ARTICLE 23
by
Stephen M. Phillips
Flat 4, Oakwood House, 117119 West Hill Road. Bournemouth.
Dorset BH2 5PH. England.
Website: http://smphillips.mysite.com
Abstract
The 2dimensional (polygonal) form of the inner Tree of Life
consists of two identical sets of seven enfolded, regular polygons. Their
beautiful properties expressed in terms of the integers 1, 2, 3 & 4
symbolised by the Pythagorean tetractys are prescribed by the Godnames assigned
to the 10 Sephiroth of the outer Tree of Life. Constructed from 47 tetractyses,
the seven enfolded polygons contain 264 yods. 120 yods are on their boundaries,
inside which are 144 yods. The 120 yods shaping the inner form of the Tree of
Life denote the 120 faces of the disdyakis triacontahedron, the Catalan solid
with the most faces. The 144 internal yods symbolise the 144 faces of a
polyhedron (‘144 Polyhedron’) formed by attaching tetrahedra to the 48 faces of
the disdyakis dodecahedron, another Catalan solid. The interior and exterior of
the 144 Polyhedron are made up of 720 geometrical elements that surround an axis
passing through two opposite vertices. They are symbolised by the 720 yods that
surround the centres of the seven separate polygons when their sectors are
divided into three tetractyses. It is evidence of the Tree of Life nature of
this polyhedron. Constructed from tetractyses, 650 (=65×10) yods fill up the
faces of the 144 Polyhedron, showing how ADONAI, the Godname of Malkuth with
number value 65, prescribes its form. The sectors of its interior and exterior
triangles have 1370 edges. The number of yods in the (7+7) enfolded polygons is
1370. This is further evidence of the Tree of Life nature of the 144 Polyhedron.
It also demonstrates how the polyhedron embodies the number 137 determining the
finestructure constant known to physicists. There are also 1370 hexagonal yods
on the edges of the 540 tetractyses in each half of the 144 Polyhedron, as well
as 1370 yods lining tetractyses in its faces. The 360:540 division of vertices
& edges in each half conforms to the archetypal pattern of the Platonic
Lambda Tetractys The numbers of vertices surrounding the axes of the two
polyhedra, their edges and their triangles have the same proportion of 6:5. For
each polyhedron, the proportion of numbers of geometrical elements in their
exterior and interior is 3/2, the tone ratio of the perfect fifth in music.
Their 660 triangles have 138 vertices. 138 is the sum of the powers of 1, 2, 3
& 4 on the raised edges of the tetrahedral Lambda Tetractys. The assignment
of successive, odd integer multiples of 10 to the 22 Paths of the Tree of Life
generates the number of geometrical elements surrounding the axes of the two
polyhedra. The number of corners of the sectors of the 1400 polygons enfolded in
100 overlapping Trees of Life is the number of geometrical elements in the two
polyhedra. This demonstrates that they are the polyhedral version of the Tree of
Life, which can be differentiated into, firstly, 10 overlapping Trees of Life
and, secondly, 100 overlapping Trees of Life.

1
Table 5. Gematria number values of the 10 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.

2
(All numbers from this table that are referred to in the article are
written in boldface).
1. The outer
& inner Trees of Life
The outer form of the Tree of Life (Fig. 1) is a 3dimensional object consisting of 16 triangles with 10 corners
joined by 22 straight lines, where 10 = 1 + 2 + 3 + 4, 16 = 4^{2} and 22 =
1^{4} + 2^{3} + 3^{2} + 4^{1}. The number of vertices, sides
& triangles = 48, the number value of Kokab, the Mundane
Chakra of Hod (see Table 1).
As we shall see later, this is also a basic geometrical parameter of the
inner form of the Tree of Life (Fig. 2), which consists of two identical sets of
seven regular polygons — the triangle, square, pentagon, hexagon, octagon,
decagon and dodecagon. They are enfolded in one another so that they share a common edge called
the ‘root edge.’ When the 16 triangles of the outer Tree of Life are transformed into
tetractyses, they contain 70 yods.
They denote degrees of freedom whose counterparts in the inner Tree of Life
are the 70 corners of the 14 enfolded polygons (Fig. 3).
3
The outer and inner forms of the Tree of Life are analogous, respectively,
to the human cell and its nucleus containing the DNA that encodes its replication. They both
embody ‘bits’ of information of various kinds in the form of the different types of
yods in the tetractyses from which they can be
constructed. The yods on the boundaries of the seven separate polygons
become
yods on the boundaries of the seven enfolded polygons, i.e., 24 (=1×2×3×4)
such yods disappear in the enfolding (Fig. 4). The seven centres of the separate polygons and the 144 internal yods
that surround them become 144 yods inside the boundaries of the enfolded polygons, that is,
seven internal yods disappear in the enfolding. (24+7=31) yods disappear in
the enfolding of the seven polygons. 31 is the number value of EL (“God”),
the Godname of
Chesed (Table 1), showing how it prescribes the properties of the inner Tree of Life.
The 96 corners of both sets of separate polygons become the 70 corners of the 14 enfolded
polygons, that is, 26 corners disappear in the enfolding. According to
Table 1, 26 is the number value of YAHWEH, the Godname of
Chokmah, which is the Sephirah above Chesed on the Pillar of Mercy. There are
288 = 1^{1} + 2^{2} + 3^{3} + 4^{4} =
1!×2!×3!×4!
=


3 5
7 9
11 13
15
17
19 21
23 25
27 29
31
33 
(33=1!+2!+3!+4!) 
yods inside the two sets of enfolded polygons.^{1} This illustrates how the properties of the inner Tree of Life are
expressed in terms of the Pythagorean integers 1, 2, 3 & 4.
The number of yods in the seven enfolded polygons = 144 + 120 = 264. 260
yods are outside the root edge with four yods. This is the number of yods in
26 tetractyses. It is powerful evidence of how the creative archetype embodied
in YAHWEH prescribes the yod population of the inner form of the Tree of Life (see the author’s
Article 4^{2} for how other Godnames of the Sephiroth prescribe the inner Tree of
Life). The 120 boundary yods comprise 36 corners, where 36
is the number value of ELOHA, the Godname of
4
Table 2. Properties of the Archimedean & Catalan solids.
5
Geburah, and 84 hexagonal yods, where
36 is the sum of the first four even integers and
the first four odd integers, again illustrating how the Tetrad Principle formulated in
Article 1^{3} defines properties of the Tree of Life.
We will now ascertain what these 264 yods signify by investigating a class
of polyhedra called the Archimedean solids. There are 13 of these solids^{1}. Two — the snub cube and the snub dodecahedron — exist in two
chiral forms, so that, if their enantiomorphs are included, there are 15
Archimedean solids.
This shows how the Divine Name YAH with number value 15
(see Table 1) prescribes the number of such solids. Their duals, generated by
interchanging vertices and faces, are called the Catalan solids. The 26
basic Archimedean & Catalan solids are prescribed by YAHWEH with number value
26. Table 2 lists the numbers of vertices, edges & faces in these solids. It
indicates that the last Archimedean solid (truncated icosidodecahedron) has 120
vertices.
_______________________________
^{1} An Archimedean solid is a convex polyhedron having at least two
different regular polygons as faces.
6
However, none of the 30 solids has 144 vertices. Moreover, if we placed
pyramids on their faces, creating a new vertex for each face , none could create a polyhedron
with 144 vertices because inspection of Table 2 reveals there is no Archimedean or Catalan
solid with C + F = 144. This means that the yods cannot symbolise vertices. Can they symbolise
faces? The only member of the family of Archimedean & Catalan solids with 120 faces is the
disdyakis triacontahedron. There is no Archimedean or Catalan solid with 144 faces. However,
there is one, and only one, polyhedron with F triangular faces, where 3F = 144, i.e., F =
48, namely, the disdyakis dodecahedron with 26 vertices and
72 edges. If a tetrahedron is stuck on each of its faces, a polyhedron is
created that has (48+26=74) vertices, (72 +
3×48 = 216) edges and 144 faces (Fig. 6). In the case of Archimedean solids with some nontriangular faces,
there are only two other possible candidates possessing 144 triangular faces when pyramids
are attached to them with bases that match the shapes of their faces: the truncated
cuboctahedron and the cuboctahedron (Table 3). There are 144 sectors in the 26 faces of the
former that become triangular faces when pyramids are attached to them (Fig. 7). The latter has 48 sectors which become 144 faces
when square pyramids and tetrahedra are attached
Table 3. Numbers of sectors in faces of the Archimedean solids.
Archimedean solid

