ARTICLE 25
by
Stephen M. Phillips
Flat 4, Oakwood House, 117119 West Hill Road. Bournemouth.
Dorset BH2 5PH. England.
Website: http://smphillips.mysite.com
Abstract
This article elucidates the scientific meaning of the
patterns of layers of the three types of vertices of the disdyakis
triacontahedron. This Catalan solid was identified in Articles 22–24 as the
3dimensional (polyhedral) version of the 2dimensional (polygonal), inner form
of the Tree of Life. Its 62 vertices are the counterpart of the 62 Sephirothic
emanations of 10 overlapping Trees of Life that are shared with their inner
forms. Its vertices are arranged in either 7, 11 & 15 sheets perpendicular
to axes that join, respectively, pairs of opposite A, B & C vertices. The 33
sheets correspond to the 33 tree levels (rungs of the biblical ‘Jacob’s ladder’)
of 10 overlapping Trees of Life. Within the inner form of a single Tree of Life,
they correspond to the 33 corners outside the shared edge of the set of 7
enfolded polygons that, being intrinsic to that set, represent geometrical
degrees of freedom that are unique to any Tree of Life system. The 32 B & C
vertices of the golden rhombic faces of the triacontahedron underlying the
disdyakis triacontahedron span 26 sheets prescribed by the Divine Name YHVH. The
number value 15 of YH is the number of sheets orthogonal to a CC axis and the
number value 11 of VH is the number of sheets orthogonal to a BB axis. The 7
sheets orthogonal to an AA axis correspond to the 7 tree levels spanning the
lowest of the 10 Trees of Life; they determine the 84 circularly polarised
oscillations made by a whorl of the E_{8}×E_{8}' heterotic
superstring during onequarter of a revolution about its spin axis. The 11 BB
sheets correspond to the next 11 tree levels; they determine the 168
oscillations in a halfrevolution of a whorl. The 15 CC sheets correspond to
the final 15 tree levels in 10 overlapping Trees of Life; they determine the 240
gauge charges associated with the nonzero roots of the superstring gauge
symmetry group E_{8} that are spread along the length of the heterotic
superstring. Assigning 4, the Pythagorean Tetrad, to the 62 vertices of the
disdyakis triacontahedron generates the dimension 248 of E_{8}.
Assigning 4 to the 420 yods in the 120 tetractyses needed to construct the
sacred geometry of the disdyakis triacontahedron from two diametrically opposite
A vertices generates the number 1680. This is the number of circularly polarised
oscillations in each whorl of the heterotic superstring, as well as the number
of hexagonal yods forming the seven types of polyhedra leading to the disdyakis
triacontahedron. This polyhedron therefore embodies the dynamics and oscillatory
form of the E_{8}×E_{8}' heterotic superstring as the
manifestation of the Tree of Life blueprint in the subatomic world.

