ARTICLE 44
by
Stephen M. Phillips
Flat 4, Oakwood House, 117119 West Hill Road. Bournemouth.
Dorset BH2 5PH. England.
Website: http://www.smphillips.mysite.com
Abstract
Previous articles have presented considerable evidence
that correlates the properties of the disdyakis triacontahedron with the
inner form of the Tree of Life. It established beyond reasonable doubt that
this Catalan solid is the polyhedral counterpart of the latter. Its
prescription by the Kabbalistic Divine Names supports this conclusion. Just
as the root structure of the heterotic superstring symmetry group
E_{8}×E_{8} is embodied in the inner Tree of Life, so it
manifests in the pattern of geometrical elements of the disdyakis
triacontahedron. Both its seven layers of vertices and its 15 layers of
vertices form sets of polygons with the remarkable attribute of being
geometrically analogous to this root structure. As confirmation of their
holistic character, the archetypal pattern of Plato’s Lambda Tetractys,
which generates the tone ratios of the Pythagorean musical scale, appears in
the geometrical composition of the polyhedron and its 15 polygons. The
disdyakis triacontahedron is shown to be also the polyhedral representation
of the Cosmic Tree of Life (CTOL), which maps the seven cosmic planes of
consciousness. Half of the polyhedron maps the cosmic physical plane and its
other half maps the six cosmic superphysical planes. The polyhedron provides
a simple explanation of why the Hebrew word ‘Malkuth,’ which signifies the
physical aspect of the Tree of Life, has the gematria number value 496 and
why, as the dimension of E_{8}×E_{8}, it appears at the
heart of superstring physics, for this number has a more profound meaning
than that understood by researchers in superstring theory. When constructed
from tetractyses, the 15 polygons and the polyhedron have 1680 yods
surrounding its centre other than its two opposite vertices. This is the
number of turns in each helical whorl of the basic unit of matter
paranormally described over a century ago by the Theosophists Annie Besant
and C. W. Leadbeater. It is also the number of geometrical elements
surrounding the axis of the polyhedron that are required to construct it.
The number 168 is the gematria number value of the Hebrew name of the
Mundane Chakra of Malkuth. The division of the formbuilding elements into
780 edges and 900 vertices & triangles reproduces the gematria number
values 78 and 90 of the Hebrew words making up this name, making unlikely
the possibility of coincidence. The vertices, edges and triangles in the
polyhedron and its 15 polygons that are not shared with each other reproduce
the gematria number value of Tiphareth, the Sephirah at the centre of the
Tree of Life. The number of vertices making up both is 137, which is the
number that determines the finestructure constant measuring the strength of
the coupling of the electromagnetic field to the electron. This confirms its
status as the polyhedral Tree of Life because a number of previous articles
have shown that 137 is — like 168 — one of the parameters that define
holistic systems.

1
1. Introduction In Article 27, it was shown
that the Catalan solid (1) with the most faces — the disdyakis triacontahedron, which possesses
62 (2) vertices, 180 edges & 120 faces — embodies the superstring structural
parameter 1680 discussed in many previous articles. This is because it is the polyhedral
version of the universal Tree of Life blueprint governing holistic systems, such as the
heterotic superstring described by the Theosophists Annie Besant and C. W. Leadbeater
(Fig. 1). When its 180 internal triangles, having the centre of the
polyhedron as a vertex and its edges as their
outer sides, are divided into their sectors, 1680 vertices, lines &
triangles surround an axis joining any pair of diametrically opposite vertices:
The 1680 geometrical elements consist of 780 edges and 900 vertices &
triangles. This 78:90 division is the polyhedral manifestation of the gematria number values
(Table 1) of the Hebrew words Cholem and Yesodeth making up
the name of the Kabbalistic Mundane Chakra of Malkuth, the Sephirah signifying the physical
manifestation of the Tree of Life on a cosmic scale:
This is a remarkable conjunction of three different contexts: the paranormal
observation by Besant & Leadbeater of the basic units of matter, the gematria aspect of
Kabbalah and the geometrical properties of the object that is the polyhedral counterpart of
the universal Tree of Life blueprint (3). Even so, the diehard sceptic may feel tempted to dismiss it as coincidence
— however implausible his suggestion may seem to more impartial minds. As a rebuttal, this
article will demonstrate that the disdyakis triacontahedron also embodies the dimension
496 of the E_{8}×E_{8} heterotic superstring gauge symmetry
group. In other words, it will prove that the polyhedron embodies both the
fundamental structural and
2
Table 1. 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.