Types of faces

Number of sectors

cuboctahedron 
8 triangles + 6 squares 
8×3 + 6×4 = 48 
icosidodecahedron 
20 triangles + 12 pentagons 
20×3 + 12×5 = 120 
truncated tetrahedron 
4 triangles + 4 hexagons 
4×3 + 4×6 = 36 
truncated cube 
8 triangles + 6 octagons 
8×3 +6×8 = 72 
truncated octahedron 
6 squares + 8 hexagons 
6×4 + 8×6 = 72 
truncated dodecahedron 
20 triangles +12 decagons 
20×3 + 12×10 = 180 
truncated icosahedron 
12 pentagons + 20 hexagons 
12×5 + 20×6 =180 
rhombicuboctahedron 
8 triangles + 18 squares 
8×3 + 18×4 = 96 
truncated cuboctahedron 
12 squares + 8 hexagons + 6
octagons 
12×4 + 8×6 + 6×8 = 144 
rhombicosidodecahedron 
20 triangles + 30 squares +
12 pentagons 
20×3 + 30×4 + 12×5 = 240 
truncated
icosidodecahedron 
30 squares + 20 hexagons +
12 decagons 
30×4 + 20×6 + 12×10 = 360 
snub cube 
32 triangles + 6 squares 
32×3 + 6×4 = 120 
snub dodecahedron 
80 triangles + 12 pentagons 
80×3 + 12×5 = 300 
7
to them. So there are two Archimedean solids and one Catalan solid which can
potentially create a polyhedron with 144 triangular faces (henceforth to be referred to as the
‘144 Polyhedron’). The latter provides the simplest transformation, but one needs stronger
reasons than that to regard it as the correct basis for the 144 Polyhedron. The 144 yods inside
the boundaries of the seven enfolded polygons signify its faces, but their distribution
provides no obvious way of differentiating between the three candidates. In order to regard the
disdyakis triacontahedron as the 26th one prescribed by YAHWEH with number
value 26, counting has to start with the truncated tetrahedron, the simplest
Archimedean solid, whether we count down the list of each family or proceed by taking in turn
each pair of solids that are dual to each other. Including the five Platonic solids makes the
disdyakis triacontahedron the 31st one prescribed by EL with number value
31, whilst it is the 15th in the complete family of Catalan
solids (chiral partners included) and the 26th in the family of Archimedean
& Catalan solid, when chiral partners are excluded. Including the Platonic solids makes the
disdyakis dodecahedron the 21st one, counting pairs of dual polyhedra at a
time and the 21st one within the set of 26 Archimedean
& Catalan solids, when the former are first counted. In the latter case, it becomes the
26th one when one starts counting with the Platonic solids. It is highly
significant that the disdyakis dodecahedron and the disdyakis triacontahedron are picked out by
three Godnames. It means that the 144 Polyhedron must be seen as generated from the disdyakis
dodecahedron, not from either the cuboctahedron or the truncated cuboctahedron, even though
this is geometrically possible. Its properties will now be examined.
2. The 144 Polyhedron
Formed from the disdyakis dodecahedron with 26 vertices,
72 edges & 48 faces by attaching tetrahedra to
its faces, it has (26+48=74) vertices,
(72 + 3×48 = 216) edges and
(3×48=144) triangular faces. 72 is the number value of
Chesed and 216 is the number value of Geburah, the next Sephirah after
Chesed. Euler’s formula relates the C vertices, E edges & F faces of a convex
polyhedron:
C – E + F = 2.
As expected, the 144 Polyhedron obeys this equation because
74 – 216 + 144 = 2.
Starting with the triakis tetrahedron, the simplest Catalan solid, and
counting pairs of dual polyhedra downwards in a zigzag fashion in Table 2, the disdyakis dodecahedron is the 15th, so that it
is prescribed by YAH, the older Godname assigned to Chokmah with number value
15. The complete Godname YAHWEH prescribes the 144 Polyhedron because it
possesses (216+144=260=26×10) edges & triangles. It
has (74+216+144=434) vertices, edges & triangles, that is, 432
geometrical elements surround its axis. They comprise 216 elements and
their 216 mirror images. The 144 Polyhedron has
(216+144=360=36×10) edges & triangles, where
36 is the number value of ELOHA, the Godname of Geburah.
72 vertices and (72+144=216)
vertices& triangles surround its axis, where 72 is the number
value of Chesed and 216 is the number value of Geburah. 288 vertices
& edges surround the axis, where
288 = 1^{1} + 2^{2} + 3^{3} +
4^{4}.
This shows the beautiful way in which the integers 1, 2, 3 & 4
symbolised by the four rows of the Pythagorean tetractys express properties of holistic
systems.
A polyhedron with triangular faces constructed from tetractyses has (C+2E+F)
yods in its faces. The 144 Polyhedron has 650 (=65×10) such yods, where
8
We see that ADONAI, the Godname of Malkuth with number value
65, appropriately prescribes the outer form of one of the two
polyhedra that are the counterparts of the inner Tree of Life. The number of yods needed to
create its 144 faces is the sum of the Godnames of the first four Sephiroth. It is
also the
sum of the squares of the first 12 squares arranged along the edges of a
square, a symbol of the Tetrad. The Decad defines the number 650 because 650 = 10×sum of first
10 integers after 1. The polyhedron has (650–74=576) yods in its faces other than vertices,
where 576 = 1^{2}×2^{2}×3^{2}×4^{2}, showing again how the
integers 1, 2, 3 & 4 express its properties.
There are 74 edges of 216 triangles created inside the
polyhedron when all its vertices are joined to its centre. The geometrical composition of its
exterior and interior is:

Vertices

Edges

Triangles

Total

Exterior: 
74

216

144

434

Interior: 
1

74

216

291

Total = 
75

290

360

725

288 edges and 288 internal edges & triangles surround its axis, which
comprises three vertices and two edges, i.e., five geometrical elements. (725–5=720)
geometrical elements surround the axis made up of
(72+216=288) vertices & edges of faces and 432 triangles
& internal; edges. Compare this with the fact that 288 yods surround the centres of the
seven separate polygons of the inner Tree of Life when each of their 48
9
sectors is a tetractys and 720 yods surround their centres when each sector
is divided into three tetractyses, i.e., 432 yods are added (Fig. 8). The seven centres are the counterparts of the axis of the 144
Polyhedron. The 720 geometrical elements have the composition:

Vertices

Edges

Triangles

Total

Exterior: 
72

216

144

432

Interior: 
0

72

216

290

Total = 
72

288

360

720

The 72 vertices comprise the 48
peaks of the tetrahedra stuck on the faces of the disdyakis dodecahedron and the 24 of its
26 vertices that surround an axis passing through two of them. Their
counterpart in the seven polygons are the 72 yods surrounding the centre of
the dodecagon, which comprise 24 corners & centres of tetractyses and 48
hexagonal yods on their edges. The counterparts of the 216 edges of the 144
Polyhedron are the 216 yods
surrounding the centres of the first six polygons. They comprise
72 vertices & centres of their 36 tetractyses and 144
hexagonal yods on the 72 edges. They correspond to the 72
edges of the disdyakis dodecahedron and to the 144 edges added by attaching tetrahedra to its
48 faces. The 360 triangles consist of (48×3=144) external
triangles and 216 internal ones. Their counterparts in the seven polygons with
three tetractyses as each sector are the (48×3=144) hexagonal yods at centres
of tetractyses and the 216 hexagonal yods on internal edges of tetractyses in
the first six polygons. The 72 internal edges correspond to the
72 hexagonal yods on internal edges of tetractyses in the dodecagon.
Such detailed correlation is remarkable evidence that the polyhedron whose
144 faces correspond to the 144 yods inside the boundaries of the seven enfolded polygons
embodies the archetypal pattern not of the inner Tree of Life but of its separate
polygons before they become organised into the former. This is a vital point, because the
disdyakis triacontahedron is the polyhedral counterpart of these polygons when they are
enfolded in one another (Fig. 1). The enfoldment reduces the 295 yods of the seven separate polygons
to 264 yods, i.e., the Godname EL with number value 31 prescribes the
31 yods that disappear. As four of these yods are located at Tiphareth,
Chokmah,
10
Chesed & Netzach (or, rather, their projections onto the plane of the
polygons), 260 (=26×10) yods are unshared. 260 yods are also outside the root
edge. This is how YAHWEH with number value 26 prescribes the inner form of the
Tree of Life.
There are two hexagonal yods on each of the 74 internal edges of the 144
Polyhedron, one hexagonal yod at the centre of each internal triangle and one yod at the centre
of the polyhedron. The number of internal yods = 74×2 + 216 + 1 = 365. There
are 360 (=36×10) internal yods surrounding its axis. The Godname ELOHA
prescribes both the 360 triangles that make up the polyhedron and the number of internal yods
surrounding its axis. The total number of yods = 365 + 650 = 1015, i.e., 1008 yods surround the
seven yods on its axis. Each half of the 144 Polyhedron contains 504 such yods. This is the
number of yods surrounding the centre of a heptagon with 2ndorder tetractyses as sectors
(Fig. 9a). It is also the number of yods surrounding the centre of a dodecagon
whose sectors are constructed from nine tetractyses (Fig. 9b). The Decad determines the 720 vertices, edges & triangles
surrounding the axis of the 144 Polyhedron because a decagon has 720 yods surrounding its
centre when its sectors are 2ndorder tetractyses (Fig. 10). The Decad determines its yod population because the dodecagon is
the tenth regular polygon. The two halves of the polyhedron are the polyhedral
counterpart of the two dodecagons. The significance of this is that the dodecagon is itself
equivalent to the outer Tree of Life. Both contain 60 hexagonal yods when their triangles or
sectors are turned into tetractyses (Fig. 11). They symbolise the 60 vertices that surround the axis of the
disdyakis triacontahedron — the minimal number of points distributed in space that create
its form. As the polygonal Tree of Life, the dodecagon has the same number of yods
surrounding its centre, when its sectors are each divided into three sectors and
these sectors further divided into three tetractyses, as
each half of the 144 Polyhedron has surrounding its axis. It demonstrates
how the polyhedron has a structure analogous to the pair of dodecagons in the inner Tree of
Life. Indeed, so does the disdyakis triacontahedron, for the (60+60=120) hexagonal yods in a
pair of dodecagons symbolise its (60+60) faces, one set of 60 faces being the mirror images of
the other set.
11
12
These properties are powerful pieces of evidence supporting the claim made
in Articles 22 that the disdyakis triacontahedron with 120 faces is the polyhedral
manifestation of the exterior of the 2dimensional, inner form of the Tree of Life, as
indicated by the 120 yods on its boundaries. The 144 Polyhedron generated from the disdyakis
dodecahedron is the polyhedral manifestation of its interior, the 144 internal yods of which
are the counterpart of its 144 triangular faces.
3. Triangulation of the 144 Polyhedron Instead
of regarding the internal triangles and faces of the 144 Polyhedron as single triangles,
let us divide them into their sectors and then turn each sector into a tetractys. In
other words, we shall construct the polyhedron from triangles and then turn each one into
a tetractys. The geometrical elements making up a polyhedron with C vertices, E edges and
F triangular faces are:

Vertices

Edges

Triangles

Total

Exterior: 
C+F

E+3F

3F

C+E+7F

Interior: 
E+1

C+3E

3E

C+7E+1

Total = 
C+F+E+1

C+4E+3F

3(F+E)

2C+8E+7F+1

Using Euler’s polyhedral formula for a convex polyhedron:
C − E + F = 2,
the number (N) of geometrical elements can be written as:
N = 2C + 8E + 7F + 1 = 10E + 5F + 5 = 15E – 5C +
15 = 10C + 15F – 15.
Notice that the number value 15 of the Godname YAH is both
a proportionality constant and an additive constant in the expression in terms of two
geometrical elements.
Table 4 displays the numbers for the 144 Polyhedron with C = 74, E =
216 and F = 144:
Table 4. Geometrical composition of the 144 Polyhedron.

Vertices

Edges

Triangles

Total

Exterior: 
218

648

432

1298

Interior: 
217

722

648

1587

Total = 
435

1370

1080

2885

There are 216 vertices in the faces surrounding an axis and
216 internal vertices, where 216 (=6^{3}) is the
number value of Geburah, the sixth Sephirah. There are also 216
triangles in the 72 faces of each half of the polyhedron. 72
is the number value of Chesed, the preceding Sephirah. 360 (=36×10)
vertices are not vertices of the polyhedral, where 36 is the number value of
ELOHA, the Godname of Geburah. 2880 geometrical elements surround the axis made up of five
elements. It is expressed by the Tetrad because 2880 =
(1+2+3+4)×(1^{1}+2^{2}+3^{3}+4^{4}). This is the number of yods
surrounding the centres of the 10 sets of seven separate polygons associated with 10
overlapping Trees of Life, each a representation of a Sephirah. The 1080 tetractyses have 1370
edges. This is remarkable, because the (7+7) enfolded polygons with their 94 sectors divided
into three tetractyses contain 1370 yods^{4} (Fig. 12). It demonstrates that the polyhedron possesses a Tree of Life
structure. 1370 = (1+2+3+4)×137, where 137 is the 33rd prime number and 33 = 1! + 2! + 3! +
4!. This is how the polyhedron embodies the number 137. It is known to physicists as one of
the most important numbers in Nature because its reciprocal is approximately equal to the
fine structure constant e^{2}/ħc ≈ 1/137. This sets the energy scale of atomic
physics because it measures the strength of the coupling of the electron to the
electromagnetic field.
1800 vertices & edges surround its axis, 900 in each half. 900 =
30^{2} = (1^{2}+2^{2}+3^{2}+4^{2})^{2}.
13
The 900 vertices & edges comprise 360 internal edges and 540 vertices
& external edges. This composition conforms to the 36:54 division
displayed by the 10 numbers in the Lambda Tetractys^{5}:
This is highly significant, as it shows the archetypal nature of the 144
Polyhedron paired with the disdyakis triacontahedron as, respectively, the ‘yang’ (male) and
‘yin’ (female) components of the polyhedral Tree of Life. There are 650
(=65×10) vertices & triangles in the faces, showing how ADONAI prescribes
the form of the polyhedron.
Now suppose that each triangular sector is a tetractys. A polyhedron has the
yod composition:

Vertices

Hexagonal yods
on edges

Hexagonal yods
at centres

Total

Exterior: 
C+F 
2(E+3F)

3F

C+2E+10F

Interior: 
E+1 
2(C+3E) 
3E

2C+10E+1

Total = 
C+E+F+1 
2C+8E+6F

3(E+F)

3C+12E+10F+1

Using Euler’s equation, the number (n) of yods can be written as:
n = 3C + 12E + 10F + 1 = 15E + 7F + 7 = 15C + 22F – 23 = 22E
– 7C + 21.
Notice the presence of the number values of YAH (15) and
EHYEH (21). The number of yods surrounding the axis, which has seven yods on
it, is 15E + 7F. Table 5 lists the various yod populations for the 144 Polyhedron:
Table 5. Yod population of the 144 Polyhedron.

Vertices

Hexagonal
yods on edges

Triangles

Total

Exterior: 
218

1296

432

1946

Interior: 
217

1444

648

2309

Total = 
435

2740

1080

4255

There are 1370 hexagonal yods lying on the edges of the 540 tetractyses in
each half of the polyhedron. There are also (218+1296–144=1370) yods on edges of tetractyses in
its faces other than their centres. The number of yods on the boundaries of the 1008
tetractyses = 435 + 2740 = 3175. There are 3100 (=31×10×10) boundary yods
except for the centre of the polyhedron and its 74 vertices. This is a remarkable demonstration
of how EL, the Godname of Chesed with number value 31, prescribes the number
of yods other than vertices needed to shape the 1370 edges of its tetractyses.
4. The polyhedral Tree of Life The geometrical
composition of the faces of the disdyakis triacontahedron and the 144 Polyhedron is:

C 
E 
F 
Total 
Disdyakis triacontahedron:

62

180

120

362

144 Polyhedron: 
74

216

144

434

Total = 
136

396

264

796

14
The number of vertices = 136. This is the sum of a 4×4 array of the first 16
integers:
It demonstrates how the Tetrad Principle determines a parameter of both
polyhedra. If one imagines the 144 Polyhedron inside the disdyakis triacontahedron and
sharing the same centre, their internal and external triangles will have 137 vertices.
They embody the finestructure number 137. Including the four edges of internal triangles
that define their axes, there are (796+4=800=80×10) geometrical elements
(400 vertices & triangles, 400 edges) surrounding their centres. 80
is the number value of Yesod (“Foundation”), the penultimate Sephirah of the Tree of Life.
It is the number of yods in the 1tree — the lowest of any set of overlapping Trees of
Life (Fig. 13). Assigning the Decad to each yod generates the number of
geometrical elements in the faces and axes of the two polyhedra that surround their centres.
It is more evidence of their Tree of Life nature.
The numbers of geometrical elements in the faces surrounding each axis
are:

C

E

F

Total

Disdyakis triacontahedron:

60

180

120

360

144 Polyhedron: 
72

216

144

432

Total = 
132

396

264

792

Notice that the numbers of vertices, edges & triangles bear the same
proportion of 6/5:
72/60 = 216/180 = 144/120 = 6/5.
The integers 6 & 5 appear in the number value 65 of
ADONAI, the Godname of Malkuth! Notice also that for each polyhedron (and therefore for both)
the number of edges is three times the number of vertices surrounding the axis, whilst the
number of triangles is twice the latter. Hence, if the numbers
of vertices that surround an axis in the disdyakis triacontahedron and 144
Polyhedron are written, respectively, as ‘X’ and ‘Y’, the seven possible combinations of
geometrical elements number nX and nY, where n =1, 2, 3, 4, 5, 6 and Y= 6X/5. If the integers
1, 2 & 3 are assigned to the corners of a triangle, not only do the latter denote the three
types of geometrical elements in the two polyhedra but also all combinations of the integers
generate the numbers of all combinations of these elements.
Including the interior geometrical elements, the numbers of elements
surrounding each axis are:
15

Vertices

Edges

Triangles

Total

Disdyakis triacontahedron 




exterior: 
60

180

120

240

interior: 
0 
60 
180 
360 
Subtotal = 
60

240

300

600

144 Polyhedron 




exterior: 
72

216

144

432

interior:

0

72

216

288

Subtotal = 
72

288

360

720

Total = 
132

528

660

1320

Notice that, for either polyhedron, the total number of geometrical elements
is 10X or 10Y, dividing as 6X:4X or 6Y:4Y, that is, as 3:2, for the relative proportion of
external and internal elements. The ratio is the tone ratio of the perfect fifth in the
Pythagorean musical scale — the note whose pitch is the arithmetic mean of those of
the first and last notes.
Including the 10 elements in the axes of the two polyhedra, there are 1330
geometrical elements (138 vertices, 532 edges & 660 triangles). There are 670
(=67×10) vertices & edges, showing how the number value
67 of Binah determines how many points and lines make up the two polyhedra.
Below Binah in the 1tree are 67 yods (Fig. 15). This property illustrates the meaning of Binah as the source of
archetypes governing the form of holistic systems that embody the divine paradigm. The
yod is the tenth letter of the Hebrew alphabet. The points of the
tetractys should be called yods because they denote the number 10, not simply
because the yod has the approximate appearance of a dot. Assigning the Decad to each yod
below Binah generates the number of
vertices & lines needed to construct the disdyakis triacontahedron and
the 144 Polyhedron. Amazingly, 67 is the 19th prime number, where 19 is the
tenth odd integer and the number of yods in the triangle when its sectors are
tetractyses (Fig. 16). Hence, the
16
simplest, geometrical figure determines numerically the geometrical
composition of the polyhedral Tree of Life. 132 vertices surround the axes of both polyhedra,
66 in their separate halves. 66 is the 65th integer after 1, showing how the
Godname ADONAI with number value 65 prescribes the pair of polyhedra. 134
vertices surround their centres, 67 in both halves. This shows again how the
number value 67 of Binah determines the shapes of the two polyhedra. The
number of yods below Binah in the 1tree is the number of points in 3dimensional space needed
to construct half of each polyhedron, starting with another point. It is the minimal
number because every vertex in these polyhedra has a mirror image, so that the other halves are
their mirror images.
1330 is the sum of the squares of the first ten odd integers ending in
19:
The Decad therefore determines both the number 670 and the number 1330. This
is a spectacular example of how the Decad defines properties of holistic systems (in this case,
the polyhedral Tree of Life). ‘1^{2}’ denotes the centre of the 144 Polyhedron, the sum
(530) of 13^{2} and 19^{2} at the two other corners of this tetractys array is
the number of internal geometrical elements in both polyhedra surrounding their axes plus the
two edges on the axis of the 144 Polyhedron:
13^{2} + 19^{2} = 530 = 240 + 288 + 2,
and the sum (799) of the seven other squares is the number of elements in
their faces plus the centre and two edges on the axis of the disdyakis triacontahedron:
3^{2} + 5^{2} + 7^{2} + 9^{2} +
11^{2} + 15^{2} + 17^{2} = 799 = 362 + 434 + 1 +
2.
Additional evidence that the pair of polyhedra constitute an archetypal
system is the fact that the sum of the ten powers of 1, 2, 3 & 4 on three edges of the
tetrahedral Lambda Tetractys^{6} is 138 (Fig. 17):
This number is the sum of the least number of integers in the tetrahedral
array of 20 integers that are needed to generate the complete set, for the 10 other integers
are created by extrapolation of them, just as the Lambda Tetractys is the extrapolation of
17
the seven numbers lining Plato’s Lambda. The minimal character of the number
138 is illustrated in the 138 vertices in the 660 internal and external triangles
making up the 144 Polyhedron and the disdyakis triacontahedron. The 1:137 differentiation
between the integer 1 at the ‘peak’ of the tetrahedral ‘mountain’ and the integers on its
slopes correspond to the centre of the 144 Polyhedron (the geometrical starting point of the
generation of the polyhedral Tree of Life) and the 137 vertices that create it.
The (7+7) enfolded polygons of the inner Tree of Life have 94 sectors with
80 corners. Six of them coincide with the positions of the Sephiroth on the side
pillars, leaving 74 corners (denoted by the red dots in Fig. 18) that are unshared. When the 16 triangles of the Tree of Life are
turned into tetractyses, there are 70 yods, including the six shared yods. This leaves
64 unshared (blue) yods. The inner and outer Trees of Life have (74+64=138)
yods or corners that are unshared with each other. These independent points are the Tree of
Life basis for the 138 vertices in the two polyhedra.
5. Triangulation of the polyhedral Tree of
Life Tabulated below is the geometrical composition of the two
polyhedra when their faces and internal triangles are divided into their sectors