1
1. Tree levels as spacetime dimensions Just
as the DNA molecule in the nucleus of a cell encodes the properties of the biological
organism made up of such cells, so the inner form of the Tree of Life (Fig. 1) encodes
how its outer form replicates to represent the whole cosmos — both physical
and superphysical. By exploiting the isomorphism between a regular, nsided polygon
divided into tetractyses and n overlapping Tree of Life up to Chesed of the nth tree, the
author has proved rigorously that the two sets of seven separate polygons contain a
unique subset of 12 polygons that is isomorphic to a whole number of overlapping Trees of
Life. One set of seven polygons is isomorphic to 49^{1}such trees and the remaining five polygons (the pentagon,
hexagon, octagon,
decagon and dodecagon of the second set of polygons) are isomorphic to 42 trees. Encoded,
therefore, in the regular polygons making up the inner Tree of Life are 91 overlapping
Tree of Life. The 49 trees represent the 49 subplanes of the seven
planes of consciousness — one of the teachings of Theosophy that can now be mathematically
proved. These seven planes represent the outer evolutionary journey of the soul from one
incarnation to another towards absorption in the inner Divine Life. They constitute but
the lowest physical plane of seven cosmic planes of consciousness, the six higher planes
of which are similarly divided into seven subplanes, making a total of 42 subplanes. These
are represented by the 42 trees encoded in the five polygons belonging to the second set
of seven polygons. Accordingly, the two halves of the inner Tree of Life signify the
physical and superphysical domains of the spiritual cosmos, which the author has called
the ‘Cosmic Tree of Life’ (CTOL). The boundary between these two domains is prescribed by
EL CHAI, the Godname of Yesod, which has number value 49 (the other Godnames
also prescribe it in ways that need not be described here).
The seven lowest trees in CTOL represent the seven levels of biological
consciousness. In their normal physical activity, humans are focussed in the third tree but are
evolving as well into ‘etheric’ realms of consciousness that are mapped by the next four trees.
Defining the ‘ntree’ as the lowest n trees in CTOL, physical consciousness can, potentially,
extend over the 7tree. When trees overlap, all the Sephirothic emanations (called ‘Sephirothic
levels,’ or SLs) of one tree are SLs of the next higher and lower tree except Chesed and
Geburah. These two types of SLs play a preformative role in the manifestation of the next
lower tree as a level of consciousness. They represent gaps, or critical jumps, in the spectrum
of awareness as it shifts from one modality to the next represented by the SLs of the next
higher tree.
2
Each level of consciousness is a band of types of awareness characterised by
the seven fundamental Sephiroth of Construction. The lowest four relate to what Carl Jung
called the four ‘psychic functions’ of sensing, feeling, thinking and intuiting. These
Sephirothic variations define seven ‘tree levels’ within each tree/level of consciousness (Fig.
2). The fourth has special evolutionary significance because it expresses the quality of
Tiphareth. As the central Sephirah of the Tree of Life, it is the interface between the
egocentred and transpersonal, egoless levels of being. The number of tree levels in the
ntree ≡ T(n) = 3n + 4. Hence, T(7) = 25. They
3
denote the 25 spatial dimensions of the 26dimensional spacetime
predicted by quantum mechanics for spinless strings. The 26th tree level (the lowest one
above the 7tree) maps the time dimension. It is marked by Yesod of the ninth tree, the
50th SL. This shows how ELOHIM, Godname of Binah with number value 50,
prescribes the 26 dimensions of spacetime, which are prescribed directly by YAHWEH,
Godname of Chokmah, because it has number value 26. YAH, the older version of YAHWEH,
has number 15. It prescribes the 25 dimensions of space because the top of the seventh
tree marked by the 25th tree level is the 47th SL, where 47 is the 15th prime number.
The Godname EL of Chesed also prescribes these 25 dimensions because its number value 31
is the number of stages of vertical descent of what Kabbalists call the ‘Lightning Flash’ from
the apex of the seventh tree to the nadir of CTOL, namely, Malkuth of the first tree.
There are three types of tree levels: Malkuths (M) of trees, Yesods (Y) and
Hod Netzach Paths (P) (apart from the lowest tree, the lattermost are BinahChokmah Paths of
the next lower tree). Article 2 explained the role of these tree levels in creating the ten
whorls of the UPA described by Annie Besant and C.W. Leadbeater in their book Occult
Chemistry. The author has identified the UPA as a heterotic E_{8}×E_{8}'
superstring,^{2} each of its ten whorls resulting from the curling into a circle
of a dimension represented by a tree level (Fig. 3). Five M and five Y tree levels belonging
to the uppermost four trees define a 10dimensional space S×C' whose curling up generates
the 10 whorls. Five P tree levels define a 5dimensional space C whose symmetries generate
the E_{8} gauge symmetry group. The remaining 10dimensional space is that of
supergravity theories, one of the dimensions being a finite segment between the two
