer Tree of
Life
A Type B triangle is composed of 9 triangles with 3 external corners and 4 corners that are inside it. The 16 Type B triangles of the outer Tree of Life have 10 corners on their 22 sides and (16×4=64) internal corners of their (16×9=144) triangles with (16×12=192) sides. The outer Tree of Life comprises its root, its trunk and its branches. In terms of the 74 corners, the root consists of two corners whose projection onto the plane containing the (7+7) enfolded polygons of the inner Tree of Life are the endpoints of their shared root edge. One corner is located at Daath, which is the centre of the triangle formed by Chokmah, Binah & Tiphareth. The other corner coincides with Tiphareth, which is the centre of the Tree of Life in both a geometrical and a metaphysical sense. It is where the root joins the trunk of the Tree of Life, which is the sequence:
pointlinetriangletetrahedron
symbolising the integers 1, 2, 3 & 4,
Tiphareth being located at the lowest corner of the triangle in this sequence. The trunk is
composed of the 9 red corners of the 16 primary triangles, 5 green centres of
its 5 primary triangles and 15 blue
corners of their (5×9=45) triangles. The branches are the remainder of the outer Tree of
Life. They comprise 11 primary triangles with 10
green centres and (3×11=33) blue corners.
The 74 corners comprise the two black
corners forming the root and the 72 corners of the trunk (29
corners) and branches (43 corners). The 16 primary
triangles have (10+16=26) corners & centres, where
and
so that
This compares with
the 26 dimensions predicted by quantum mechanics for spinless strings being made
up of the time dimension, the longitudinal
dimension and 24 transverse dimensions comprising 9 dimensions of the
11dimensional spacetime predicted by Mtheory and the 15 additional, bosonic string dimensions. The
root of the outer Tree of Life is analogous to the time and longitudinal dimensions of such
a string in 26dimensional spacetime. Neither of these participates in creating
shape. Rather, in each case they are its rootlike
source. It means that the 9
dimensions are the trunk of spacetime and the 15 bosonic
string dimensions are its branches.
Inner Tree of
Life
The 7 enfolded, Type B polygons have 47 sectors with 41 corners. Their (47×3=141) triangles have (41+47=88) corners (86 outside the root edge). The (7+7) enfolded, Type B polygons have (2×141=282) triangles with (2 + 2×86 = 174) corners. 282 is the number value of Aralim, the Order of Angels assigned to Binah. The centre, top and bottom of each hexagon coincide with corners of triangles belonging to the outer Tree of Life. These six white corners are shared and (174−6=168) corners are unshared. (168/2=84) corners (turquoise or pink) are associated with each set of 7 enfolded polygons, which have 85 unshared corners, where
84 = 1^{2} + 3^{2} +
5^{2} + 7^{2}
and
85 = 4^{0} + 4^{1} +
4^{2} + 4^{3}.
The (47×2=94)
sectors of the (7+7) enfolded polygons
have 80 corners. 87 corners of (47×3=141)
triangles are associated with each set of 7 enfolded polygons when they are Type B. 80 is the number value
of Yesod and 87 is the number
value of Levanah, its Mundane Chakra. The
separate outer and inner Trees of Life have (74+174=248) corners of
(144+282=426) triangles. Excluding the root, there
are 246 corners, where 246 is the number
value of Gabriel, the Archangel of Yesod. 248 is the
dimension of the rank8, exceptional Lie
group E_{8} appearing in E_{8}×E_{8}′ heterotic
superstring theory. The 8 white or black corners symbolise the 8 simple roots of
E_{8}; the (72+168=240) other corners symbolise its 240 roots.
The 72 corners in the trunk and branches symbolise
the 72 roots of E6, the rank6, exceptional
Lie group that is a subgroup of E8. The 168 unshared
corners in the inner Tree of Life denote the 168 roots
of E8 that do not belong to E6. The 84:84 division of the superstring structural
parameter 168 is
characteristic of sacred geometries (see Article 64).
The 426 triangles in the combined outer
& inner Trees of Life have 240 corners other than those forming the root.
Alternatively, they have 240 corners other than the two endpoints of the root
edge. This embodiment of the number 240 is characteristic of sacred geometries (see
#1) because it quantifies the 240 roots of
E_{8}, which this website demonstrates is part of the divine
paradigm.