Vertices

Edges

Triangles

Total

Disdyakis triacontahedron 




exterior: 
60+120+2=180+2

180+360=540

3×120=360

1080+2

interior: 
180+1

60+540+2=600+2

540

1320+3

Subtotal = 
360+3

1140+2

900

2400+5

144 Polyhedron 




exterior: 
72+144+2=216+2

216+432=648

3×144=432

1296+2

interior: 
216+1

72+648+2=720+2

3×216=648

1584+3

Subtotal = 
432+3

1368+2

1080

2880+5

Total = 
792+6

2508+4

1980

5280+10

(‘1’ denotes the centre and ‘2’ denotes either the top & bottom
vertices or the two edges forming the axis).
Disdyakis
triacontahedron Surrounding the axis of the disdyakis
triacontahedron are 360 (=36×10) corners of 90 triangles. This
shows how the Godname ELOHA with number
value 36 prescribes this polyhedron. Including the axis, it
is made up of 363 vertices. 363 is the number value of
SHADDAI EL CHAI, the complete Godname of Yesod. 181 vertices in each half of the disdyakis
triacontahedron surround its centre. The counterpart of this in the inner Tree of Life is
18
the 181 yods in each dodecagon with its sectors divided into three
tetractyses (Fig. 19). Their centres denote the top and bottom vertices, the 60 yods
lying on the edges of the 12 sectors denote the centres of the 60 faces in each half of the
polyhedron and the 120 other yods in each dodecagon denote the 30 polyhedral vertices and 90
vertices of the 270 triangles in each half. Such exact matching indicates that the disdyakis
triacontahedron is the Malkuth (outer, physical) aspect of the polyhedral Tree of Life. That
is why it is symbolised by the last of the enfolded polygons.
The 900 triangles in this polyhedron are divided into the 360 triangles in
its faces and 540 internal triangles. This pattern is the polyhedral manifestation of the
Lambda Tetractys whose 10 numbers add up to 90. The sum of the integers at its corners is
36 and the sum of the seven other integers is 54. We saw
earlier that the 144 Polyhedron has 900 vertices & edges in each half
that surround its centre that comprise 360 internal edges and 540 vertices & external
edges. The archetypal pattern manifests in both polyhedra. 1500 (=15×10×10)
vertices & edges surround the axis of the disdyakis triacontahedron. This shows how YAH
with number value 15 prescribes it. 2400 vertices, edges & triangles
surround its axis. The counterpart of this property in the inner Tree of Life is the 240
hexagonal yods in the seven separate polygons (Fig. 21). Its counterpart in the 1tree is the 240 yods other than Sephiroth
that its 19 triangles contain when their sectors are tetractyses (Fig. 20). Its significance in the context of superstring theory will be
discussed in more detail in later articles. What is being encoded here is the 240 nonzero
roots of E_{8}, the largest exceptional Lie group that is part of the mathematics of
superstring theory.
The number of vertices & triangles surrounding the axis is
1260. This is the number value of Tarshishim, the Order of Angels
assigned to Netzach.
144 Polyhedron We commented
on its properties in Section 3. Here are some additional features. 216
vertices surround its axis in each half. 216 vertices are in its faces and
216 vertices are internal. 1296 vertices, edges & triangles surround its
axis composed of three vertices and two edges. Hence, there are 1300 geometrical elements
either on its faces or in its axis surrounding its centre, where
1300 = 1^{5} + 2^{5} + 3^{5} + 4^{5}.
19
This is a beautiful illustration of how the integers 1, 2, 3 & 4 express
properties of holistic systems. 1300 = 26×50, where
26 is the number value of YAHWEH and 50 is the number value
of ELOHIM, the Godnames of the Sephiroth at the top of the pillars of Mercy and Judgement. 650
(=65×10) such elements are in each half, showing how ADONAI with number value
65 prescribes the geometrical composition of each half of the 144 Polyhedron.
As 650 = 26^{2} – 26, it is the sum of the first 25
even integers up to 50. This means that the number 1300 is the sum of
50 even integers. This shows how ELOHIM prescribes the 144 Polyhedron.
144 Polyhedron & disdyakis
triacontahedron 3300 vertices & edges (1500 external, 1800 internal)
surround their axes. YAH with number value 15 prescribes both polyhedra
because 1500 = 15×10×10. The number 33, which is the 16th odd integer after 1,
where 16 = 4^{2}, is the sum of the number of permutations of 10 objects arranged in a
tetractys: 33 = 1! + 2! + 3! + 4!. Hence:
3300 =
(1!+2!+3!+4!)(1^{3}+2^{3}+3^{3}+4^{3}).
This is another example of the expression of parameters of holistic systems
by the integers 1, 2, 3 & 4. As 3300 = 60×55, where
55 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10,
this number can be represented by assigning the tenth triangular
number 55 to the 60 yods surrounding the centre of a decagon (Fig. 22). It demonstrates
how the Decad determines the geometrical composition of the polyhedral Tree
of Life.
1650 vertices & edges in half of the two polyhedra surround their axes,
where
and
165 = 1^{2} + 3^{2} + 5^{2} + 7^{2} +
9^{2}.
We see that the number 1650 is the sum of (10×5=50) squares
of the first five odd integers. ELOHIM with number value 50 arithmetically
prescribes the number of vertices & edges in half of the polyhedral Tree of Life. In fact,
ELOHIM prescribes the
20
number of vertices & edges in both halves because 3300 =
10×330, where
330 = 6^{2} + 7^{2} + 8^{2} + 9^{2} +
10^{2},
which means that 3300 is the sum of 50 squares.
The disdyakis triacontahedron with 120 faces has 1200 (=120×10) geometrical
elements in each half surrounding its axis. The 144 Polyhedron has 1440 (=144×10) such
geometrical elements. This reproduces the boundary and interior of the seven enfolded polygons
with the Decad assigned to each of their 264 yods (see Fig. 5). This is an amazing property that demonstrates their Tree of Life
character. We saw in Section 3 during the discussion of the triangulation of the 144
Polyhedron that the number of geometrical elements in a polyhedron with C vertices, E edges
and F faces when its triangular faces and internal triangles are divided into their sectors
is N = 10E + 5F + 5. There are (10E+5F) elements surrounding its axis, i.e., (5E+5F/2)
elements in each half. The condition for this to be equal to 10F is E/F = 3/2. In other
words, the ratio of the number of edges to the number of faces is the tone ratio of the
perfect fifth. This property is found for the tetrahedron, octahedron, icosahedron and
seven Catalan solids, but not for any Archimedean solids (see Table 2). The only Catalan solid that has this property and possesses 120
faces is the disdyakis dodecahedron. The Decad determines this number because
120 = 11^{2} – 1 = 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19,
i.e., it is the sum of the first ten odd integers after 1. The
Decad determines the number 144 because 144 = 12^{2}, where 12 is the number of corners
of the dodecagon, the tenth regular polygon, i.e., 12 is the tenth integer
that can be represented by a shape.
There are 5280 geometrical elements surrounding the axes of the disdyakis
triacontahedron and the 144 Polyhedron. As 528 = 24×22 = 23^{2} – 1 = sum of the first
22 odd integers after 1, 5280 =10×sum of the first 22 odd integers after 1. This relates the
geometry of the polyhedral Tree of Life to the traditional Tree of Life with 10 Sephiroth and
22 Paths, to which the letters of the Hebrew alphabet are assigned in Kabbalah. In
21
terms of the integers, 1, 2, 3 & 4 symbolised by the four rows of the
Pythagorean tetractys, 5280 = 10×24×22 =
(1+2+3+4)(1×2×3×4)(1^{4}+2^{3}+3^{2}+4^{1}).
Including the 10 geometrical elements on their axes, the number of
geometrical elements is 5290. This is 10×23^{2}. It is therefore determined by the 22nd
integer after 1. Again, the number 22 is primary. As 288 = 17^{2} – 1 = 3 + 5 + 7 + … +
33, 2880 is ten times the sum of the first 16 odd integers after 1. This is the number of
geometrical elements surrounding the axis of the 144 Polyhedron. Therefore, the number of
elements surrounding the axis of the disdyakis triacontahedron is the sum of the last six odd
integers in:
5290 = 10×23^{2} = 10[1 + (3 + 5 + 7 +…+ 33) + 35 + 37 + 39 + 41 + 43
+ 45],
i.e.,
2880 = 30 + 50 + 70 + … + 330
and
2400 = 350 + 370 + 390 + 410 + 430 + 450.
Assigning the integer 1 to each Sephirah and successive odd integer
multiples of 10 to the Paths of the Tree of Life (Fig. 23), we find that the odd integers assigned to the first 16 Paths add
up to 2880 and that the next six odd integers assigned to the last six Paths add up to 2400.
The latter are the edges of a tetrahedron — the simplest Platonic solid and the starting
point of solid geometry. The 16:6 differentiation of Paths corresponds to the difference
between the geometrical compositions of the 144 Polyhedron and the disdyakis triacontahedron
— or more accurately, their metaphysical meaning in representing the bipolar, divine,
creative process (in Tantra, the union of Shiva and Shakti to create the phenomenal world).
The former represent the polyhedral version of the upper part of the Tree of Life — its
superphysical aspect — before the emergence in Malkuth of threedimensional form. This
division appears in string theory, which predicts the existence of 22 higher dimensions of
space, it marks the division between the 16 dimensions whose compactification generates the
superstring with a further six compactified dimensions. In view of this, it is not a
coincidence that the number 2400 is the number of spacetime components of the 240
10dimensional gauge fields associated with the 240 nonzero roots of the superstring gauge
symmetry group E_{8}. The integer 1 assigned to each Sephirah denotes a geometrical
element in the axes of the two polyhedra, with the following correspondences between
elements and Sephirah:
Kether 

centre 
Chokmah 

apex 
Binah 

nadir 
Chesed 

upper edge on axis 
Geburah 

lower edge on axis 
Tiphareth 

centre 
Netzach 

apex 
Hod 

nadir 
Yesod 

upper edge on axis 
Malkuth 

lower edge on axis 
Tiphareth, the centre of the Tree of Life in both a geometrical and
metaphysical sense, is the starting point of material creation embodied in the disdyakis
triacontahedron. The distinction between the functions of the two polyhedra is that between the
first five Sephirah and the last five Sephirah. We should not expect the 144 Polyhedron to
22
embody any number that relates to the structure of matter. That is the
function of the disdyakis triacontahedron, as later articles will prove. Just as matter is the
end product of superphysical causes, so the 144 Polyhedron and the disdyakis triacontahedron
can be thought of as representing, respectively, Spirit and Matter.
The fact that the number 23 determines the geometrical composition of the
pair of polyhedra is highly significant, because it provides an analogy with the 23 pairs of
chromosomes of the human cell. One pair is the X and Y sex chromosomes. The human cell displays
the same 1:22 division as that which appears numerically in the polyhedra, whereby the
geometrical composition of their axes is determined by the integer 1 and the composition of
their shapes is determined by the next 22 odd integers! The X and Y chromosomes are analogous
to the axes and the 22 other chromosomes, which determine the characteristics of the body, are
analogous to the geometry of both polyhedra. This pattern appears in the pair of dodecagons of
the inner Tree of Life