10dimensional spacetimes of ordinary and shadow matter superstrings.^{3}
The dimensionalities of the various spaces into which the author’s theory
predicts 26dimensional spacetime bifurcates are the letter values of YHVH (YAHWEH),
the
later version of the Godname of Chokmah given by God to the Jewish people,
according to their religious beliefs. The value 10 of Y (yod) denotes the 10 dimensions of S×C'
whose curling up creates the ten whorls of the UPA, the value 5 of H (he) denotes the
5dimensional space C whose geometrical symmetries generate E_{8}, the value 6 of V
(vav) denotes the curledup, 6dimensional space of superstrings and the value 5 of the second
H denotes the Einsteinian, 4dimensional spacetime and the dimensional segment separating the
two superstring spacetimes. According to Leadbeater’s description of the UPA, each whorl is a
closed helix with 1680 turns
4
(“1storder spirillae”). Each turn is a helix with seven circular turns
(“2ndorder spirillae”); each of the latter is a smaller helix with seven turns (“3rdorder
spirillae”) and so one. The six higher orders of spirillae described by Leadbeater represent
the winding of strings around successively smaller, compact, 1dimensional spaces, i.e., the 6
dimensional, compactified space of superstrings is a 6d torus. Figure 3 shows that tree levels
of type P and Y define these curledup dimensions.
The superstrings of ordinary and shadow matter are confined to two
spacetime sheets separated by a gap. This tenth dimension acts as a wall that prevents either
kind of superstring entering the other’s universe. It is represented by the tenth tree level
(M in Fig. 3). This is Malkuth of the fourth highest tree in the 7tree mapping the
physical plane, that is, 26dimensional spacetime). This 15:11 division between
the dimensionalities of supergravity spacetime and the higher, 15dimensional space
that generates both the ten stringlike whorls of the UPA/heterotic superstring and its gauge
symmetry group E_{8} has its counterpart in the combinatorial properties of ten objects
labelled A, B, C…. J when arranged in the pattern of a tetractys:
The four objects G, H, I and J have (2^{4}–1=15)
combinations. The three objects D, E and F have (2^{3}–1=7) combinations, the two
objects B and C have (2^{2}–1=3) combinations and the single object A has one
combination. The six objects A–F therefore form 11 combinations. When arranged in a tetractys,
all ten objects have 26 combinations when only objects from the same row are
considered.
This has a simple geometrical realisation. Letting the objects be points in
space, the four rows of the tetractys of ten points signify the sequence of point, line,
triangle and tetrahedron. Combinations of two points denote the line joining them, combinations
of three points signify the triangles with these points as their corners and the combination of
four points signifies the tetrahedron with these points as its vertices. There are 26
vertices, lines, triangles and tetrahedra in the 4stage sequence of generation from the point
of the simplest Platonic solid — the tetrahedron. This illustrates in a very simple way the
mathematical archetype embodied in the Divine Name YAHWEH.
n objects have n! permutations. The permutations of the objects in a
tetractys are:
5
There are 26 combinations and 33 permutations of the objects arranged
in the four rows of a tetractys when only objects in the same row are grouped together.
The tetractys and the Tree of Life are different representations of the
10fold nature of holistic systems designed according to the universal blueprint. Each Sephirah
of the Tree of Life has in turn a 10fold differentiation, so that ten overlapping Trees of
Life is the next higher differentiation of a single Tree of Life. In the same way, the 10tree
is the next higher
differentiation of the 1tree. It has 65 SLs (Fig. 4). 65 is
the number value of the Godname ADONAI assigned to Malkuth. In fact, its very letter values
signify the numbers of various types of SLs in the 10tree. This is borne out by the
representation of the 10tree as a decagon divided into 10 tetractyses and surrounded by a
square, the letter values denoting the yods that are counterparts of the SLs. The 64 SLs
of the ten overlapping trees span 33 tree levels. We saw earlier that the lowest 50 SLs
span 26 tree levels signifying the 26 dimensions of spacetime predicted by
quantum mechanics for spin0 strings. The 7tree mapping the physical plane and the 25 spatial
dimensions/tree levels emanates from the 50th SL, which, as Daath, represents
in Kabbalah the ‘Abyss’ between the Supernal Triad represented by the three uppermost trees of
the ten overlapping trees and the seven Sephiroth of Construction represented by the seven
lowest trees. Hence, the division of tree levels in ten overlapping Trees of Life:
33 = 7 + 26
6
expresses in Kabbalistic terms the distinction between the seven Sephiroth
of Construction and the triple Godhead of Kether, Chokmah and Binah.
The number of corners of the 7n regular polygons enfolded in n overlapping
Trees of Life is
C(n) = 35n + 1,
where “1” denotes the uppermost corner of the hexagon enfolded in the tenth
tree (the hexagon is the only one of the seven regular polygons that shares its corners with
