ARTICLE 34
by
Stephen M. Phillips
Flat 4, Oakwood
House, 117119 West Hill Road. Bournemouth. Dorset BH2 5PH. England.
Website: http://smphillips.mysite.com
Abstract
Earlier articles gathered evidence that the disdyakis
triacontahedron is the polyhedral counterpart of the inner form of the
Kabbalistic Tree of Life. The 60 vertices between two diametrically opposite A
vertices are arranged in seven layers as two 8sided polygons, two 10sided
polygons, two 6sided polygons and one 12sided polygon. Some vertices form 15
Golden Rectangles. The regular counterparts of these four types of polygons in
the inner Tree of Life are shown to encode the 126 paired bones of the
appendicular skeleton and the 34 single bones and 46 paired bones of the human
axial skeleton. These enfolded polygons are defined by 137 geometrical elements,
this being the integer defining the fine structure constant known to physicists.
The seven polygons formed by vertices of the disdyakis triacontahedron have 240
geometrical elements surrounding their centres. This corresponds to the 240
hexagonal yods surrounding the centres of the seven separate polygons of the
inner Tree of Life. They represent, respectively, the polyhedral and polygonal
encoding of the 240 roots of the superstring gauge symmetry group E_{8}
and seven of its eight simple roots, the eighth one corresponding to the apex of
the disdyakis triacontahedron directly above the seven centres. The first two
polygons have 72 elements and the last five have 168 elements, correlating with
the 72 roots of E_{6} and the 168 other roots of E_{8}. The four
enfolded polygons that correspond to the four types of polygons and encode the
human skeleton have 168 yods (the superstring structural parameter) outside
their root edge on the edges of their tetractys sectors. The 168:168 division of
geometrical elements belonging to the first three and last three polygons
corresponds to the same division of yods outside the root edge in the inner Tree
of Life that are not vertices of polygons. The seven layers of polygons display
the same geometrical pattern as that shown by the I Ching table, demonstrating
the holistic character of the disdyakis triacontahedron. The pattern of
geometrical elements in its faces and interior conforms to the pattern of
numbers in Plato’s Lambda Tetractys, whose ratios define the intervals of the
Pythagorean musical scale. This is exhibited also by the seven polygons, which
encode geometrical information embodied in the polyhedron.

1
Table 1. The number values of the 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.
Numbers from this table will be written in boldface in
the article.