Vertices

hexagonal yods on edges

central hexagonal

Total

Exterior: 
C

2E

F

C+2E+F

Interior: 
1

2C

E

2C+E+1

Total = 
C+1

2(C+E)

F+E

3C+3E+F+1

(Fig. 24a). They have 22 corners and 23 edges. The X and Y chromosomes are
symbolised by the pair of hexagonal yods in the root edge that generates the (7+7) polygons.
A hexagonal yod in one dodecagon and its mirror image in the other one denote a pair of
chromosomes (N.B. the two hexagonal yods on an edge do not denote a pair of chromosomes
because there are 22 such pairs, not (11+11) pairs). Assigning the odd integers (×10) to
these yods in one dodecagon generates as their sum the total number of geometrical elements
in the 144 Polyhedron and the disdyakis triacontahedron. The human chromosomal pattern is
analogous to the arithmetic behind the geometry of the polyhedral Tree of Life because
humans are the terrestrial manifestation of the divine prototype called Adam Kadmon
(“Heavenly Man”), whose geometrical representation is the Tree of Life.
6. Yod
composition of the polyhedral Tree of Life
Consider, firstly, the case when triangular faces and internal triangles are tetractyses. The
yod composition of a polyhedron with triangular faces is shown below:
23
Table 6 lists the yod population of the disdyakis triacontahedron and the 144
Polyhedron.
Table 6. Yod population of the polyhedral Tree of Life.


Vertices

Hexagonal yods
on edges

Central
hexagonal yods

Total

Disdyakis triacontahedron 
Exterior 
60+2

180×2=360

120

540+2

Interior 
1

120+4

180

300+5

Subtotal = 
60+3

480+4

300

840+7

144 Polyhedron 
Exterior 
72+2

2×216=432

144

648+2

Interior 
1

2×72+ 4 = 144+4

216

360+ 5

Subtotal = 
72+3

576+4

360

1008+7


Total = 
132+6

1056+8

660

1848+14

(‘1’ denotes the centre, ‘2’ denotes the top & bottom vertices &
‘4’ denotes the four hexagonal yods on the axis).
As we found during the analysis of its geometrical composition, the
polyhedral Tree of Life has 660 tetractyses with 138 vertices. 480 hexagonal yods line the
edges of the disdyakis triacontahedron, 240 in each half. They symbolise the 240 nonzero roots
of E_{8}×E_{8}, the gauge symmetry group governing one of the five types of
superstrings called the E_{8}×E_{8} heterotic superstring. This property will
be analysed in more detail in later articles. There are 840 yods surrounding the seven yods on
its axis. They comprise 480 hexagonal yods on edges of tetractyses and 360 vertices or
hexagonal yods at centres of faces. This pattern conforms to the Lambda Tetractys:
because the sum of the integers at the corners of the hexagon is
48 and the sum of the integers at the corners of the tetractys is
36. The number 840 is defined by the Tetrad because 840 = 10×84 =
(1+2+3+4)×(1^{2}+3^{2}+5^{2}+7^{2}). It is the sum of the first
28 odd integers:
840 = 3 + 5 + 7 + … + 57.
where 28 is the second perfect number^{7} (the first perfect number is 6, the centre of the Lambda
Tetractys). Together with the numbers 496 (the third perfect number and the
dimension of E_{8}×E_{8}) and 168 (=6×28), 840 is a
parameter of superstring physics, as has been discussed in previous articles.
780 hexagonal yods surround the axis of the disdyakis triacontahedron. 78 is
the dimension of E_{6}, the exceptional subgroup of E_{8} that physicists have
investigated as a potential candidate for accommodating the current Standard Model of particle
physics in superstring theory. The significance of the number 780 will become apparent when we
analyse the triangulated polyhedral Tree of Life with all sectors regarded as tetractyses.
The 144 faces of the 144 Polyhedron are composed of 650
(=65×10) yods, showing how ADONAI with number value 65
prescribes its form. 360 (=36×10) internal yods
24
(216 hexagonal yods at centres of tetractyses) surround its
axis, where 36 is the number value of ELOHA, the Godname of Geburah, whose
number value is 216. There are 1008 yods surrounding its axis, 504 yods in
each half. As we saw earlier, this number is defined by the Decad because the dodecagon has 504
yods surrounding its centre when each sector is divided into three triangles, each of which is
then divided into three tetractyses. The 144 Polyhedron has 576 hexagonal yods on edges
surrounding its axis, where 576 = 1^{2}×2^{2}×3^{2}×4^{2}.
Surrounding the axes of both polyhedra are 132 vertices and 1056 hexagonal
yods that line the edges of 660 tetractyses, i.e., (132+1056+14=1202) yods line all edges,
including axes. 1200 such yods shaping the polyhedra surround their centres. As 120 is the sum
of the first 10 odd integers after 1:
This demonstrates the beautiful way in which the Decad defines the form of
the Polyhedral Tree of Life. It is demonstrated geometrically by the pair of decagons in the
inner Tree of Life. With its sectors turned into tetractyses, each decagon has 60 yods
surrounding its centre (Fig. 25). Assigning the
integer 1 to the two centres and 10^{2} to each of the 120 yods in
the pair of decagons generates the number 1202. Each decagon represents half of the two
polyhedra and the two centres denote their centres.
1848 yods surround the 14 yods on the axes of the two polyhedra, i.e.,
(1848+12=1860)
Table 7. Yod populations of the polyhedral Tree of Life.


Vertices

Hexagonal yods
on edges

Central
hexagonal yods

Total

Disdyakis triacontahedron 
Exterior 
60+2+120=180+2

360+720=1080

360

1620+2

Interior 
180+1

120+1080+4=1200+4

540

1920+5

Subtotal = 
360+3

2280+4

900

3540+7

144 Polyhedron 
Exterior 
72+144+2=216+2

432+864=1296

432

1944+2

Interior 
216+1

144+1296+4=1440+4

648

2304+ 5

Subtotal = 
432+3

2736+4

1080

4248+7


Total = 
792+6

5016+8

1980

7788+14

25
yods surround their centres. A pentagon with its sectors turned into
tetractyses has 31 yods, where 31 is the number value of EL,
the Godname of Chesed. Assigning 31 to the 60 yods surrounding the centres of
a pair of pentagons generates the number 1860.
Imagine the arms of a 10pointed star as a straight line of
187 dots. The total number of dots = 1 + 10×186 = 1861. This is the number
of yods in the two polyhedra when their centres coincide. 187 is the
number value of Auphanim, the Order of Angels assigned to Chokmah.
Now let us consider the triangles in both polyhedra divided into their
sectors and each sector then turned into a tetractys. Table 7 lists their yod populations. The disdyakis triacontahedron has
363 vertices. This is the number value of SHADDAI EL CHAI, the Godname of
Yesod. Surrounding its axis are 2280 (=228×10) hexagonal yods that line edges of 900
tetractyses. 228 is the 227th integer after 1, where 227 is the 49th prime
number and 49 is the number value of EL CHAI. (360+2280=2640) yods on edges
of 900 tetractyses surround its axis. They are made up of 360(=36×10)
vertices and 2280 (=228×10) hexagonal yods. Compare this with the 36
corners and 228 other yods in the seven enfolded polygons of the inner Tree of Life. This
amazing correlation confirms the Tree of Life nature of thedisdyakis triacontahedron. The
360 vertices comprise the 120 (=12×10) vertices in its faces, 60 (=6×10) polyhedral vertices
& 180 (=18×10) internal vertices. This compared with the 12 corners of the dodecagon,
the six corners of the first three polygons outside the root edge and the 18 corners of the
next three polygons.
2740 hexagonal yods line the 1080 tetractyses of the 144 Polyhedron. 1370
hexagonal yods line the 540 tetractyses in each half. This demonstrates its Tree of Life
nature, for we saw earlier that the 282 tetractyses making up the (7+7)
enfolded polygons of the inner Tree of Life contain 1370 yods (see Fig. 12).
The 144 Polyhedron has 1946 yods in its 144 faces. 1946 =
36×54, where 36 is the sum of the integers at the corners of
the Lambda Tetractys and 54 is the sum of its remaining integers.
The pair of polyhedra has 7788 yods surrounding their axes, i.e.,
(7788+12=7800) yods surround their centres. 7800 = 50×156, where
50 is the number value of ELOHIM and 156 is the 155th integer
after 1, where 155 is the number value of ADONAI MELEKH, the complete Godname
of Malkuth. This shows how these two Godnames
26
27
determine the number of yods needed to construct the two concentric
polyhedra, starting with their shared centre. 156 is also the sum of all the different
combinations of the letter value of YAHWEH.
YHVH = 26
Y = 10, H = 5, V = 6
Y + H + V =