polygons enfolded in adjacent trees. It results in its being picked out by the above formula).
Therefore, C(10) = 351 = 1 + 2 + 3 +…. + 26. This shows how YAHWEH with number
value 26 prescribes the inner form of ten overlapping Trees of Life. 351 is the
number value of Ashim (“Souls of Fire”), the Order of Angels assigned to Malkuth.
The number of corners of
the 7n polygons outside their n root edges ≡ C'(n) – 2n = 33n + 1. C'(n+1) –
C'(n) = 33. In other words, there are 33 corners per set of polygons outside their root edge
(Fig. 5). The emanation of successive Trees of Life generates 33 new geometrical degrees of
freedom associated with their inner form. The significance of this property for the
disdyakis triacontahedron will be discussed in the next section.
The 10tree has 34 tree levels, whilst ten overlapping Trees of Life have 33
tree levels. Their counterparts in the inner Tree of Life are its 34 corners outside the root
edges, 33 per set of polygons. Any system or structure that conforms to the blueprint of the
Tree of Life possesses 33 independent degrees of freedom. The counterpart of this in the
tetractys is the 33 permutations of objects in its four rows.
The seven polygons enfolded on one side of the central Pillar of Equilibrium
have corners shared with Chokmah, Daath, Chesed and Netzach. The other seven polygons
7
have corners shared with Binah, Tiphareth, Geburah and Hod. The top corner
of the hexagon is the bottom corner of the hexagon enfolded in the next higher tree. This means
that there are (3+1) corners associated with each set of polygons, “1” denoting the shared, top
corner of the hexagon. Enfolded in n overlapping Trees of Life are 7n polygons on either side
of the central pillar with (3n+1) corners that coincide with SLs. The 10tree has 70 polygons
on each side with 31 corners shared with it. Hence, there are 62 SLs of the
10tree that are shared with its inner form. The 62 vertices of the disdyakis
triacontahedron correspond to these SLs (Fig. 6), the 31 independent vertices
and their 31 inverted images corresponding to the 31 SLs that coincide with
corners of each set of polygons. 31 is the number value of EL (“God”), the Godname of
Chesed, and 62 is the number value of Tzadkiel (“Benevolence of God”), the
Archangel of Chesed — the same Sephirah.
8
2. Vertex planes in the disdyakis
triacontahedron The following evidence was presented in Article 22
that the disdyakis triacontahedron is the 3dimensional realisation of the 2dimensional
(polygonal) form of the inner Tree of Life:
1. the 120 shapeforming yods on the boundaries of the seven enfolded
polygons symbolise the 120 faces of this polyhedron;
2. the disdyakis triacontahedron completes the 10step sequence of
generation of polyhedra from the mathematical point:
3. the 480 hexagonal yods in the 120 faces of the disdyakis triacontahedron
correspond to the 480 hexagonal yods in the two sets of seven regular polygons constituting the
inner Tree of Life;
4. the 62 vertices of the disdyakis triacontahedron correspond to the
62 SLs of 10 overlapping Trees of Life that coincide with corners of their 140
enfolded polygons and to the 62 corners of the 14 polygons enfolded in one Tree of Life
that do not coincide with its Sephiroth;
5. the seven polyhedra in the above sequence are made up of 1680 hexagonal
yods — the number of yods below the top of the 10tree when its triangles are divided into
three tetractyses. This is the structural parameter of the E_{8}×E_{8}'
heterotic superstring, as established in many previous articles and in the author’s books;
6. the hexagonal yod population of the 28 Platonic and Archimedean solids in
the disdyakis triacontahedron is 3360. This is the yod population of the seven enfolded
polygons when constructed from 2ndorder tetractyses.^{4}
Further evidence in support of this conclusion is provided by the patterns
of planes of vertices of the disdyakis triacontahedron. Its 62 vertices are
made up of the 32 vertices of a rhombic triacontahedron, which form 30 Golden Rhombi, and 30
raised centres of the latter. The long diagonals of each Golden Rhombus are the edges of an
icosahedron and its short diagonals
are the edges of a dodecahedron. Labelling the vertices of the 120
triangular faces A, B & C (Fig. 7), there are 30 A vertices, 12 B vertices (corners of the
icosahedron) and 20 C vertices (corners of the dodecahedron).
9
Remarkably, this pattern of types of vertices appears also in the inner Tree
of Life in the 62 corners of the two sets of enfolded polygons that are unshared with Sephiroth
of its outer form (Fig. 8). The two dodecagons have 20 corners (red yods) outside the root
edge. The pair of octagons has 12 corners outside the root edge and the pair of squares,
pentagons, hexagons
and decagons have 30 corners outside the root edge. We find the following
correspondence:
30 corners of squares, pentagons, hexagons and decagons 30 A vertices
12 corners of octagons 12 B vertices
20 corners of dodecagons 20 C vertices
For the sake of clarity, only three representative A, B and C vertices are
shown in the disdyakis triacontahedron in Fig. 7. It may be argued that the numbers of unshared
corners in pairs of polygons:
square