(All numbers in this table referred to in the article are written in
boldface).
3
dynamical parameters of the E_{8}×E_{8} heterotic
superstring in a natural way, thereby rendering the possibility of coincidence as highly
implausible.
2. The polyhedral basis of E_{8}×E_{8} in
the seven polygons The 120 faces of the disdyakis triacontahedron are
the sides of 30 pyramids. Their bases are the Golden Rhombic faces of a rhombic
triacontahedron formed by the 20 C vertices of a dodecahedron and the 12 B vertices of an
icosahedron, leaving 30 A vertices as their peaks. The vertices are arranged in either
seven parallel sheets perpendicular to an axis passing through two opposite A vertices,
11 sheets perpendicular to an axis joining two opposite B vertices or
15 sheets perpendicular to an axis joining two opposite C vertices.
Fig. 2 displays the projection of the vertices onto the central sheet.
The topmost A vertex is at the centre of the graph in the XY plane.
4
Four types of polygons are formed by 36 vertices that are
prescribed by ELOHA, Godname of Geburah with number value 36:
8sided polygons:
10sided polygon:
6sided polygon:
12sided polygon: 
4 A, 2 B & 2 C vertices
4 A, 2 B & 4 C vertices
4 A & 2 C vertices
4 A, 4 B & 4 C vertices. 
The topmost three polygons have 24 vertices made up of 12 A vertices, four B
vertices & eight C vertices; similarly for the lowest three polygons. The central
polygon comprises four A vertices, four B vertices & four C vertices. The four types of
polygons consist of 16 A vertices, eight B vertices & 12 C vertices. All seven polygons
have 28 A vertices, 12 B vertices & 20 C vertices, with A vertices at the apex and
nadir.
The numbers of geometrical elements in faces and interior polygons that
surround an AA axis of the disdyakis triacontahedron are set out below:
The first three layers have 216 internal and external
geometrical elements, where 216 is the number value of Geburah (see
Table 1); similarly for the last three layers. Surrounding the centre of the
central polygon are 48 elements (24 internal, 24 external), totalling
49 elements. This shows how EL ChAI, the Godname of Yesod with number value
49, prescribes the central polygon. 48 is the number value
of Kokab, the Mundane Chakra of Hod. As explained in Article 40, it is a defining
parameter of all holistic systems, as, indeed, is its 24:24 division displayed by the 24
geometrical elements in each half of the 12sided polygon. Surrounding the AA axis are 264
geometrical elements in the four layers of half the polyhedron, of which 36
are vertices, where 36 is the number value of ELOHA, the Godname of
Geburah.
Compare this with the fact that the seven enfolded polygons making a half of
the inner Tree of Life have 36 vertices and 264 yods when their 47 sectors
are tetractyses (Fig. 3). The number 264 defines half of the polyhedron, just as it does for
its polygonal counterpart. The 12 vertices in the central polygon correspond to the 12
vertices of the first four enfolded polygons and the 24 vertices in the first three polygons
correspond to the 24 vertices of the last three enfolded polygons.
The seven polygons defined by the vertices of the disdyakis triacontahedron
have 480 geometrical elements made up of 240 elements and their 240 mirror images on the
opposite side of the polyhedron. Each set of 240 elements consists of 168
elements in the faces (24 vertices, 60 triangles & 84 edges) above or below the plane of
the central polygon and 72 elements that comprise six vertices and six
external edges of the fourth polygon, 30 internal edges and 30 internal triangles. This
72:168 division reflects a particular division of roots of the superstring
gauge symmetry group E_{8}. The 72 elements correspond to the
72 roots of its subgroup E_{6} and the 168
elements correspond to the 168 roots of E_{8} that are not roots of
E_{6}. Inspection of Fig. 2 reveals that two edges of the 10sided polygon in the second layer of
vertices and all six edges of the sixsided polygon in the third layer are not
edges of the disdyakis
5
triacontahedron, i.e., the two polygons have eight edges (denoted in
Fig. 2 by black lines) that do not belong to the polyhedron. Their
corresponding polygons below the XY plane similarly have eight unshared edges. Hence, the
faces of the disdyakis triacontahedron and its seven polygons have in total
248 geometrical elements and their 248 mirror images, that
is, 496 elements, 16 of which are polygonal edges that do not manifest as
part of the form of the polyhedron. Its geometry therefore reflects the holistic pattern of
the 496 roots of the E_{8}×E_{8} heterotic superstring, the
direct product of the two identical groups E_{8} with dimension 248
reflecting the inversion symmetry of the polyhedron, namely, the fact that every geometrical
element has its mirror image. The eight unshared edges of polygons in each half correspond
to the eight simple roots of each group. As the number value of Malkuth and the dimension of
E_{8}×E_{8}, 496 is simply the number of geometrical
elements in the faces of the polyhedron and in the seven polygons formed by its vertices
that surround its AA axis. This is the polyhedral basis of the heterotic superstring gauge
symmetry group E_{8}×E_{8}. The property does not hold for any type of axis
because, for example, in the case of the 15 polygons perpendicular to the
CC axis, none of their edges is an edge of the polyhedron. As we shall discover later, this
configuration embodies not only the basic dynamical parameter of the
E_{8}×E_{8} heterotic superstring but also its basic structural
parameter.
There are 30 vertices, 128 edges & 90 triangles in each half of the
polyhedron, i.e., 128 edges and 120 vertices & triangles. One half geometrically encodes
the forces between superstrings of ordinary matter, its mirror image half encodes the forces
between superstrings of shadow matter. The 60 faces above the central plane are defined by
168 geometrical elements (84 edges, 84 vertices & triangles) that are
analogous to the 168 circularly polarized oscillations made by each whorl
of the UPA in a halfrevolution, the 168 elements below the central plane
corresponding to the 168 oscillations made in its other halfrevolution —
the mirror image of the first. The two sets of 60 faces, defined by two sets of
168 geometrical elements above and below the central plane, correspond to
halfrevolutions in the outer and inner halves of the UPA. The 168
tendimensional gauge fields corresponding to the 168 gauge charges of
E_{8} that do not belong to E_{6} have 1680 spacetime components. Is this
the basis of the 1680 helical turns of each whorl? Not quite. In discussion in earlier
articles of the inner Tree of Life basis of the 240 E_{8} gauge charges, wherein
each whorl is represented by a Tree of Life, the first six enfolded polygons of whose inner
form possess 24 corners denoting E_{8} gauge charges, it was suggested that each
whorl carries 24, not 168, E_{8} gauge charges. In other words, the