21

YH + YV + HV + HH =

52

YHV + YHH + YVH + HVH =

57

YHVH =

26

Total =

156

7800 (=78×100) yods surround the centres of the two polyhedra. The number 78
is determined by the Tetrad because it is the sum of the first 12 integers that can line the
edges of a square, four to each edge:
As a square contains 25 yods when its sectors are tetractyses, 100 is the
sum of the Tetrad assigned to each yod:
The significance of the number 78 is that it is the number of corners of the
94 sectors of the (7+7) enfolded polygons outside their root edge. This means that if the
number 1 is assigned to the endpoints (symbolising the centres of the two polyhedra) of the
root edge and the square of the Decad is assigned to the 78 external corners of the 94 sectors
of the 14 enfolded polygons, the sum of the 80 numbers is 7802 yods — the yod
population of the two polyhedra (Fig. 26). The 3900 yods in each pair of halves that surround their centres
correspond to the assignment of 10^{2} to the 39 external yods in each half of the
inner Tree of Life. Each set of seven enfolded polygons corresponds to the upper or lower
halves of the two polyhedra.
As
780 = 39×20 = 1 + 2 + 3 + … + 39,
7800 = 10 + 20 + 30 + … + 390.
Assigning the integer 1 to the endpoints of the root edge and the integers
10, 20, 30 etc to the 39 external corners of the seven enfolded polygons generates the number
7802 as their sum. It is a remarkable demonstration of the harmonious conjunction of number and
sacred geometry in determining the properties of holistic systems. (39+2=41) integers make up
this sum, where 41 is the 21st odd integer. This shows how EHYEH, the Godname
of Kether with number value 21, prescribes the yod population of the
polyhedral Tree of Life. ELOHA, the Godname of Geburah with number value 36,
prescribes it because the seven enfolded polygons have 36 corners.
28
The number of yods in the ntree when its triangles are tetractyses is Y(n)
= 50n + 30.^{8} Successive Trees of Life have 50 yods, where
50 is the number value of ELOHIM, the Godname of Binah. The
15tree has Y(15) = 780 yods. The number of triangles in
the ntree is T(n) = 12n + 7. The 15tree has 187
triangles, where 187 is the number value of Auphanim, the Order of
Angels assigned to Chokmah whose Godname is YAH with number value 15. The
number of corners of triangles in the ntree is S(n) = 6n + 5. The 15tree
has S(15) = 95 corners, where 95 is the
number value of Madim, the Mundane Chakra of Geburah, and 282
corners & triangles, where 282 is the number value of Aralim,
the Order of Angels assigned to Binah. If we imagine each yod in the
15tree as the Decad, the sum of the integers is 7800. This is how YAH
prescribes the yod population of the polyhedral Tree of Life.
As the top corners of the two hexagons coincide with the bottom corners of
the two hexagons enfolded in the next higher Tree of Life, there are 78 corners of the 94
sectors of the set of 14 polygons that are unshared. The number of corners of the 14n polygons
enfolded in n overlapping Trees of Life is N(n) = 78n + 2. A Tree of Life can be
represented by 10 overlapping Trees of Life because each Sephirah can be represented by a Tree
of Life. This is the meaning of ADONAI, for its number value 65 is the number
of Sephiroth in the lowest 10 Trees of Life. Each of the Trees of Life in the set of 10 can in
turn be represented by 10 Trees of Life, so a single Tree of Life is equivalent to 10,
10^{2}, 10^{3}, etc Trees of Life. The inner form of (10^{2}=100) Trees
of Life consists of 1400 polygons whose 9400 sectors have N(100) = 7802 corners (Fig. 27). Enfolded in the 50tree are 700 polygons whose
4700 sectors have N(50) = 3902 corners, 3900 being intrinsic to them
because none coincide with the corners of polygons enfolded in higher Trees of Life. This is
the number of yods in one pair of halves of both polyhedra, excluding their centres. This is
how the Godname ELOHIM with number value 50 prescribes the number of yods
making up half of both polyhedra. As those in the other halves are merely their mirror
images, this is how ELOHIM mathematically determines the structure of the polyhedral Tree of
Life. The 700 polygons enfolded in the next 50 Trees of Life have 4700
sectors with 3902 corners. This is the number of yods in the other pair of halves, including
their centres. The number ‘2’ in the above formula always refers to the two corners of the
14 polygons that are shared with those enfolded in the next higher Tree of Life, namely, the
topmost corners of the pair of hexagons We see that the topmost pair of corners of the
hexagons enfolded in the 100th Tree of Life correspond to the centres of the two polyhedra.
The 50:50 division corresponds to the 5:5 division of the
Sephiroth of the Tree of Life. In a fundamental sense, the halves that contain the centres
are the first stage of creation of the polyhedral Tree of Life and correspond to the five
Sephiroth of the Upper Face of the Tree of Life, whilst the remaining halves, which are
their mirror reflections, correspond to the five Sephiroth of its Lower Face.
Table 6 indicates that the 120 faces of the disdyakis triacontahedron have
300 hexagonal yods in their centres and 480 hexagonal yods on edges surrounding its axis,
that is, 780 hexagonal yods. The 940 sectors of the 140 polygons enfolded
in the 10tree have N(10) = 782 corners, 780 of them being intrinsic to them because they
are not shared with polygons enfolded in the 11th Tree of Life. There are 780 corners of
polygons enfolded in every 10 overlapping Trees of Life representing a Sephirah. We see that
the exterior of the disdyakis triacontahedron embodies a Tree of Life pattern. Later
articles will confirm that the polyhedron is the polyhedral version of 10 Trees of Life
because its properties are analogous to their properties. The first (6+6) enfolded polygons
constitute a Tree of Life pattern in themselves because they have 50
corners prescribed by ELOHIM and because the Godnames determine their other
properties.^{9}
29
They have 48 unshared corners. There are 30 corners that
are either the 20 external corners of the two dodecagons or the 10 centres of polygons that are
not also corners, The 120 polygons of the first six types enfolded in 10 Trees of Life have 480
corners intrinsic to them, leaving 300 corners that are either corners of dodecagons or
centres. According to Table 6, they correspond to the 480 hexagonal yods on edges of tetractyses
and to the 300 hexagonal yods at centres of tetractyses. This remarkable correlation between
the inner form of 10 Trees of Life and the hexagonal yods in the disdyakis triacontahedron
demonstrates its Tree of Life character. It is confirmed by the following: Table 7 indicates that 3540 (=354×10) yods surround the axis of this
polyhedron. The integers 1, 2, 3 & 4 express 354 as
354 = 1^{4} + 2^{4} + 3^{4} + 4^{4}.
Hence, 3540 =
(1+2+3+4)(1^{4}+2^{4}+3^{4}+4^{4}). This is a powerful sign of
mathematical design underlying the disdyakis triacontahedron. Ten overlapping Trees of Life
have 124 triangles with 64 corners and 166 edges, i.e., 354 corners, edges
& triangles. The same number characterises the geometrical composition of 10 overlapping
Trees of Life and the yod population of the disdyakis triacontahedron. It demonstrates that it
is equivalent to 10 Trees of Life. The number of yods in the faces and at the centre of this
polyhedron is 3541. This is the 496th prime number, where 496
(Fig. 28) is the number value of Malkuth and the dimension of the heterotic
superstring gauge symmetry group E_{8}×E_{8}! Here is an astounding example
of the mathematical design of the outer form of the polyhedral Tree of Life. According to
Table 6, the 360 tetractyses making up its 120 faces have 1622 yods. Starting
at the apex of the disdyakis
triacontahedron and counting yods, an extra 1621 yods are needed to
construct its surface. 1621 is the 257th prime number, where 257 is the 55th prime number and
55 is the sum of the first 10 integers:
55 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10.
This demonstrates how the Decad determines the shape of the outer form of
the polyhedral Tree of Life.
The 264 faces of the two polyhedra each has 10 ten yods inside its edges.
The number of yods that are needed to construct them without filling in the faces = 7802 –