pentagon

hexagon

octagon

decagon

dodecagon

4

6

4

12

16

20

allow other combinations of polygons to generate the numbers 30, 12 and 20,
e.g., the pair of squares, pentagons and dodecagons have 30 corners. This is not true of the
number 12, which can be associated only with the pair of octagons because it is not the sum of
any of the above numbers. The dodecagons are the simplest choice for the number 20 because
other choices require combinations of two or three polygons. Any argument over uniqueness,
however, misses the point being demonstrated here, namely, that correspondence can be
established not only between the numbers of vertices of the disdyakis triacontahedron and the
corners of the inner Tree of Life but also between the numbers of each type of vertex and
numbers of corners of different
10
11
polygons. Such precise correspondence cannot be dismissed as due to chance.
It is further evidence that the disdyakis triacontahedron is the 3dimensional form of the
inner Tree of Life.
Table 1 lists the Cartesian coordinates of the 62 vertices.
Their coordinates are the first three powers of φ, the Golden Ratio. They are arranged in nine
vertical layers, the
vertical axis being defined as the line joining the top and bottom A
vertices indicated by the two white rows. The 60 vertices between them are distributed in seven
horizontal planes
coloured according to the seven prismatic colours. The 62 sets of
three coordinates are really 31 sets, as the other 31 vertices are the inversions
of those in the first set, the coordinate values of one vertex being the negatives of its
inverted counterpart. This is a clear example of the prescriptive power of the
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mathematical archetype embodied in the Godname EL (“God”) associated with
Chesed, the first Sephirah of Construction. Its letter vales E = 1 and L = 30 denote,
respectively, the topmost vertex and the 30 vertices below it that have mirror image
counterparts.
Fig. 9 indicates that the 30 A vertices are arranged in nine layers.
The seven sheets between the outermost vertices are separated by distances that alternate as 1,
φ, 1, φ, this pattern being reversed for the last four sheets.
Fig. 10 indicates that the 12 B vertices (the corners of an
icosahedron) are arranged in 11 sheets between the opposite B vertices defining the axis at
right angles to them. Fig. 11 shows that the 20 C vertices (corners of a dodecahedron)
belong to 15 sheets between the opposite C vertices defining the axis. The
disdyakis triacontahedron has therefore (7+11+15=33) sheets.
3. Tree levels signify
sheets of vertices
Comparing the sheet patterns of the three types of vertices with the tree levels of 10
overlapping Trees of Life shown in Fig. 4, we see that there are as many sheets as there are
tree levels. Moreover, there are as many sheets (7) perpendicular to an AA axis as there are
tree levels in 10 overlapping Trees of Life above the 26th tree level marking the
50th SL, there being 26 sheets perpendicular to a BB and a CC axis. This
is the number value of YAHWEH. The distinction between the 11 BB sheets and the
15 CC sheets is expressed in the letter values of YAHWEH:
It prescribes the distribution of vertices perpendicular to BB and CC
axes, whilst YAH, the older version of this divine name with number value 15, prescribes
the distribution of vertices perpendicular to a CC axis.
The fact that the same three numbers 7, 15 and 11 appear in the
contexts of 10 overlapping Trees of Life and the disdyakis triacontahedron is additional
confirmation that the latter is equivalent to the inner form of the Tree of Life, its
62 corners unshared with its outer form being the counterpart of the
62 SLs of 10 trees that are
13
unshared with its enfolded polygons.
The pentagon is the polygonal representation of the Godnames of the four
highest Sephiroth (Fig. 12). Constructed from tetractyses, it comprises 31 yods. This is
the representation of EL, Godname of Chesed, because its central yod symbolises the letter
value 1 of E (aleph) and the 30 yods surrounding its centre symbolise the letter value 30 of L
(lamed). YAH is
represented by the boundary of the pentagon because the letter value 10 of Y
(yod) is the number of hexagonal yods between its five corners, which denote the number value 5
of H (heh). The number value 26 of YAHWEH is the number of radial and boundary yods. The
number value 21 of EHYEH is the number of yods on the boundary of the pentagon, at its
centre and at the centres of its five sectors.
It is wellknown that the pentagon contains the Golden Proportion as the
ratio of the lengths of its diagonal and side (Fig. 13). The very regular polygon whose
proportions define the shape of the faces of the rhombic triacontahedron underlying the
disdyakis triacontahedron embodies the Godname numbers 15 and 26 that
characterise the numbers of sheets of B and C vertices defining the shape of these faces.
Listed below are the numbers of A, B and C vertices within each of the seven
planes between two outermost A vertices:
The fourth, central plane contains 12 vertices (six vertices and their
images inverted through the centre of the disdyakis triacontahedron). There are
(1+8+10+6+6=31) independent vertices in the uppermost vertex and the four planes
containing half of the polyhedron. This is how the Godname EL of Chesed with number value
31 prescribes its vertex population. The other 31 vertices are their
mirror images. The total number of vertices in the four planes = 8 + 10 + 6 + 12 = 36.
This shows how the Godname ELOHA of the next Sephirah with number value 36 prescribes
the polyhedron.
14
The 31 vertices are made up of 15 A vertices, six B vertices
and 10 C vertices. In a pentagon divided into five tetractyses, there are 15 yods on its
boundary, six yods at the centres of the tetractyses or the pentagon and 10 yods on the sides
of tetractyses
15
inside it (Fig. 14). Half of the polyhedron contains 37 vertices. This is
the number of yods in a hexagon constructed from tetractyses, its central yod corresponding to
the uppermost A vertex. The tetractys therefore reveals how these two polygons embody numbers
defining the disdyakis triacontahedron as the inner form of the Tree of Life.
The number of vertices needed to create the disdyakis triacontahedron is not
the number that actually constitutes it. The former is 31; these vertices are symbolised
by the 31 corners of the seven enfolded polygons unshared with the outer Tree of Life
(see Fig. 8). The number constituting the first or last four planes is 36; these are
symbolised by all 36 corners of the seven enfolded polygons forming the inner Tree of
Life.