2
1. The seven layers of vertices in the
disdyakis triacontahedron
Articles 2230 provided evidence that the disdyakis triacontahedron
(Fig. 1) — the Catalan solid with the most faces — is the polyhedral version
of the polygonal, inner Tree of Life. It has 62 vertices, 180 edges and 120
triangular faces, that is, 362 geometrical elements in its faces. Therefore, 360 elements
surround the central axis joining any two diametrically opposite vertices. Because it
is the polyhedral counterpart of the inner form of the Tree of Life, it constitutes
sacred geometry par excellence. This means that the geometrical (indeed,
all mathematical) properties of this polyhedron are prescribed by the gematria
number values of the Hebrew names of the ten Sephiroth of the Tree of Life, their Hebrew
Godnames, Archangelic Names, Orders of Angels and Mundane Chakras. Table 1 lists the English versions of these names (their literal English
translations appear within brackets) and the number values of all the Hebrew names. It
indicates the remarkable fact that ELOHA,^{*} the Godname of Geburah with a number value of 36,
prescribes the shape of the polyhedron because the number of geometrical elements
surrounding its axis is 360, which is the number of yods in 36 tetractyses.
The Godname EL (“God”) of Chesed with number value 31 also prescribes it
because every one of 31 vertices in onehalf of the polyhedron has its
mirror image diametrically opposite it. Remarkably, the gematria number value of
Tzadkiel, the Archangel of Chesed, is 62 — the number of vertices
in the disdyakis triacontahedron. Other ways in which the Godnames, etc define its
properties were discussed in previous articles. This article will explain how they
characterise the layering of vertices.
The 62 vertices are of three types: 12 B vertices of an
icosahedron, 20 C vertices of a dodecahedron (these form the 32 vertices of a rhombic
triacontahedron (Fig. 2)) and 20 A vertices created by raising the
centres of the Golden Rhombic faces of the
latter polyhedron. Table 2 displays the x, y & zcoordinates of the
62 vertices of the disdyakis triacontahedron. All coordinate values belong
to the set (0, ±φ, ±φ^{2}, ±φ^{3},
±2φ^{2}), where φ = 1.6180…. is the Golden Ratio. The topmost vertex
A is at a height z = 2φ^{2} above the XY plane. Eight vertices are at a height
φ^{3}, 10 vertices are at a height φ^{2} and six vertices are at a height φ.
Counterparts of these three layers of vertices lie below the XY plane, which contains 12
vertices. The vertices of the polyhedron are arranged in seven layers. In Fig. 2, each layer is assigned one of the seven colours in white light.
Vertex 1 is the apex of the polyhedron and vertex 62 is its nadir. The dark
line divides the first 31 vertices from their mirror images. Vertices are
ordered so that the nth vertex at (a,b,c) is the mirror image of the (63–n)th vertex at
(–a,–b,–c).
Whether viewed along the X, Y or Zaxes, the polyhedron has seven layers
of vertices between the A vertices at its apex and nadir. Counting from the top, the numbers of
vertices in the layers are:
1. 8 (4A, 2B & 2C);
2. 10 (4A, 2B & 4C);
3. 6 (4A, 2C);
4. 12 (4A, 4B & 4C);
5. 6 (4A, 2C);
6. 10 (4A, 2B & 4C)
7. 8 (4A, 2B & 2C).
__________________________________
^{*} Godnames will be written in upper case and
Archangelic Names, Angelic Names & Mundane Chakras will be italicized.
3
4
5
The four uppermost layers have 36 vertices made up of 16 A
vertices, eight B vertices and 12 C vertices. The three lowest layers have 24 vertices
comprising 12 A vertices, four B vertices and eight C vertices. Figure 3 shows the projection of the uppermost four layers onto the XY plane,
looking downwards along the Zaxis. The apex is shown as a black circle. The uppermost layer
of eight vertices is in red, the next layer of 10 vertices is in violet, the third layer of
six vertices is in blue and the fourth layer of vertices lying in the XY plane is in green.
The black gridlines mark distances ±φ, ±φ, ±φ^{2} & ±2φ^{2}. Four A
vertices in the first layer are at the corners of a Golden Rectangle, as are four A vertices
in the third layer, although the size of the rectangle is different. Similarly, four B
vertices in the fourth layer lie at the corners of the same sized Golden Rectangle rotated
by 90°. No four vertices in the second layer form a Golden Rectangle. The seven layers
therefore contain five Golden Rectangles. Because of the mirror symmetry of the disdyakis
triacontahedron, there is a vertex below every one shown in Fig. 3 that belongs to one of the three layers below the XY plane.
Figure 4 displays the projection of the vertices onto the XZ plane as viewed
in the positive direction of the Yaxis. The distribution of vertices is the same as that
seen along the Zaxis after rotating the disdyakis triacontahedron through 90°. The
projection of vertices onto the YZ plane as viewed along the positive Xaxis is identical to
the projection onto the XY plane shown in Fig. 3. Hence, (3×5=15) Golden Rectangles are formed by
vertices in the three perpendicular sets of seven layers totalling 21
layers. This shows how the Godname EHYEH of Kether with number value 21 and
the Godname YAH with number value 15 prescribe the orthogonal layering of
vertices in the disdyakis triacontahedron.
2. Encoding of the human skeleton in the seven
layers According to Kabbalah, Adam Kadmon, or Heavenly Man, is the
universal prototype for all holistic systems, whether a subatomic particle or a human
being. Genesis 1, verse 27, states: “So God created man in his own image, in the image of
God created he him; male and female created he them.” This ‘image’ is the Tree of Life. It is the
cosmic blueprint governing matter, man and the universe. As the representation of God, it
manifests in the four Worlds of Atziluth (Archetypal World), Beriah (Creative World),
Yetzirah (Psychological World) and Assiyah (Physical World). Godnames, Archangels, Angels
and Mundane Chakras are the Sephiroth of the four overlapping Trees of Life mapping these
worlds, which, of course, are not literally places but functional divisions of the Divine
Life. A human being at the Assiyatic level is the body composed of bones, flesh and vital
organs. The Tiphareth of the body viewed as the lowest of the four Trees of Life is the
central nervous system, a highly complex organisation of nerves that allow twoway
communication between the brain (representing Kether) and all the parts of the body. Its
Malkuth is the skeleton, which in the adult is composed of 206 bones. They form two
groups: the 80 bones of the axial skeleton (bones of the head,
vertebral column, ribs and breastbone or sternum) and the 126 bones of the appendicular
skeleton (the upper and lower limbs and their attachments, or girdles) (Fig. 5).
Some bones occur singly and others exist in pairs, with one belonging to the
left side of the body and the other belonging to its right side. Listed below are the numbers
of single bones and pairs of bones:

Single

Pairs

Axial skeleton (80 bones) 


Skull 
6

11

Hyoid: 
1


Vertebral column: 
26


Thoracic cage: 
1

12

Subtotal: 
34

23

Appendicular skeleton (126 bones) 


Pectoral girdles: 

2

Upper extremity 

30

Pelvic girdle: 

1

Lower extremity: 

30 
Subtotal: 
0

63

Total: 
34

86

The human skeleton is composed of 34 bones that occur singly and 172 bones
that appear in pairs. The
6
80 bones of the axial skeleton consist of 34 single bones
and 46 bones that occur in pairs. The 126 bones of the appendicular skeleton comprise 63 pairs
of bones.
It was shown in Articles 32^{1} and 33^{2} that the axial skeleton is the manifestation of the ‘trunk’ of
the Tree of Life. This is the downward sequence of point, line, triangle and tetrahedron —
what mathematicians call the first four ‘simplexes.’ With its triangles constructed from
three tetractyses, the trunk is made of 80 yods, each one denoting a bone
of the axial skeleton. The rest of what is the lowest of the four Trees mapping the human
body — what may be called its ‘branches’ — has 171 yods, 126 of which denote the 126 bones
of the appendicular skeleton. The 34 types of single bones are encoded in the inner Tree of
Life as the
34 yods in its pair of triangles. The 23 types of pairs of bones are encoded
as the 23 pairs of yods in the pair of squares. With their sectors transformed into three
tetractyses, the pair of joined triangles and the pair of joined squares contain 206 yods — the
number of bones in the human skeleton, whilst a separate triangle and a square contain
80 yods when similarly transformed, which is the number of bones in the axial
skeleton (Fig. 6).
Let us return to the seven layers of polygons in the disdyakis
triacontahedron. They are of four types: an 8sided polygon, a 10sided polygon, a 6sided
polygon and a 12sided polygon. They correspond in the polygonal Tree of Life (Fig. 7) to the octagon, decagon, hexagon and dodecagon, although the polygons
formed by the layers of vertices are not regular. It is trivial to prove that the number of
yods in an nsided polygon (regular or irregular) whose n triangular sectors are tetractyses
with four yods on each side is 6n+1. The hexagon (n=6) has 37 yods, of which (37–4=33) yods
are outside the root edge,^{*} the octagon (n=8) has
49 yods (45 outside the root edge), the decagon (n=10) has 61 yods (57
outside the root edge) and the dodecagon (n=12) has 73 yods (69 outside the
root edge). The hexagon and octagon have (33+45=78) yods outside the root edge. Two yods of
the latter are associated with one set of seven polygons and the two other yods of the root
edge are associated with the other set of seven polygons (Fig. 8). (78+2=80) yods are associated with the hexagon and
octagon on each side of the root edge. They consist of (33+1=34) yods associated with each
hexagon and (45+1=46) yods associated with each octagon. They symbolize, respectively, the
34 single bones of the axial skeleton and its 46 bones that exist as pairs, one on the
lefthand side of the body and the other on the righthand side. The (57+69=126) yods of the
decagon and dodecagon symbolize the 126 bones of the appendicular skeleton.
_________________________________
^{*} The root edge is the vertical edge shared by all the
14 polygons of the inner Tree of Life.
7
Layers 1 and 3 of the disdyakis triacontahedron have 14 vertices arranged as
an 8sided polygon and as a 6sided polygon. They correspond in the inner Tree of Life to,
respectively, the octagon and the hexagon associated with which are 80 yods
symbolizing the 80 bones of the axial skeleton. Layers 2 and 4 have 22
vertices arranged as a 10sided polygon and as a 12sided polygon. Their counterparts in the
inner Tree of Life are the decagon and the dodecagon, which have 126 yods outside the root edge
symbolizing the 126 bones of the appendicular skeleton. The oddnumbered layers therefore
determine the axial skeleton and the evennumbered layers determine the appendicular skeleton.
Layers 5, 6 & 7 are the respective mirror images of layers 3, 2 &1.
Of the 264 yods in the seven enfolded polygons, the 206 yods associated with
the last four polygons symbolize bones, leaving 58 yods in the first three enfolded
polygons. The last (4+4) enfolded polygons have 58 corners, showing how this number
defines the last four polygons in two different ways. Eight of these corners coincide with
Sephiroth of the lowest tree mapping the human body (Fig. 9), leaving 50 corners that are unshared with it. This
shows how the Divine Name ELOHIM with number value 50 prescribes the last
(4+4) polygons and therefore the number of bones in the human skeleton. The 70 triangular
sectors of the last (4+4) enfolded polygons have 66 corners, where 66 is the
65th integer after 1. This shows how ADONAI, the Godname of Malkuth with
number value 65, prescribes these polygons embodying information about the
human body — the Malkuth aspect of Adam Kadmon. 33 such corners are associated with each set
of the last four polygons. The centre of the hexagon is a corner of the triangle and the
centre of the decagon is a corner of the pentagon. This leaves (33–2=31)
corners of the 35 sectors of the last four polygons that are intrinsic to them
alone because they do not belong to the first three polygons. EL, the Godname of Chesed with
number value 31, therefore prescribes the last four polygons of the inner
Tree of Life that encode the bone composition of the human skeleton.
The 35 sectors of the last four enfolded polygons have 34 corners and 69
sides, totalling 138 geometrical elements. The number of elements associated with each set of
four polygons is 137. This integer defines the finestructure constant e^{2}/ħc ≈
1/137.036, making it perhaps the most important number in physics because its reciprocal
measures the strength of the coupling of the electromagnetic field to electric charges and thus
the size of atoms and the energy scale of their chemical bonding. 137 is the 33rd prime number.
As mentioned earlier, 33 corners of sectors are associated with the last four polygons, where
33 = 1! + 2! + 3! + 4!, i.e., the sum of the numbers of permutations of 10 objects arranged in
the four rows of a tetractys. Of the 34 corners, eight are either centres of polygons, shared
with the Tree of Life or both. This leaves 26 corners, showing how
YAHWEH, the Godname of Chokmah with number value 26 prescribes the last
four enfolded polygons. 26 = 1 + 3 + 7 + 15, where
Table 3. Geometrical composition of the seven polygons (single triangles as
sectors).