264×10 = 5162. Starting from the centre of each, 5160 more such yods are needed, 2580 (=258×10)
yods in either pair of halves. 256 is the 129th even number. This shows
how
30
YAHWEH SABAOTH, the Godname of Netzach with number value
129, prescribes the two separate polyhedra.
Coaxial polyhedra
Suppose, finally, that the two polyhedra are concentric and share the same axis, so that their
pairs of opposite vertices on the axis coincide. There are now 792 vertices surrounding the
three vertices, that is, 794 vertices surrounding the shared centre. Of these, 134 are
polyhedral vertices, 67 in each pair of
halves, where 67 is the number value of Binah. There are
5016 hexagonal yods on edges surrounding the four hexagonal yods on the axis, that is, 5020
hexagonal yods in total. Each pair of halves contain 2510 (=10×251) hexagonal yods. This is
remarkable because there are 251 yods in the 1tree when its 19 triangles are each divided into
three tetractyses (Fig. 29).
31
If we imagine the 1tree constructed with the Decad as building block, the
resulting number that it represents is the number of hexagonal yods lying on edges of the 1980
tetractyses. There are 1980 hexagonal at the centres of tetractyses, 990 in each half. This is
expressed by the Decad because 990 = 10×(10^{2}–1) = 10^{3} – 10. The total
number of hexagonal yods is 5020 + 1980 = 7000. Each pair of halves has 3500 (=10×350)
hexagonal yods. The sum of the 20 integers in the tetrahedral Lambda Tetractys is 350
(Fig. 30), so this number is ten times this archetypal number. The sum of the
36 integers on the sides of the four separate tetractyses in the
tetrahedral Lambda tetractys is 700, so that the number of hexagonal yods is ten times the
sum of these integers:
The number of corners of the 7n polygons enfolded in the ntree is C(n) =
35n + 1, where ‘1’ refers to the topmost corner of the hexagon in the nth Tree of Life, which
coincides with the lowest corner of the hexagon enfolded in the (n+1)th Tree of Life. Hence,
there are 35n corners that are intrinsic to polygons enfolded in the ntree. The 70 polygons
enfolded in the 10tree have 350 intrinsic corners (Fig. 30). Assigning the Decad to these corners generates the
number of hexagonal yods in each pair of halves of the polyhedral Tree of
Life.
The ‘trunk’ of the outer Tree of Life is defined as the point (Kether), line
(Path joining Chokmah and Binah), triangle (ChesedGeburahTiphareth) and tetrahedron
(NetzachHodYesodMalkuth). It contains 35 yods, the ‘branches’ containing 35 yods as well
(Fig. 31). The division of the outer Tree of Life into its trunk and branches
corresponds to the two halves of the inner Tree of Life, each of which has 35 corners. This
division manifests in the polyhedral Tree of Life as one half and its mirror image because
assigning the square of the Decad to each yod or corner generates the number of hexagonal
yods in each half. Such correlation is highly convincing evidence for the disdyakis
triacontahedron and the 144 Polyhedron being the polyhedral Tree of Life.
32
Including the 794 vertices (397 in each half) that surround the centre of
the two concentric polyhedra, there are (397+3500=3897) yods in each half surrounding the
centre, that is, 7794 yods in total. Of these, 67 are vertices of the two
polyhedra. Hence, there are (3897–67=3830) yods that are not their vertices.
The polyhedral Tree of Life has (2×3830=7660) yods that are not polyhedral vertices, 3830
(=10×383) in each half. 383 is the 76th prime number.
YAHWEH ELOHIM, the Godname of Tiphareth with number value
76, prescribes the number of yods other than the 67 vertices
that are needed to construct each half of the pair of concentric polyhedra from 990
tetractyses. This is an amazing property, further demonstrating how the Godname numbers define
prime numbers characterising the polyhedral Tree of Life.
There are 7794 yods surrounding the centre of the coaxial pair of polyhedra.
(7794– 6=7788) yods surround the axis. 7788 = 236×33. There are 236 yods on the boundary of the
(7+7) enfolded polygons (Fig. 32). Assigning the number 33 to each yod generates the number of yods
surrounding the axis of the polyhedral Tree of Life. Moreover, the root axis with four yods
generates the number 132, which is the number of polyhedral vertices surrounding the axis
shared by the two polyhedra. This reveals the importance of the number 33
(its importance in Freemasonry is wellknow, but here is a more profound reason for why it is
significant). In the Pythagorean context, it is the number of permutations of the
26 combinations that can be made when selecting 10 objects arranged in the
four rows of a tetractys: 33 = 1! + 2! + 3! + 4!.
Referring to the geometrical composition of the two separate polyhedra
detailed on page 18, they have 798 vertices, 2512 edges and 1980 triangles. When they share the
same axis, centre and their opposite vertices coincide, they have (798–3=795) vertices
33
and (2512–2=2508) edges, that is, 3303 vertices & edges, 2775 vertices
& triangles, 4492 edges & triangles and 5284 vertices, edges & triangles. 5280
geometrical elements surround their centre. Hence, each half of the polyhedral Tree of Life
(which has 264 faces) has 2640 (=264×10) geometrical elements surrounding its centre. This
number is generated by assigning the Decad to the 264 yods making up the seven enfolded
polygons of the inner Tree of Life. It is remarkable proof of the Tree of Life nature of the
concentric disdyakis triacontahedron and 144 Polyhedron (Fig. 33). Given two joined dodecagons divided into their 24 sectors, 264
more yods are needed to divide each sector into three tetractyses. This property illustrates
how the last
polygon of the inner Tree of Life is its single polygonal
counterpart. The 20 dodecagons enfolded in 10 overlapping Tree of Life need 2640 more yods to
construct their sectors from three tetractyses, whilst there are 2640 yods in the 70 polygons
enfolded in ten separate Trees of Life. Once again, here is proof that each half of the
polyhedral Tree of Life is equivalent to 10 Trees of Life.
The coaxial polyhedral Tree of Life contains 7795 yods (see above). Of
these, 1980 are hexagonal yods at centres of tetractyses. There are (7795–1980=5815) yods
lining their edges. Hence, (5815–7=5808) yods line edges outside the axis, 2904 yods in each
half. Including the axis, (2904+3=2907) yods in each half lie on edges. As 2907 =
19×153, this is the sum generated by assigning the number value
153 of ELOHIM SABAOTH, the Godname of Hod, to the 19 yods in a triangle
divided into three tetractyses (Fig. 34). The shaping character of edges of tetractyses is
reflected in the fact that Hod lies on the Pillar of Judgement. Alternatively, as 2907 =
9×17×19 = 9×(18^{2}–1) = 9(3 + 5 + 7 +… + 35), we that 2907 is the sum of
(9×17=153) odd integers.
References ^{1} Proof: There are
120 yods on the boundary of the seven enfolded polygons, which contain 264 yods. The
number of yods inside the boundary of both sets of polygons = 2×(264–120) = 2×144 =
288.
^{2} Phillips, Stephen M. Article 4: “The Godnames Prescribe the Inner
Tree of Life,” (WEB, PDF).
^{3} Phillips, Stephen M. Article1: “The Pythagorean Nature of
Superstring and Bosonic String Theories,” (WEB, PDF), p. 4.
^{4} Proof: the 7 enfolded polygons have 47 sectors with 41 corners
and 88 sides. The number of yods on these sides when each sector is divided into 3 tetractyses
= 41 + 88×2 = 217. There are 10 yods inside each sector. The number of yods in the (47×3=141)
tetractyses = 217 + 47×10 = 687. Four yods lie on the edge shared by the polygons. The number
of yods in the 7 enfolded polygons outside this edge = 687 – 4 = 683. The number of yods in the
(7+7) enfolded polygons = 4 + 2×683 = 1370.
^{5} Phillips, Stephen M. Article 11: “Plato’s Lambda — its Meaning,
Generalisation & Connection to the Tree of Life,” (WEB, PDF), p. 3.
^{6} Ibid, p. 8.
^{7} A perfect number is an integer that is the sum of its
divisors.
^{8} Proof: the number of corners of the triangles in the ntree is
S(n) = 6n + 5. The number of edges is P(n) = 16n +9. The number of triangles is T(n) = 12n + 7.
Two hexagonal yods are on each edge and one hexagonal yod is at the centre of each triangle
when it becomes a tetractys. The number of yods in the
34
ntree = S(n) + 2P(n) + T(n) = 50n + 30.
^{9} Ref. 2, p. 4.
35