n overlapping Trees of Life consist of (12n+4) triangles. Transformed into
tetractyses, they contain (50n+20) yods, where 50 is the number value of ELOHIM.
10 Trees of Life have 520 yods in 124 tetractyses. The ntree consists of (12n+7) triangles.
Changed into tetractyses, they contain (50n+30) yods. The 5tree has 280 yods
(Fig. 15). This is the number value of Sandalphon, Archangel of Malkuth. As the apex of
the 5tree is Malkuth of the seventh tree, we see that Sandalphon prescribes the 7tree
representing the physical plane, the 25 tree levels of which are the 25 spatial dimensions
predicted by quantum mechanics (hence the appropriateness in this context of the Sephirah
Malkuth). The number value 26 of YAHWEH is the number of yods between successive SLs on
the central pillar and the number value 50 of ELOHIM is the
16
number of yods between corresponding SLs on this pillar. Chesed of the 5th
tree — the first Sephirah of Construction — is the 31st SL. Fig. 15 shows that there are
248 yods up to the level of the 31st SL. This is how the Godname EL with number
value 31 prescribes the dimension 248 of the superstring gauge symmetry group
E_{8}. Each yod up to the 31st SL denotes a physical particle — a gauge boson of
E_{8}. The 1tree contains 80 yods, where 80 is the number
value of Yesod. There are, therefore, (248–80=168) yods above the 1tree
up to the 31st SL. This is the number value of Cholem Yesodeth,
(“Breaker of the Foundations”), the Mundane Chakra of Malkuth. As pointed out in Section 1 and
as discussed in many previous articles, 168 is the structural parameter of the
E_{8}×E_{8}' heterotic superstring, being the number of circularly polarised
oscillations of each of its whorls during onehalf revolution.
The number of yods in n overlapping Trees of Life up to the level of Chesed
of the nth tree = 50n – 3.^{5} There are 496 yods up to (but not including) Chesed of the
tenth tree (Fig. 16). They include 60 SLs, of which the lowest 31 SLs have 248
yods associated with them. Therefore, there are 248 yods above the 31st SL up
to Chesed of the tenth tree. The E_{8}×E_{8}' heterotic superstring has a
unified interaction transmitted by 496 gauge bosons, of which 248 particles
are the gauge bosons of the first E_{8} group and 248 particles are the gauge
bosons of the second E_{8}' group. The Godname EL prescribes the direct product
structure of the gauge symmetry group governing the interactions of heterotic superstrings.
The product reflects the division of the ten Sephiroth of the Tree of Life into the five
Sephiroth of its Lower Face and the five Sephiroth of its Upper Face.
It was found in Article 22 that the disdyakis triacontahedron has 480
hexagonal yods in its 120 faces. This is the number of yods in the 9tree, which has 31
tree levels, showing how the Divine Name EL prescribes the 480 hexagonal yods in the disdyakis
triacontahedron that symbolise the 480 nonzero roots of E_{8}×E_{8}'.
There are 15 sheets of vertices orthogonal to a CC axis and
26 sheets of vertices perpendicular to AA & BB axes, totalling 33 sheets. Against
the backdrop of CTOL— the map of physical and superphysical reality — the 15tree has
780 yods, i.e., the yods in 78 tetractyses, and the 26tree has 1330 yods, i.e., the
yods in 133 tetractyses. 78 is the dimension of E_{6}, the rank6 exceptional group,
and 133 is the dimension of E_{7}, the rank7 exceptional group. The two numbers
15 and 26 that specify the number of sheets of vertices that create the shapes of
the faces of the disdyakis triacontahedron define the dimensions of two exceptional subgroups
of the exceptional group E_{8}. The 33tree has 1680 yods. As pointed out in Article
22, this is the number of hexagonal yods in the sequence of seven polyhedra that evolve into
the perfect disdyakis triacontahedron:
It is the number of 1storder spirillae in a whorl of the UPA. Here,
therefore, is the scientific meaning of the numbers 15, 26 and 33 defining the
three types of groups of sheets of vertices. It is revealed by representing sheets by Trees of
Life. Malkuth of the 33rd Tree of Life in CTOL is the 65th SL on the central Pillar of
Equilibrium. This is
17
18
how the Godname ADONAI of Malkuth with number value 65 prescribes the
number 1680. Another way whereby ADONAI determines this fundamental structural parameter of the
E_{8}×E_{8}' heterotic superstring is as follows: as proved in Article 22, when
the (12n+7) triangles of the ntree are each divided into three tetractyses, the number of yods
below the top of the nth tree is^{6}
N(n) = 158n + 100.
The number of yods below Kether of the tenth tree (the 65th SL) is
N(10) = 1680, that is, the number of yods in 168 tetractyses, where 168 is the
number value of Cholem Yesodeth, the Mundane Chakra of Malkuth (Fig. 17). This is how
ADONAI prescribes the structural parameter 1680. The astounding result involves both the
Godname and the Mundane Chakra of the same Sephirah and so it is implausible that it is
coincidental. Instead, it reveals the amazing nature of the Tree of Life as the blueprint that
determines the nature of microscopic as well as macroscopic reality. Moreover, as shown in
Article 22,^{7} the 1680 yods belong to 385 tetractyses, where
Such a simple, yet beautiful, mathematical property is not an accident.
Instead, it is an eloquent expression of mathematically perfect design.
There are 33 sheets of vertices in the disdyakis triacontahedron between its
diametrically opposite and outermost vertices. They contain 60 vertices. This is the
3dimensional counterpart of the outer Tree of Life having 60 yods generated when its 16
triangles are turned into tetractyses. A decagon with its ten sectors turned into tetractyses
has 60 yods surrounding its centre (Fig. 18). In other words, starting with the mathematical
point, 60 more points are needed in 2dimensional space to construct a decagon from
tetractyses. Similarly, starting with ten
points as Sephiroth, 60 more points are required to create their arrangement
in space as the 16 triangles of the Tree of Life. These 60 formative degrees of freedom appear
in the disdyakis triacontahedron as the 60 vertices between any two outermost A vertices that
are diametrically opposite. This is yet more evidence for its Tree of Life nature. It is truly
remarkable that there should be as many yods (1680) spanning 33 tree levels below the
65th SL
19
when the triangles of the trees are divided into three tetractyses as there
are in the 33tree with its triangles turned into tetractyses and with 65 SLs on the
central pillar up to Malkuth of the 33rd tree. The reason for this correspondence is that the
geometry of the Tree of Life has a fractallike quality of being characterised by the same set