UPA has ten whorls simply because it possesses 240 E_{8} gauge charges distributed
24 to a whorl. That is what is indicated in the geometry of the inner form of ten
overlapping Trees of Life that represent the ten whorls of the UPA, because 24 corners are
associated with a set of the first six enfolded polygons, and this is repeated in each set
enfolded in an overlapping Tree of Life, making 240 corners for the ten Trees of Life.
Symbolizing independent degrees of freedom, these corners physically manifest as the 240
gauge charges and associated gauge fields of a superstring corresponding to the 240 roots of
E_{8}.
The seven polygons in the disdyakis triacontahedron have 60 internal edges
surrounding the axis and 16 outer edges that are not polyhedral edges, i.e.,
76 edges. YAHWEH ELOHIM, the Godname of Tiphareth with number value
76, prescribes the seven sheets of vertices. Adding the eight unshared
edges to the 36 internal edges and 96 external edges in the first four
polygons creates 140 edges. 140 is the number value of
Masloth, the Mundane Chakra of Chokmah (also the number value of Malachim,
the Order of Angels assigned to Tiphareth). 248 is the number value of
Raziel, the Archangel of Chokmah.
There are nine vertices on the axis, namely, the centres of the seven
polygons and the A vertices at the top and bottom. Hence, the total number of geometrical
elements is (496+9=505). 504 elements surround the centre of the
polyhedron. As we saw in Article 35 (4), this is the number of geometrical elements in the threedimensional Sri
Yantra when each of its 42 triangles is divided into its sectors! It is striking evidence
that the Sri Yantra and the disdyakis triacontahedron are equivalent
representations of divine archetypes. They are not identical in terms of their geometrical
composition because the 504 elements in the polyhedron consist of 68 vertices, 256 edges
& 180 triangles, whereas the 504 elements in the 42 primary triangles of the Sri Yantra
comprise 126 vertices, 252 edges & 126 triangles (5). What counts as equivalence in this context is not mathematical isomorphism
between corresponding geometrical elements but their total numbers, which are the
same because, although dissimilar in their form, both sacred geometries embody numerically
the same bits of information, spatially represented by points, lines and triangles.
These three types of geometrical elements represent independent degrees of freedom. Any
triangle exhibits seven of them: three vertices, three edges and the space inside it. The
120 triangular faces (Fig. 1) of the disdyakis triacontahedron are identical, each consisting of A,
B & C vertices, edges AB, BC & AC and the triangle ABC. They play the role of the
seven Sephiroth of Construction in the Tree of Life, which are similarly divided into two
triads: ChesedGeburahTiphareth & NetzachHodYesod and the single Sephirah, Malkuth.
The threefold nature of both vertices and edges reflects the active, passive &
homeostatic pillars of the Tree of Life.
6
The Cosmic Tree of Life (CTOL) was discussed in Article 5 (6) as the Tree of Life of the spiritual cosmos. It consists of 91 overlapping
Trees of Life with 550 Sephirothic levels (SLs). There are 504 SLs in CTOL down to the top
of the 7tree (the lowest seven Trees of Life) mapping the physical plane, the Malkuth level
of CTOL. The centre of the polyhedron corresponds to the universe of space and time — the
centre of the spiritual cosmos — mapped by the 7tree. The 504 geometrical elements
surrounding it are the counterpart of the 504 SLs that map superphysical levels of reality.
There are 496 SLs down to the 55th SL from the bottom, which is Chesed of
the ninth Tree of Life. The eight vertices on the AA axis surrounding the centre of the
polyhedron are the counterpart of the next eight SLs down to the top of the 7tree. The
248 geometrical elements in one half of the polyhedron (they, of course,
include half the central, 12sided polygon) correspond to the 248 SLs in
CTOL down to Binah of the 50th Tree of Life, which is prescribed by ELOHIM,
Godname of Binah with number value 50 (see Table 1). The 248 mirror images of these elements, reflected
through the centre of the polyhedron, correspond to the next 248 SLs down
to Chesed of the ninth Tree of Life, whose Malkuth is the top of the 7tree and the 25th
tree level, marking the first of the 25 dimensions of the physical universe.
Adding the 16 (=4^{2}) unshared edges to the 240 edges listed above,
the disdyakis triacontahedron has 496 geometrical elements surrounding its
AA axis that comprise 256 edges and 240 vertices & triangles, where 256 =
4^{4}, 240 = 1×2×3×4(1+2+3+4), and
This exemplifies how the Tetrad expresses the properties of holistic
systems. As 496 = 16×31 =
16(1+15+15), we see that assigning the square of 4 to the
31 yods in a pentagon whose five sectors are tetractyses generates the
496 geometrical elements in the disdyakis triacontahedron, the central
number, 4^{2} = 16, denoting the number of unshared edges, the 15
internal squares of 4 adding to 240, which is the number of other edges, and the
15 squares of 4 on its boundary adding also to 240, the number of vertices
& triangles. The Tetrad reveals a beautiful harmony between number and geometry
that is always present in holistic systems like the disdyakis triacontahedron. The
geometrical composition of the polyhedron is
prescribed by EL, whose number value is 31 and by YAH,
whose number value is 15. EL also prescribes the polyhedron because it
has 31 vertices and their 31 mirror
images.
Of the 504 elements surrounding its centre, 62 are
polyhedral vertices, leaving 442 other elements. The
7
221 such elements in each half comprise 90 triangles, three centres of
polygons & 128 edges, i.e., 131 vertices & edges.
131 is the number value of Samael, Archangel of Geburah. The (7+7)
enfolded polygons of the inner TOL have 444 hexagonal yods (Fig. 4). This means that each set of seven enfolded polygons has 221
hexagonal yods outside the root edge, which has two hexagonal yods. They symbolize the extra
221 geometrical elements needed to construct the faces of the disdyakis triacontahedron and
the seven polygons, starting from the vertices. The (7+7) polygons enfolded in each
overlapping Tree of Life have 70 corners, of which 68 are outside the root edge. They
comprise six corners that coincide with SLs of the outer Tree of Life (three on each side
pillar), when the latter is projected onto the plane of the polygons, and
62 that belong solely to its inner form. The 62 unshared
corners correspond to the 62 vertices of the disdyakis triacontahedron as
the polyhedral version of the inner Tree of Life.
3. The embedding
of E_{8}×E_{8}' in 10 Trees of Life According to the table in Fig. 2, the 248 geometrical elements surrounding an AA axis
in each half of the polyhedron comprise 168 elements above the central
polygon in its 60 faces and 80 other elements. The former comprise 24