n

Number of vertices
n+1

Number of edges
2n

Number of triangles
n

Total
4n+1

1 

8

8+1

16

8

32+1

2 

10

10+1

20

10

40+1


Subtotal 
18

18+2

36

18

72+2

3 

6

6+1

12

6

24+1

4 

12

12+1

24

12

48+1


Subtotal 
36

36+4

72

36

144+4

5 

6

6+1

12

6

24+1

6 

10

10+1

20

10

40+1

7 

8

8+1

16

8

32+1


Total 
60

60+7

120

60

240+7

‘1’ is number of combinations of one object, ‘3’ is the number of
combinations of two objects, ‘7’ is the number of combinations of three objects and
‘15’ is the number of combinations of four objects. The
8
9
permutational and combinatorial properties of ten objects arranged in the
form of a tetractys are therefore connected to the shapes of the last four of the seven
enfolded polygons that make up the inner Tree of Life. When regarded as separate polygons, the
last four polygons have (6+8+10+12=36) corners, where 36 is
the number value of ELOHA, the Godname of Geburah. Their 36 sectors have
(36+4=40) corners, where
This demonstrates the basic role of the Tetrad (4) in defining properties of
any geometrical object that conforms to the universal pattern of the Tree of Life — in this
case the 40 corners of the 36 sectors of the first four layers of vertices of
its polyhedral form — the disdyakis triacontahedron.
3. Geometrical composition of the seven
polygons Table 3 tabulates the numbers of vertices, edges and triangles making up
each polygon in the seven layers of the disdyakis triacontahedron. The seven polygons
comprise 67 vertices, where 67 is the number
value of Binah. This is very appropriate in view of the meaning of the Sephirah Binah as
embodying the archetype of form, for this number does, indeed, quantify the shapes
of the seven polygons when constructed from tetractyses. In fact, when the 19
triangles of the 1tree are
converted into tetractyses, there are 67 yods below the
horizontal path joining Binah to Chokmah (Fig. 11). They include the seven yods that coincide with the seven Sephiroth
of Construction, leaving 60 other yods. This 60:7 division in the 1tree constructed from
tetractyses parallels the division of vertices in the seven polygons into 60 vertices that
surround their seven centres. The correspondence is striking evidence of the Tree of Life
nature of the disdyakis triacontahedron.
According to Table 3, there are 247 geometrical elements in the seven polygons. 240 of
these elements surround the seven centres. This important property, the superstring
significance of which will be discussed later, corresponds in the polygonal form of the
inner Tree of Life to the 240 hexagonal yods of the seven separate polygons that surround their seven centres (Fig. 10. The second subtotal row in the table above for the four layers of
vertices making up half of the
disdyakis triacontahedron shows that they comprise 148
elements. 148 is the number value of Netzach, the fourth Sephirah of
Construction. 144 of these elements surround the centres of the four polygons, where 144 =
12^{2} = 12th Fibonacci number. 36 of them are vertices of polygons,
showing how the Godname ELOHA with number value 36 prescribes
10
the shapes of the four polygons, just as we saw earlier that it does for
their separate, regular counterparts in the inner Tree of Life. The four polygons have
72 edges and 36 triangles, where 72 is the
number value of Chesed, the fourth Sephirah from the top of the Tree of Life.
Including the apex of the disdyakis triacontahedron (an A vertex), there are 41 vertices from
this ‘North pole’ to the equatorial plane containing the 12 vertices. 41 is the
21st odd integer, showing how EHYEH, the Godname of Kether with number value