of parameters whenever it maps sections of CTOL that are counterparts of each other in a way
that need not be discussed here.
20
We have seen that the 33 sheets of vertices in the disdyakis triacontahedron
correspond to the 33 tree levels of 10 overlapping Trees of Life. The question now arises: how
should the three types of groups of sheets be ordered when correlating them with tree levels?
We know that the 1tree has seven tree levels (Fig. 1), so it seems natural to correlate them
with the 7 sheets of vertices perpendicular to an AA axis. The 1tree is the Malkuth level of
10 overlapping Trees of Life representing the 10 Sephiroth. The seven sheets of vertices
represent the lowest tree and therefore the Malkuth aspect of the disdyakis triacontahedron as
the polyhedral counterpart of these ten trees. There are 80 yods in the 1tree
and, as Fig. 15 indicates, there are four yods outside the 1tree up to the level of its apex.
The seven tree levels therefore span 84 yods (Fig. 19), where
is the sum of the squares of the first four odd integers. The 5tree has
280 yods, of which seven yods are above the BinahChesed path of the 5th tree.
This leaves 273 yods in the five
trees up to the level of Chokmah of the 5th tree — the 18th tree level.
Below Yesod of the 3rd tree are 105 yods (see Fig. 15). There are (273–105=168) yods in
the 11 tree levels from the 8th to the 18th. Finally, as the 10 overlapping Trees of Life have
520 yods, there are (520–280=240) yods in the 15 tree levels between the 5tree
and the top of the 10th tree.
The superstring significance of the numbers 84, 168 and 240 is as
follows: each whorl of the heterotic superstring twists five times around its central axis (see
Fig. 17), making 1680 circularly polarised oscillations. There are 1680/5 = 336 oscillations
per revolution, 168 oscillations per halfrevolution and 84 oscillations per
quarterrevolution. The 84:84 pattern manifests in the inner form of the Tree of Life as the 84
yods that line the first six polygons outside the root edge on each side (Fig. 20). This
pattern has been discussed in the context of octonions^{8} and the Klein Configuration.^{9} The seven sheets of vertices between diametrically opposite A
vertices of the disdyakis triacontahedron define the 84 oscillations made by a whorl in a
quarter of a revolution. The 11 sheets of vertices perpendicular to a BB axis define the
168 oscillations in half a revolution. According to heterotic string theory, its
E_{8} gauge charges are spread around its length. However, it is the 240 charges
corresponding to its nonzero roots that are spread
21
around the 10 whorls, 24 per whorl. There are 168 such charges along
the seven minor whorls and 72 carried by the three major whorls. The former correspond
to the 168 nonzero roots of E_{8} that are not also nonzero roots of
E_{6}, one of its exceptional subgroups, whilst the latter correspond to the nonzero
roots of E_{6}. The 15 sheets of vertices perpendicular to a CC axis correspond
to the 15 tree levels spanning the 240 yods between the top of the 5tree and the top of
the 10th tree. These yods denote the gauge charges corresponding to the 240 nonzero roots of
E_{8} that are spread around the heterotic superstring. Information about the dynamics
and oscillatory form of the heterotic superstring is thus encoded in the sheets containing the
three types of vertices. The sheets of vertices perpendicular to an AA axis and a BB
axis encode through their Tree of Life counterparts the numbers of
circularly
polarised oscillations made in, respectively, a quarterrevolution and a
halfrevolution of a whorl, whilst the CC sheets encode the number of E_{8} gauge
charges that are spread around all ten whorls.
The disdyakis triacontahedron is made up of 62 vertices, 180 edges
and 120 triangles, that is, 362 geometric elements. This attribute is encoded in the pair of
dodecagons — the last of the regular polygons making up the inner Tree of Life — because the
division of their 24 sectors into three tetractyses produces 362 yods (Fig. 21). Each yod
denotes a geometric element of the disdyakis triacontahedron. The two centres denote any pair
of diametrically opposite A vertices and the 360 yods surrounding them denote the 60 vertices,
180 edges and 120 triangles between these opposite poles. The archetypal ‘form’ quality of the
Godname ELOHIM assigned to Binah in the Tree of Life is illustrated by the fact that its number
value 50 is the number of corners of the 72 tetractyses in the pair of
dodecagons. The same applies to the Godname ELOHA assigned to Geburah below Binah in the Tree
of Life because its number value is 36, which is the number of tetractyses in
a dodecagon. Each one has 25 corners and 60 edges, a total of 121 geometric elements, where 121
= 11^{2}
22
“1” denotes the centre of the dodecagon. This shows how the decagon
symbolising the perfect Pythagorean Decad expresses the tetractysdivided dodecagon.
As well as the Godname YAH assigned to Chokmah, the Godname EHYEH of Kether
with number value 21 prescribes the superstring grouptheoretical number 240. It is the
sum of the first 21 binomial coefficients other than 1 in Pascal’s Triangle:
As 24 = 5^{2} – 1 = 3 + 5 + 7 + 9, the number has the tetractys
representation:
In terms of the integers 1, 2, 3 & 4 symbolised by the tetractys, 240 =
(1+2+3+4)1×2×3×4, whilst 33 (=1!+2!+3!+4!) is the number of tree levels in 10 overlapping Trees
of Life, the counterparts of which in the disdyakis triacontahedron are the 33 sheets of its
three types of vertices.
These arithmetic expressions for geometrical parameters of holistic objects
such as this polyhedron are examples of the Tetrad Principle at work.^{10} It also determines the E_{8}×E_{8}' heterotic
superstring structural parameter 84 because
84 = 1^{2} + 3^{2} + 5^{2} + 7.^{2}
Indeed, the fact that the disdyakis triacontahedron has 120 vertices is a
clear illustration of this fundamental mathematical principle because the 30 Golden Rhombic
faces of the rhombic triacontahedron are each divided into four faces to produce 120 faces,
where
30 = 1^{2} + 2^{2} + 3^{2} + 4^{2}
and
120 = 4×30 = 2^{2}×(1^{2} + 2^{2} + 3^{2} +
4^{2})
=
2^{2} + 4^{2} + 6^{2} +
8^{2}.
As 248 = 4×62, the disdyakis triacontahedron with the Tetrad
assigned to each of its 62 vertices represents the 248 E_{8} gauge fields
that transmit superstring forces, two diametrically opposite A vertices signifying the eight
gauge fields that correspond to the eight zero roots of E_{8} and the 60 vertices in
the seven sheets between them generating the number 240 as the 240 nonzero roots of
E_{8} and therefore signifying their corresponding 240
23
Table 2. The gematria number values of the ten Sephiroth in the four Worlds.