vertices (12A, 4B & 8C), 84 edges (28 of each kind) and
60 triangles, i.e., 84 edges and 84 vertices or triangles. This 84:84
division in the superstring structural parameter 168 is characteristic of
holistic systems in which it appears. The 80 elements are made up of the
eight edges of polygons that are not polyhedral edges and 72 elements that
comprise the six vertices in half the central polygon, 36 edges and 30
internal triangles. The latter correspond to the 72 roots of E_{6},
one of the five exceptional groups. As stated earlier, the eight unshared edges correspond
to the eight simple roots of E_{8}. The six nonpolyhedral edges of the sixsided
polygon correspond to the six simple roots of E_{6} and the two nonpolyhedral edges
of the 10sided polygon correspond to the two extra simple roots of E_{8}.
The disdyakis triacontahedron exhibits the same division:
248 = 80 + 168,.
as is found for the 80 yods in the 1tree and the
168 yods above it as far as Chesed of the fifth TOL, the
31st SL prescribed by EL, the Godname of Chesed (Fig. 5). The 80 elements comprise 30 internal triangles and
50 vertices & edges, where 50 is the number value of
ELOHIM, Godname of Binah. This 30:50 division appears also in the 1tree
because the yod population of the ntree is given by:
Y(n) = 50n + 30,
so that the (Y(1)=80) yods of the 1tree comprise the 30
yods in its Lower Face and the 50 yods above it. Half of the polyhedron
with 248 geometrical elements surrounding its axis is therefore the
counterpart of the 248 yods in the 5tree of CTOL up to its highest Chesed
(the first Sephirah of Construction). Its mirror image half with 248
elements is the counterpart of the next 248 yods up to (but not including)
Chesed of the tenth Tree of Life. As this is the 61st SL (again prescribed by EL because 61
is the 31st odd integer), the 496 yods contain 60 SLs as
corners of triangles, just as the 496 geometrical elements include the 60
vertices of the polyhedron! The latter therefore are the counterpart of the SLs. The number
of triangles in the ntree is
T(n) = 16n + 9.
The 9tree has (T(9)=153) triangles, where
153 is the number value of ELOHIM SABAOTH, Godname of Hod. There are three
more triangles above the top of the 9tree up to the 61st SL, so that the
496 yods belong to 156 tetractyses. As 156 is the 155th
integer after 1, this shows how ADONAI MELEKH, the Godname of Malkuth with number value
155, prescribes the dimension 496 of
E_{8}×E_{8}. ELOHA, the Godname of Geburah with number value
36, prescribes the dimension 248 of E_{8} because
there are 36 hexagonal yods on the two side pillars up to the
31st SL. YAHWEH ELOHIM, Godname of Tiphareth with number value
76, prescribe this number because there are 76 tetractyses
up to the 31st SL prescribed by EL.
8
4. Polyhedral
embodiment of the superstring structural parameter 1680 We saw
earlier that the disdyakis triacontahedron embodies the superstring structural parameter
1680 as the number of geometrical elements surrounding any axis when its
internal triangles are divided into their sectors. As confirmation that this is not
merely a coincidence, we found that
the number is made of 780 edges and 900 vertices & triangles,
reproducing (apart from a factor of 10) the gematria number values 78 and 90 of,
respectively, Cholem and Yesodeth, the Kabbalistic name of the Mundane
Chakra of Malkuth. It will now be shown how 1680 is the yod population of both the faces
of the polyhedron and the 15 parallel polygons formed by its 60 vertices
surrounding a CC axis.
Fig. 6 displays the projection of the vertices of the polygons onto the XY
plane occupied by the central polygon. They are triangles or 6sided polygons, the central
(eighth) polygon being a hexagon. Unlike in the case of the seven polygons perpendicular to
an AA axis discussed earlier, the vertices of a given polygon are of the same type. It
means that none of their edges is an edge of the polyhedron. The significance of this is
that, when the sectors of the polygons are turned into either tetractyses or divided into
three tetractyses, none of the yods on the edges of polygons other than their vertices is
shared with the polyhedron when its faces are similarly transformed. The only yods shared
when its 180 internal triangles are turned into tetractyses are all vertices and the
hexagonal yods on the internal edges of its sectors.
There are eight polygons in half the polyhedron with the following sets of
vertices, commencing from the top:
The order in the polygons below the XY plane is reversed. The uppermost C
vertex (black circle) at the origin of the XY coordinate system is below the level of the
triangle formed by the three A vertices (red circles). The lowest C vertex is above their
inverted counterpart.
9
Now suppose that the 60 sectors of the 15 polygons are
divided into three tetractyses. The number of yods in an nsided polygon with its sectors so
divided is given by:
N(n) = 15n + 1.
(Notice that the number 15 of YAH, Godname of Chokmah, is
the number of yods per sector). Triangles have (N(3) = 46) yods. 42 yods other than corners
of the triangle surround its centre. A 6sided polygon has (N(6)=91) yods. 84 yods other
than corners surround its centre. There are 840 yods surrounding the centres of the
15 polygons other than their 60 vertices (Fig. 7). The five 6sided polygons have 30 corners and (5×84=420) yods other
than corners surrounding their centres, as do the ten triangles with 30 corners. This equal
division of 420 yods is prescribed by the Godname YAH (YH) because the letter value 5 of H
(heh) specifies the number of 6sided polygons and the letter value 10 of Y (yod) specifies
the number of triangles. This is yet more evidence of the archetypal nature of the disdyakis
triacontahedron and its prescription by the Divine Names.
Suppose that the 120 faces of the disdyakis triacontahedron are turned into
tetractyses. There is one hexagonal yod at the centre of each face and two hexagonal yods on
each of its 180 edges. The number of yods in its faces surrounding the CC axis = 60 +120 +
180×2 = 540. Every edge is also the side of an internal triangle with the centre of the
polyhedron as one shared vertex. Next, suppose that the 180 internal triangles are turned
into tetractyses. One hexagonal yod is at the centre of each tetractys and two hexagonal
yods lie on each of the 62 internal edges ending on the vertices of the
polyhedron. The number of yods in internal triangles that surround the CCaxis = 180 + 60×2
= 300. Therefore, (540+300=840) yods surround this axis (this is true for any type of axis).
The disdyakis triacontahedron is the Catalan solid with the most faces. Table 2 shows that, when all the Catalan solids with triangular faces are
constructed from tetractyses, only the disdyakis triacontahedron has 840 yods surrounding an
axis joining two diametrically opposite vertices:
Table 2. Various yod populations of the Catalan solids that have triangular
faces.
Catalan solid