Correspondence between the geometrical or yod compositions of the first four Platonic solids, the disdyakis triacontahedron, the inner & outer Trees of Life and the inner form of the 10tree 


The subquark state of the E_{8}×E_{8} heterotic superstring remoteviewed by the Theosophists Annie Besant & C.W. Leadbeater over a century ago consists of 10 closed curves, or "whorls." They bear a correspondence to the 10 Sephiroth of the Tree of Life. The three major whorls correspond to the Supernal Triad and the seven minor whorls are the counterpart of the seven Sephiroth of Construction. Each whorl is a helix with 1680 circular turns. The three major whorls have (3×1680=5040) turns. Sacredgeometrical embodiment of 504 & 5040 Heptagon Type C dodecagon Disdyakis triacontahedron Each edge and each side of a sector in the green faces of the disdyakis triacontahedron are sides of internal grey triangles with the centre of the polyhedron as their shared corner. The (180+360=540) internal triangles have (540×3=1620) sectors with (60 + 120 + 540×3 = 1800) internal sides & 540 internal corners surrounding the centre, i.e., 3960 geometrical elements. The number of geometrical elements in the faces and interior that surround the axis = 1080 + 3960 = 5040. They include 1680 elements (red cells) either in the faces (1080) or sides (600) of sectors of internal triangles created by the edges of the disdyakis triacontahedron, leaving 3360 elements (1680 elements in each half of the polyhedron).* This is the polyhedral counterpart of the 1680 helical turns in the first major whorl and the 3360 turns in the second & third major whorls. * Alternatively, surrounding the axis are 1680 geometrical
elements comprising 180 corners of sectors in the faces, 180 edges & 1320 geometrical elements
in the internal triangles created by edges. This totals 1680 elements, leaving 3360
elements. 


3dimensional projection of a rotating 24cell. 

The 240 vertices of the 4_{21} polytope coincide with the positions of the 240 roots of E_{8}, the rank8, exceptional Lie group. The 4d projection of this 8d polytope is a compound of two 600cells. The 120 vertices of a 600cell can be partitioned into those of five disjoint 24cells. As each vertex of the 4_{21} polytope defines one of the 240 root vectors of E_{8}, there is a geometrical basis for dividing the 240 gauge charges corresponding to these roots into 10 sets of 24, each set being represented by the 24 vertices of a 24cell. The outer half of the UPA is the counterpart of one 600cell, the 120 gauge charges denoted by the 120 vertices of the five 24cells being spread out along the five halfrevolutions of the 10 whorls in this half. The inner half of the UPA is the counterpart of the other 600cell. The 2½ revolutions (five halfrevolutions) of the whorls that make up each half are the counterpart of the five 24cells in each 600cell. The 840 vertices & edges in each 600cell are the geometrical counterpart of the 840 circular turns in the five halfrevolutions of the outer or inner half of each helical whorl. The 1680 vertices & edges belonging to the compound of two 600cells in the Gosset polytope are the counterpart of the 1680 circular turns in each helical whorl of the UPA. 70 turns "carry" an E_{8} gauge charge: 1680 = 24×70. This correlation is irrefutable evidence that the UPA paranormally described over a century ago is a state of the E_{8}×E_{8} heterotic superstring (see Article 62 for more details). 



A 2ndorder tetractys contains 85 yods, where 85 = 4^{0} + 4^{1} + 4^{2} + 4^{3}. Including the yods at the centres of the six triangular gaps between the 1storder tetractyses in the 2ndorder tetractys generates a triangular array of 91 yods, where 91 = 1^{2} + 2^{2} + 3^{2} + 4^{2} + 5^{2} + 6^{2}. 13 yods line each side of the array, so that a parallelogram made up of two triangular arrays of 91 yods placed back to back contains (91+91−13=169) yods. Surrounding the centre of a 10pointed star whose points are these arrays are 1680 yods. Each 5pointed star has 840 yods. Each point of the star contains 120 corners of 1storder tetractyses and 720 hexagonal yods. The red 5pointed star contains 120 black corners of 1storder tetractyses and 720 red hexagonal yods. The blue 5pointed star contains 120 white corners of 1storder tetractyses and 720 blue hexagonal yods. The 1680 yods in the 10pointed star comprise 240 corners of tetractyses and 1440 hexagonal yods. Compare this with the fact that a 600cell is a polychoron with 120 vertices and 720 edges. The 10pointed star is a representation of the compound of two 600cells with 240 vertices and 1440 edges, each 5pointed star representing a 600cell. The 240 corners of 1storder tetractyses denote the 240 vertices of the compound and the 1440 hexagonal yods denote the 1440 edges. Each point of the star contains 24 corners of 1storder tetractyses. They correspond to the 24 vertices of a 24cell, each 5pointed star representing the compound of five disjoint 24cells that make up a 600cell. We saw in the last section that the 24 vertices of a 24cell consist of the eight vertices of a 16cell and the 16 vertices of an 8cell. In the point of the 10pointed star, they correspond to the eight corners of tetractyses outside the corner shared between star points that line two adjacent sides of the parallelogram and to the remaining 16 corners. This 10pointed star representation of the 240 vertices of a compound of two 600cells as the 4dimensional projection of the 240 roots of E_{8} mapped by the 8dimensional 4_{21} polytope is a particularly clear demonstration of the 10foldness of this number displayed by sacred geometries, as explained in #2 of 4d sacred geometries. It should not, therefore, come as a surprise that the 1680 turns of each helical whorl of the UPA/heterotic superstring are generated in 10 halfrevolutions (180°). The outer five halfrevolutions of a whorl with 840 turns are represented by the 840 yods (120 corners, 720 hexagonal yods) in one 5pointed star and its inner five halfrevolutions with 840 turns are represented by the 840 yods in the other 5pointed star. However, if we want to retain the correspondence between the 240 corners of tetractyses and the 240 vertices of the two 600cells determining the 240 roots of E_{8}, this correspondence cannot be interpreted as referring to a single whorl. Rather, each point in the 10pointed star must correspond to either a whorl or (as we concluded in earlier sections of 4d sacred geometries) a halfrevolution of all 10 whorls of the UPA, which is represented by the whole star because the UPA "carries" the 240 gauge charges of E_{8} corresponding to its roots. The counterparts of this in the inner form of 10 Trees of Life (see righthand picture) are the 1680 corners of the 2820 triangles in the (70+70) Type B polygons that are unshared with them. They comprise (120+120=240) red corners of the sectors of the 20 dodecagons and 720 remaining corners in each set of 70 enfolded polygons that are unshared with the outer Trees of Life. This demonstrates in an unequivocal way the Tree of Life basis of the 120:720 division in vertices & edges of each 600cell. 