SEPHIRAH

GODNAME

ARCHANGEL

ORDER OF ANGELS

MUNDANE CHAKRA

1 
Kether
(Crown)
620 
EHYEH
(I am)
21 
Metatron
(Angel of the Presence)
314 
Chaioth ha Qadesh
(Holy Living Creatures)
833

Rashith ha Gilgalim
First Swirlings
(Primum Mobile)
636 
2 
Chokmah
(Wisdom)
73 
YAHWEH, YAH
(The Lord)
26,
15

Raziel
(Herald of the Deity)
248 
Auphanim
(Wheels)
187 
Masloth
(The Sphere of the Zodiac)
140 
3 
Binah
(Understanding)
67 
ELOHIM
(God in multiplicity)
50

Tzaphkiel
(Contemplation of God)
311

Aralim
(Thrones)
282

Shabathai
Rest
(Saturn)
317 

Daath
(Knowledge)
474 




4 
Chesed
(Mercy)
72 
EL
(God)
31 
Tzadkiel
(Benevolence of God)
62 
Chasmalim
(Shining Ones)
428

Tzadekh
Righteousness
(Jupiter)
194 
5 
Geburah
(Severity)
216

ELOHA
(The Almighty)
36

Samael
(Severity of God)
131

Seraphim
(Fiery Serpents)
630

Madim
Vehement Strength
(Mars)
95 
6 
Tiphareth
(Beauty)
1081

YAHWEH ELOHIM
(God the Creator)
76 
Michael
(Like unto God)
101

Malachim
(Kings)
140

Shemesh
The Solar Light
(Sun)
640 
7 
Netzach
(Victory)
148

YAHWEH SABAOTH
(Lord of Hosts)
129

Haniel
(Grace of God)
97 
Tarshishim or Elohim
1260

Nogah
Glittering Splendour
(Venus)
64 
8 
Hod
(Glory)
15

ELOHIM SABAOTH
(God of Hosts)
153

Raphael
(Divine Physician)
311

Beni Elohim
(Sons of God)
112

Kokab
The Stellar Light
(Mercury)
48 
9 
Yesod
(Foundation)
80

SHADDAI EL CHAI
(Almighty Living God)
49,
363

Gabriel
(Strong Man of God)
246

Cherubim
(The Strong)
272

Levanah
The Lunar Flame
(Moon)
87 
10 
Malkuth
(Kingdom)
496

ADONAI MELEKH
(The Lord and King)
65,
155

Sandalphon
(Manifest Messiah)
280 
Ashim
(Souls of Fire)
351

Cholem Yesodeth
The Breaker of the Foundations The Elements
(Earth)
168

The Sephiroth exist in the four Worlds of Atziluth, Beriah, Yetzirah and
Assiyah. Corresponding to them are the Godnames, Archangels, Order of
Angels and Mundane Chakras (their physical manifestation). This table gives
their number values obtained by the ancient practice of gematria, wherein a
number is assigned to each letter of the alphabet, thereby giving a number
value to a word that is the sum of the numbers of its letters.

24
gauge fields. Constructed from tetractyses, the disdyakis triacontahedron
has 422 yods on the 180 edges of the 120 tetractyses,^{11} that is, 420 yods between polaropposite A vertices are needed
to define the edges of these tetractyses. Assigning the number 4 to these yods generates the
number 1680. This is the number of oscillations in each whorl of the heterotic superstring.
It is also the number of hexagonal yods symbolising the seven Sephiroth of Construction
needed to build the sequence of seven polyhedra completed with the disdyakis
triacontahedron. This is a profound meaning of the disdyakis triacontahedron: if we imagine
it built with the integer 4 as each vertex, its seven sheets of vertices express both the
number of particles transmitting the unified superstring force and the number of circularly
polarised oscillations in each whorl of the heterotic superstring. Moreover, the
arithmetic connection between 240 as the sum of the integers 4 assigned to the 60
vertices between opposite A vertices and 1680 — the sum of these integers assigned to the
yods between them — is 1680 = 7×240. Remarkably, this factorisation is identical to the
seven sets of 240 hexagonal yods in the sequence of seven polyhedra (see page 17). The
disdyakis triacontahedron embodies numbers that characterise both fundamental force
and form. It does so simply because, as shown in Articles 2224 and as further
demonstrated here, it is the 3dimensional version of the universal blueprint called the
‘Tree of Life.’ No wonder that the ancient Pythagoreans gave to the number 4 the title of
“holding the key of nature”!
References
^{1} All numbers written in boldface are the number values of the ten Sephiroth,
their Godnames, Archangels, Orders of Angels and Mundane Chakras. They are listed above in
Table 2.
^{2} Phillips, Stephen M. “ESP of Quarks & Superstrings,” New
Age International, New Delhi, India, 1999.
^{3} Horava, Petr and Witten, Edward. “Heterotic and type 1 string
dynamics from eleven dimensions,” Nucl. Phys. B460 (1996), pp. 506524.
^{4} For the definition of the 2ndorder tetractys, see p. 4 of
Article 24: “More Evidence for the Disdyakis Triacontahedron as the 3dimensional
Realisation of the Inner Tree of Life & its Manifestation in the
E_{8}×E_{8} Heterotic Superstring,” by Stephen M. Phillips, (WEB, PDF).
^{5} The ntree has (50n+30) yods. Therefore, the
number of yods in the (n–1)tree = 50(n–1) + 30 = 50n –
20. There are 17 yods beyond the (n–1)tree up to the level of Chesed of the nth tree. The
number of yods in n overlapping tree up to Chesed of the nth tree = 50n –
20 + 17 = 50n – 3.
^{6} Phillips, Stephen M. Article 22: “The Disdyakis Triacontahedron
as the 3dimensional Counterpart of the Inner Tree of Life,” (WEB, PDF), ref. 5.
^{7} Ibid, p. 16.
^{8} Phillips, Stephen M. Article 15: “The Mathematical Connection
between Superstrings and Their Micropsi Description: a Pointer Towards Mtheory,”
(WEB, PDF), p. 11.
^{9} Ibid, p. 27, and Phillips, Stephen M. Article 21: “Isomorphism
between the I Ching table, the 3×3×3 Array of Cubes and the Klein Configuration,” (WEB, PDF), p. 6.
^{10} Phillips, Stephen M. Article 1: “The Pythagorean Nature of
Superstring and Bosonic String Theories,” (WEB, PDF), p. 5.
^{11} Ref. 6, p. 9.
25