C

E

F

C'

H_{s}

H_{i}

H

N_{s}

N_{i}

N

triakis tetrahedron 
8

18

12

6

48

30

78

54

30

84

rhombic dodecahedron 




















triakis octahedron 
14

36

24

12

96

60

156

108

60

168

tetrakis hexahedron 
14

36

24

12

96

60

156

108

60

168

deltoidal icositetrahedron 




















pentagonal icositetrahedron 




















rhombic triacontahedron 




















disdyakis dodecahedron

26

72

48

24

192

120

312

216

120

336

triakis icosahedron 
32

90

60

30

240

150

390

270

150

420

pentakis dodecahedron 
32

90

60

30

240

150

390

270

150

420

deltoidal hexacontahedron 




















pentagonal hexacontahedron 




















disdyakis triacontahedron 
62

180

120

60

480

300

780

540

300

840

(C = number of vertices, E = number of edges, F = number of faces, C' =
number of vertices surrounding the axis, H_{s} = number of hexagonal yods in faces,
H_{i} = number of internal hexagonal yods surrounding the axis, H
(=H_{s}+H_{i} ) = number of hexagonal yods surrounding the axis,
N_{s} (=C'+H_{s}) = number of yods in faces surrounding the axis,
N_{i} (= 2C'+E) = number of internal yods surrounding the axis & N
(=N_{i}+N_{s}) = number of yods surrounding the axis.)
This is highly significant, because it indicates the unique status of this
polyhedron in embodying the superstring structural parameters 840 and 1680, as would be
expected if it were the polyhedral Tree of Life. The 15 polygons also have
840 yods other than vertices surrounding their centres. These, however, cannot be added to
the 840 yods in the polyhedron that surround its axis because the latter include some of
those in the central polygon, so that simply adding them together would involve some double
counting. Two hexagonal yods on each internal edge of the six sectors of the hexagon are
shared with the internal triangles of the polyhedron, although, as the outer edges of the
hexagon connecting A vertices are not edges of the polyhedron, none of its sectors coincide
with internal triangles of the latter — in fact, none of the sectors of polygons do, so that
no other yods coincide. There are 12 hexagonal yods in the central polygon that are shared,
as well as the 60 vertices and the centre of the polyhedron, i.e., 73
shared yods, where 73 is the number value of Chokmah. Of the 840 yods other
than polyhedral vertices surrounding centres of the 15 polygons,
(840–12=828) yods are unshared. Of the 840 yods belonging to the
10
polyhedron that surround the CC axis, (12+60=72) yods are
shared with polygons, leaving (840–72=768) unshared yods. Therefore, there
are (828+768=1596) unshared yods. Four hexagonal yods belonging to the internal tetractyses
of the polyhedron lie on the CC axis, as do the 14 centres of the polygons and the two C
vertices. These, too, are unshared. (1596+4+14+2=1616) yods are unshared and
73 yods are shared, totalling 1689 yods. 1680 yods surround the centre of
the polyhedron other than the two C vertices at its poles and the six A vertices at its
equator at the ends of shared edges. 840 yods belong to half the polyhedron; their 840
mirror images belong to its other half. They correspond to the 840 turns in the inner and
outer halves of each helical whorl of the UPA. The Godname EHYEH with number value
21 prescribes the 21 yods on the axis. In fact, its letter
values are the numbers of different yods:
AHIH = 21
A = 1:
H = 5:
I = 10:
H = 5: 
centre of polyhedron
centres of five triangles above the central polygon
two opposite A vertices + centres of four 6sided polygons + four hexagonal
yods
centres of five triangles below the central polygon 
This is true also for the letter values of the Godname YAH:
YH = 15
Y = 10: 
centres of ten triangles above & below
the central polygon 
H = 5: 
centres of five hexagons 
This shows how both EHYEH and YAH prescribe the polygons.
As no CC axis is unique, five hexagons with different A vertices at their
corners can be chosen from the 30 A vertices in the disdyakis triacontahedron. This means
that five axes can be drawn between opposite C vertices so that their associated central
hexagons have no A vertices in common. Each similar set of 15 polygons has
its own distinct set of 840 yods because none of them is a vertex of the polyhedron. In the
case of an AA axis, there are four A vertices in the central polygon, so that the six A
vertices they form with the apex and nadir A vertices can be chosen in five ways from the 30
A vertices available Therefore, there are ten sets of polygons with 8400 yods. For each of
the ten axes, there are 840 yods in the polyhedron, according to Table 2. Hence, there are ten copies of this primary set, that is, 8400 yods.
The ten sets of polygons and their corresponding orientations of the polyhedron generate
16800 yods, which is the number of turns in the ten whorls of the UPA. The 8400 yods in the
triangles of the ten copies of the polyhedron would correspond to the 8400 turns in the
outer half of the UPA and the 8400 yods in the ten sets of polygons would correspond to the
8400 turns in its inner half. However, this has not taken into account the fact that some of
the outer edges of the set of polygons perpendicular to an AA axis are shared with the
polyhedron, so that multiplying by 10 causes some overcounting. However, if it were
meaningful for some reason to count copies separately, then the correspondence between the
yod populations of the 110 polygons in the ten sets and the number of turns in all ten
whorls of the UPA would be established. It is intriguing that the five choices of A axis and
the five choices of C axis might be the counterpart in the Tree of Life of the first five
Sephiroth and the last five Sephiroth, with the distinction between inner and outer halves
of each whorl being due, simply, to mirror symmetry in the spatial distribution of yods.
This issue will not be explored further here. The primary result has been established,
namely, that 1680 yods surround the centre of the disdyakis triacontahedron apart from the
two opposite C vertices and the six A vertices on its equator that lie on shared edges.
Table 3. Numbers of yods in the eight polygons in half the polyhedron
(sectors as three tetractyses).
11
The 840 yods surrounding the centres of the 15 polygons
that are not polyhedral vertices consist of 60 vertices of 180 tetractyses and 780 hexagonal
yods. Similarly, the 840 yods surrounding the axis of the polyhedron comprise 60 polyhedral
vertices and 780 hexagonal yods. The 15 polygons have 900 yods surrounding
the axis, when polyhedral vertices are included, whilst the polyhedron has 780 hexagonal
yods. These two numbers cannot be added because the internal triangles of the latter share
six edges with 12 hexagonal yods on them with the edges of the sectors of the hexagon. That
apart, we see the same 90:78 division as found for the geometrical composition of the
polyhedron.
5. Polyhedral
embodiment of CTOL Table 3 shows the numbers of yods in each polygon (‘1’ denotes the centre).
The vertices refer to corners of tetractyses, not just to vertices of the polyhedron. The
seven polygons above the XY plane have 61 vertices. This is prescribed by the Godname EL
with number value 31 because 61 is the 31st odd integer.
The hexagon has 13 vertices, so that the 15 polygons have (13 + 2×61 = 135)
vertices. Including the top and bottom C vertices, we see that 137 vertices define the
disdyakis triacontahedron and its 15 polygons. This is remarkable, because
137 is known to physicists as the integer whose reciprocal defines the finestructure
constant: α = e^{2}/ħ c ≈ 1/137. In fact, the number 137 is a parameter of all
holistic systems. For example, the inner form of the Tree of Life contains 1370 yods (the
yods in 137 tetractyses) when the 94 sectors of its 14 enfolded polygons are constructed
from three tetractyses (6). As 17 vertices lie on the CC axis, there are (137–17=120) vertices
surrounding it, where
120 = 2^{2} + 4^{2} + 6^{2} + 8^{2} .
This illustrates how the Tetrad Principle determines properties of objects
possessing sacred geometry (8). Its counterpart in he inner Tree of Life is the 120 yods on the edges of
the seven enfolded polygons. YAHWEH with number value 26 prescribes the
seven polygons because they have 351 hexagonal yods, where
351 is the sum of the first 26 integers. They have 385
yods other than polyhedral vertices, where
385 =