The 4_{21} polytope has 240 vertices and 6720 edges. 
The 70 regular polygons enfolded in the inner form of 10 Trees of Life have 6720 sides when they are separate and Type C, i.e., when every sector is a Type B triangle. They correspond to the 6720 edges of the 4_{21} polytope. It is evidence that the latter is the 8d polytopic version of the inner form of 10 Trees of Life. The implication from this correlation of a hidden 10foldness in the 4_{21} polytope is confirmed by the mathematical fact that its 4d Coexeter plane projection is a compound of two concentric 600cells, each of which is a compound of five 24cells. The two halves of the 4_{21} polytope, each with 3360 edges, correspond to the two sets of 35 polygons (red & blue), each set being composed of 2160 triangles with 3360 sides. 
There are 240 white dots & white sides of triangles in every 10 overlapping Trees of Life that either belong solely to their outer form or are white centres of 100 of the 140 Type B polygons associated with these Trees that remain "pure" centres when the polygons become enfolded (note: centres of hexagons become corners of the triangles and centres of decagons become corners of pentagons). Green corners & sides of triangles in every 10 Trees become shared with enfolded polygons, whilst 40 green centres of 20 hexagons & 20 decagons coincide with corners of other polygons when they all become enfolded. 6720 corners & sides of the 2820 (=10×282) triangles in the 140 separate Type B polygons surround their centres.* 282 is the number value of Aralim, the Order of Angels assigned to Binah, and 140 is the number of Malachim, the Order of Angels assigned to Tiphareth.
Here are two inferences from this amazing correspondence:
* Proof: There are 7 corners & sides of triangles per sector, 48 sectors per set of 7 polygons and 10 sets of (7+7) polygons in the inner form of 10 Trees of Life. Total number of their corners & sides = 10×(48+48)×7 = 6720.



16800 yods outside their root edges surround the white yods at the centres of the 240 2ndorder tetractyses that make up the (10+10) dodecagons enfolded in 10 overlapping Tree of Life. The number of yods surrounding the centres of the 120 2ndorder tetractyses in each set of 10 dodecagons = 8400 = 10^{2} + 30^{2} + 50^{2} +70^{2}. They denote the 8400 turns in the outer/inner halves of the 10 whorls of the UPA.




YAH (יה) is the shorter form of YAHWEH (יהוח), which is the complete Godname of Chokmah, the second Sephirah of the Tree of Life. Its gematria number value is 15. As 15^{2} − 1 = 224, 15(15^{2}−1) = 15×224 = 3360 = 15^{3} − 15. This is the number of turns in one revolution of the 10 whorls of the UPA/E_{8}×E_{8} heterotic superstring, showing how YAH prescribes how many circularly polarised oscillations make up each revolution of these 10 whorls. Their five revolutions comprise (5×3360=16800) turns. The number 26 of YAHWEH is the dimensionality of the spacetime of bosonic strings and the number 15 of YAH is the number of dimensions beyond the 11d spacetime predicted by Mtheory, the yettobe discovered theory that encompasses both supergravity theory and the five superstring theories. 

The superstring structural parameter 16800 can be represented by a fivepointed star, with each point corresponding to one of the 5 revolutions of the 10 whorls and consisting of a 15×15 array of the number 15 of YAH arranged in a parallelogram, the 1120 numbers outside the centre of the star adding up to 16800. The 600 15s in the red outer halves of the 5 points of the star add up to 9000 (=90×100). The 520 15s in the blue inner halves of the points add up to 7800 (=78×100). 78 is the gematria number value of Cholem and 90 is the number value of Yesodoth, the two words making up Cholem Yesodoth, the Kabbalistic name of the Mundane Chakra of Malkuth. The geometry of the 5pointed representation of the superstring structural parameter 16800 naturally distinguishes between the gematria number values of these words. The UPA is par excellence the microphysical manifestation of the Mundane Chakra of Malkuth.