1^{2}
2^{2} 3^{2}
4^{2} 5^{2} 6^{2}
7^{2} 8^{2} 9^{2} 10^{2}
.

This demonstrates how the Pythagorean Decad, the measure of wholeness and
perfection, determines the number of yods in the seven polygons generated by turning their
sectors into three tetractyses. There are 66 vertices surrounding the centres of the eight
polygons in half the polyhedron. As 66 is the 65th integer after 1, this
property is prescribed by ADONAI, Godname of Malkuth with number value 65.
33 of them are polyhedral vertices, where 33 = 1! + 2! + 3! + 4!. 495 yods surround their
centres. Including the C vertex at the top of the polyhedron, there are 496
yods other than the eight centres of the polygons in half the polyhedron, that is, 504 yods
in total. Amazingly, as we found in Section 2, this is the number of geometrical elements in
the polyhedron and the seven polygons that surround its centre! Moreover, it divides into
the same two numbers: 8 & 496, for we found that 496
geometrical elements and eight points (two A vertices & six centres) on the AA axis
surround its centre! The geometrical composition of the whole polyhedron is symbolized by
the yods making up the eight polygons and the C vertex. More important still, the 504 yods
in half the polyhedron denote the 504 SLs in CTOL down to the top of the 7tree whose 25
tree levels signify the 25 dimensions of space. This is the polyhedral counterpart of the
threedimensional Sri Yantra, whose 42 triangles contain 504 yods.
According to Table 2, the central hexagon has 91 yods and the two sets of seven polygons
have (2×412=824) yods. Including the two opposite C vertices, there are 917 yods, so that
916 yods surround the centre of the polyhedron. This number is defined by the Tetrad and the
Godname ELOHIM because
where 229 is the 50th prime number. When n overlapping
Trees of Life are constructed from tetractyses, the number of yods on the central Pillar of
Equilibrium (9) is given by:
C(n) = 10n + 6.
The 91 overlapping Trees of Life of CTOL have (C(91)=916) yods on the
central pillar. 49 overlapping Trees of Life have 496 such
yods. The ntree has
C'(n) = 10n + 10
yods (10), so that the 49tree has (C'(49) = 500
= 50×10) yods, indicating how ELOHIM with number value
12
50 prescribes the seven planes of
consciousness mapped by the 49tree as well as CTOL itself. Inspection of
Fig. 5 shows that there are three yods on the central pillar between Daath of
any Tree of Life and the top of the next lower one. Therefore, there are 503 yods below
Daath of the 50th Tree of Life. The 916 yods surrounding the centre of the
polyhedron correspond to the 916 yods on the central pillar of CTOL and the 503 yods in the
polygons of half the polyhedron correspond to the 503 yods below Daath of the
50th Tree of Life. Because Binah of the 50th Tree of Life
is the 248th SL from the top of CTOL, Daath of the 50th
Tree of Life is the first of the next 248 SLs down to the
496th one — Chesed of the ninth Tree of Life. In other words, the 412 yods
above the central plane of the polyhedron define the split of the superstring number
496 into the two 248s of E_{8}×E_{8}', the
domain with 412 yods above the central polygon corresponding to the shadow matter group
E_{8}' and the lower half of the polyhedron with 504 yods (including the lowest C
vertex) corresponding to the ordinary matter group E_{8}. As a counterpart of CTOL,
the lowest eight polygons and the lowest C vertex can be thought of as representing the 504
yods up to (and including) Daath of the 50th Tree of Life prescribed by
ELOHIM, Godname of Binah with number value 50. This SL marks the
commencement of the second group of 248 SLs emanating from Binah of the
50th Tree of Life, the 248th SL from the top of CTOL. The
next 412 yods in the seven upper polygons correspond to the next 412 yods on the central
pillar up to the top of CTOL. This correspondence requires us to count the centre of the
central polygon as included in the 916 yods because it is among the 503 yods in the eight
lowest polygons, so that the 916 yods in question are not those surrounding this point but,
rather, those below the topmost C vertex, with the lowest C vertex corresponding to the
lowest point of CTOL. The eight centres of the eight lowest polygons correspond in CTOL to
the four yods above the top of the 49tree to Daath of the
50th Tree of Life and to the four yods added by turning 49
overlapping Trees of Life into the 49tree. The 495 yods surrounding their
centres correspond to the 495 yods on the central pillar up to, and including, Daath of the
49th Tree of Life.
914 yods in the 15 polygons surround the centre of the
central hexagon. This is the number of hexagonal yods lining the 457 edges of the
282 tetractyses in the 14 enfolded polygons of the inner Tree of Life when
their sectors are divided into three tetractyses (11). It is more remarkable evidence for the Tree of Life nature of the
disdyakis triacontahedron. 282 is the number value of Aralim, the
Order of Angels assigned to Binah (see Table 1). 457 is the number of yods in half the polyhedron. Each half of the
inner Tree of Life has 229 edges (the root edge being shared). As pointed out earlier, 229
is the 50th prime number, showing how ELOHIM prescribes each half.
Geometrical counterpart of
CTOL Let us now examine the geometrical elements that are intrinsic to
the 15 polygons, i.e., unshared with the polyhedron, when their sectors are
divided into three triangles. They are shown below:
548 unshared elements in the 15 polygons surround the
centre of the polyhedron, which they share with it . Including the two C vertices, which the
polyhedron does not share with the polygons, there are 550 unshared elements surrounding the
centre, consisting of 76 vertices, 294 edges and 180 triangles, that is,
474 edges & triangles. 76 is the number value of
YAHWEH ELOHIM, Godname of Tiphareth, and 474 is the number value of Daath
(“knowledge”). The 550 geometrical elements are on the CC axis or unshared with the
polyhedron. This is the geometrical counterpart of the 550 SLs of CTOL. There are
474 SLs down to the top of the twelfth Tree of Life, whose Malkuth is the
65th SL prescribed by ADONAI because it is the top of the 10tree, which
embodies information about the physics of the E_{8}×E_{8} heterotic
superstring, being the Tree Life representation of its ten spacetime dimensions. The
76 vertices in the 14 polygons or on the CC axis are the counterpart of
the 76 SLs below the top of the twelfth Tree of Life.
The 14 polygons above and below the central hexagon have 54 internal
vertices, 14 centres, 270 edges & 162 triangles, totalling 500 (=50×10)
geometrical elements. This shows how ELOHIM, Godname of Binah with number value
50, prescribes the 14 polygons. Each set of seven polygons has 250
elements. Including the two C vertices, there are 251 elements below the central hexagon,
which has 48 unshared elements surrounding its centre, and 251 elements
above it. In other words, there are (251+48=299) geometrical elements in
the eight polygons making up half the polyhedron other than its centre and 251 elements
above it. In terms of the correspondence with the 550 SLs of CTOL, this is remarkable, for
there are 299 SLs up to the top of the 49tree mapping the seven planes of
consciousness and 251 SLs in the 42 Trees of Life above it (12). What this means is that the topmost C vertex represents the top SL of
CTOL, the lowest vertex represents the lowest SL and the 48 elements in the
central polygon represent the next 48 SLs up to Chesed of the eighth Tree
of Life, which is the 49th SL prescribed by EL ChAI,
13
Godname of Yesod, whose meaning, “foundation,” is appropriate, given that
there are 25 tree levels up to this point in CTOL representing the 25 spatial dimensions of
the universe. In correlating the elements in the 14 polygons with SLs, we have to start
counting from the lowest C vertex, proceed to the central polygon, then count elements in
the lower set of seven polygons and, finally, elements in the upper set. It is truly
remarkable that exactly half the polyhedron should correspond to the seven planes of the
cosmic physical plane mapped by the 49tree and that the upper half should
correspond to the six cosmic superphysical planes mapped by the next 42 Trees of Life making
up CTOL. This is further, spectacular evidence that the polygons defined by the CC axis
constitute a map of CTOL — this time not in terms of their yods correlating with the yods on
the central Pillar of Equilibrium of CTOL but in terms of their geometrical elements
correlating with its SLs. Table 3 indicates that the eight lowest polygons in the polyhedron have
496 yods other than centres surrounding its centre when the lowest C vertex
is added. This is consistent with 496 being the number value of Malkuth,
whose cosmic manifestation is the cosmic physical plane with 299 SLs represented by these
polygons in the lower half of the polyhedron.
Lambda Tetractys pattern Let
us next consider just the 15 polygons and the outer form of the polyhedron,
ignoring the internal triangles of the latter. This means that the six internal edges of the
hexagon need no longer be considered shared. In this case, the 15 polygons
have 555 geometrical elements (75 nonpolyhedral vertices, 300 edges & 180 triangles),
whilst the polyhedron has 62 vertices, 180 edges and 120 triangles, i.e.,
362 geometrical elements. Hence, the polyhedron and its polygons have (362+555=917)
geometrical elements (137 vertices, 480 edges & 300 triangles). Remarkably, this is the
same as the number of yods on the axis and in the 15 polygons when their
sectors are each constructed from three tetractyses! 17 vertices lie on the CC axis, so
that 900 geometrical elements surround it. They comprise the 60 vertices of the polyhedron,
the 480 edges of their sectors and the 360 internal vertices & triangles. This
60:480:360 pattern is the division displayed by the Lambda Tetractys discussed in previous
articles (13):
It is convincing evidence that the polyhedron and its internal polygons
constitute an archetypal object of sacred geometry. Fittingly, the 60 polyhedral vertices
correspond to the number 6 at the centre of the tetractys, the yod symbolizing Malkuth, for
these shapedetermining vertices are the Malkuth aspect of the polyhedron. The same
36:54 pattern is exhibited in the disdyakis triacontahedron when its faces
and internal triangles are divided into their sectors, for there are, then, (120×3=360)
triangles in its faces and (180×3=540) internal triangles, totalling 900 triangles.
Musically, the ratio of these numbers: 540/360 = 3/2 defines the interval of a perfect
fifth. As Goethe said, “Geometry is frozen music.”
6. The polyhedral
basis of E_{8}×E_{8} in the 15 polygons The 550 SLs of CTOL
consist of the 496 SLs down to Chesed of the ninth Tree of Life and the
next 54 SLs. Identifying the former as being the counterpart of the 496
gauge bosons of E_{8}×E_{8}' enables us to correlate with the root structure
of this group the geometrical composition of the 15 polygons and the two C
vertices, which consist of 550 unshared elements surrounding the centre of the disdyakis
triacontahedron, as now explained. Below are displayed these geometrical elements:
14
We see that, apart from the 27 vertices in the seven upper or lower
polygons, there are 237 edges & triangles and three vertices in the central hexagon,
i.e., 240 unshared geometrical elements. They correspond to the 240 roots of E_{8}'.
The seven centres of the upper polygons and the top C vertex correspond to its eight simple
roots. The seven lower polygons and the other half of the hexagon have 240 unshared
geometrical elements corresponding to the 240 roots of the second group E_{8}. The
seven centres of the lower hexagons and the lowest C vertex correspond to the eight simple
roots of E_{8}. The reason for associating the upper half with the second group
E_{8}' rather than the first group E_{8} is that superstrings of shadow
matter are far less detectable than those of ordinary matter, so that it is apt that their
gauge symmetry group should be associated with the upper half of the polyhedron, which
corresponds to the cosmic superphysical region of CTOL.
What about the 54 vertices inside the sectors of the polygons? Together with
the lowest C vertex, they are the counterpart of the 55 SLs up to the 496th
SL from the top of CTOL. Counting down from the top C vertex and ignoring internal vertices,
the 248th geometrical element is the last of those in half the central
hexagon. It corresponds to Binah of the 50th Tree of Life specified by its
Godname, ELOHIM. The next 248 elements take us down to the lowest C vertex,
which corresponds to the 55th SL, Chesed of the ninth Tree of Life. As the lowest
26 Trees of Life in CTOL have a correspondence to the lowest
26 tree levels signifying the 26 dimensions of spacetime
predicted by quantum mechanics for bosonic strings, this SL corresponds to the ninth spatial
dimension. In other words, this parallel is indicating that the 496 gauge
bosons that correspond to the 496 SLs in CTOL down to that point manifest
in a spacetime with nine spatial dimensions, i.e., tendimensional superstring
spacetime. Malkuth — the physical universe — requires nine spatial dimensions, according to
its gematria number value, for both fermions and bosons to coexist.
The lowest three triangles have 72 edges & triangles
(45 edges, 27 triangles). This is the number of roots in its exceptional subgroup
E_{6}. The centres of the five triangles and a centre of one 6sided polygon
correspond to its six simple roots. The next four polygons have 168
vertices, edges & triangles that correspond to the 168 roots of
E_{6} that are not roots of E_{6}. They comprise 90 edges of these polygons
and 78 vertices, edges & triangles. These are the gematria number values of
Yesodeth and Cholem in the Hebrew name of the Mundane Chakra of Malkuth
(p. 2). Such matching between the numbers of geometrical elements and the values of
individual words strongly argues against coincidence, supposing that such an argument were
still needed. The 45 edges in the three lowest triangles and the 81 triangles in the seven
polygons correspond to the 126 roots of E_{7}. Its seven simple roots correspond to
the centres of the seven polygons. The lowest C vertex is the counterpart of the extra
simple root of E_{8}. Similar correspondences for the roots of E_{8}' apply
in the upper seven polygons.
Embodiment of the number of Tiphareth
(“Beauty”) The disdyakis triacontahedron has 180 edges, 120 triangles,
180 internal triangles and 60 internal edges surrounding its axis. Six internal edges are
internal edges of the central hexagon, which are shared with that of the polyhedron, so that
it has 534 unshared geometrical elements surrounding its axis. Other than the two edges
making up the axis, the polyhedron possesses 534 unshared elements surrounding its centre.
As we saw earlier, the 15 polygons have 548 unshared elements. The polygons
and polyhedron have (534+548=1082) unshared elements other than those forming its axis. This
is the 1081st integer after 1, which for the ancient Pythagoreans was not
an integer but the Monad, the source of all number, so that they would have considered the
number 1082 as really the 1081st integer. 1081 is the
gematria number value of Tiphareth, the Sephirah at the centre of the Tree of Life
(Table 1). The number value of the Hebrew word meaning “beauty” quantifies
the geometrical degrees of freedom of the polyhedron and its 15
polygons. This mathematical beauty is that of the polyhedral representation of the
divine archetype governing the form and structure of holistic systems such as CTOL and the
whorl of the E_{8}×E_{8}' heterotic superstring.
References 1.The Catalan solids
are the duals of the more wellknown Archimedean solids (polyhedra with two or more types of
regular polygons as faces), wherein the faces of the latter become vertices and their
vertices become faces.
2. Numbers in Table 1 that appear in the text are written in boldface.
3. Phillips, Stephen M. Article 22: “The disdyakis triacontahedron as the
threedimensional counterpart of the inner Tree of Life,” (WEB, PDF).
4. Phillips, Stephen M. Article 35: “The Tree of Life nature of the Sri Yantra
and some of its scientific meanings,” (WEB, PDF), Table 3, pp. 7, 8.
15
5. Ibid, p. 7.
6. Phillips, Stephen M. Article 5: “The superstring as microcosm of the spiritual
macrocosm,” (WEB, PDF), p. 2 et seq.
7. Ref. 4, p. 10.
8. Phillips, Stephen M. Article 1: “The Pythagorean nature of superstring and
bosonic string theories,” (WEB, PDF), p. 5.
9. Fig. 5 indicates that there are ten yods on the central pillar up to, and
including, Tiphareth of the 1tree. Then there are ten yods added in successive upper Faces.
Therefore, the number of yods on the central pillar of the (n–1)tree with (n–1) more Faces
= 10(n–1) + 10 = 10n. For n overlapping Trees of Life, there are six yods on this pillar
above the (n–1)tree. The total number of yods on the central pillar of n overlapping Trees
of Life = 10n +6.
10. Ten yods on the central pillar are added to the ten yods in the lowest Face
by each Upper Face of the next higher Tree of Life. The number of yods on the central pillar
of the ntree = 10n + 10.
11. There are 1370 yods in the (7+7) enfolded polygons of the inner Tree of Life
when their 94 sectors are divided into three tetractyses. These sectors have
80 vertices. Their division adds 94 vertices of tetractyses, so that the
total number of vertices is (80+94=174). Hence, there are (1370–174=1196)
hexagonal yods. A hexagonal yod is at the centre of each of the (94×3=282)
tetractyses. Therefore, there are (1196–282=914) hexagonal yods lining the
(914/2=457) edges of these tetractyses.
12 For more details about the significance of the number 251 visàvis CTOL, see
ref. 6.
13. Phillips, Stephen M. Article 30: “The equivalence of the triakis tetrahedron,
disdyakis triacontahedron and Plato’s ‘Lambda Tetractys,’” (WEB, PDF), p. 6.
16