21, prescribes the number of vertices in half of the disdyakis
triacontahedron when the polygons of
vertices in its layers are constructed from tetractyses.
The two uppermost layers are made up of an 8sided polygon and a 10sided
polygon. They contain 36 edges. 72 geometrical elements
surround their two centres. The five lowest layers are made up of an 8sided polygon, a
10sided polygon, two 6sided polygons and a 12sided polygon. They have 47 vertices, where 47
is the 15th prime number, showing how the Godname YAH of Chokmah with number
value 15 prescribes the polygons in the five lowest layers. 42 of these
vertices surround their five centres, where 42 is the 21st even integer,
showing how EHYEH, Godname of Kether with number value 21, prescribes these
five polygons. They have 84 edges, where 84 = 1^{2} + 3^{2} + 5^{2} +
7^{2}, and 42 triangles. They also have (42+42=84) vertices and triangles surrounding
their centres. The lowest five polygons therefore have (84+84=168) geometrical
elements, where 168 is the number value of the Mundane Chakra of Malkuth.
This division of the number of elements into two sets of 84 is highly significant. It
has been encountered in previous articles such as:
1. Article 15,^{3} which discussed the 84 permutations of pairs and triplets of
octonions belonging to the seven triples of octonions (e_{i}, e_{i+1},
e_{i+3}) and the 84 permutations of their inverses;
2.Article 18,^{4} which discussed the I Ching table of 64
hexagrams and the 28 hexagrams on either side of its diagonal. Each hexagram is a pair of
trigrams, so that each set of 28 offdiagonal hexagrams has (28×2×3=168)
lines and broken lines. Each set comprises 84 lines and 84 broken lines;
3. Article 21,^{5} which discussed the 84 hyperbolic triangles in the two types of
seven similar slices of the Klein Configuration, the diagram whose 168
hyperbolic triangles represent the 168 automorphisms of the equation known
to mathematicians as the Klein Quartic;
4. Article 16,^{6} which discussed, firstly, the 84 repetitions of the
Pythagoreanvalued, rising intervals between the notes of the seven types of musical scales
and their 84 falling intervals (pp. 17–18), and, secondly, the 84 yods belonging to the
first four enfolded polygons enfolded in each half of the inner Tree of Life that either are
not centres of polygons or do not coincide with Sephiroth (pp. 23–24).
Such correspondences demonstrate powerfully that the lowest five layers of
the disdyakis triacontahedron form a Tree of Life pattern in themselves. This could have
already been anticipated, because the number 168 is the structural parameter
of the superstring — the manifestation in the subatomic world of the Tree of Life blueprint.
Indeed, this is why the number 168 is the number value of the Mundane
Chakra of Malkuth, the Sephirah of the Tree of Life denoting the physical universe of space and
time.
The 84:84 division of geometrical elements in the last five polygons of
vertices, which start with the 6 sided polygon, has its counterpart in the pair of joined
hexagons belonging to the polygonal form of the inner Tree of Life. 168 more
yods (84 per hexagon) are needed to transform their sectors into three tetractyses
(Fig. 12). Furthermore, it is remarkable that the 12sided polygon in the
middle of the seven layers of vertices requires 168 more yods to construct
its sectors from three tetractyses. Figure 13 shows this property for the dodecagon. The fact that the sides of
the 12sided polygon are unequal in length is irrelevant. As 13^{2} – 1 =
168 = 3 + 5 + 7 + … + 25, assigning the first 12 odd integers after 1 to
the vertices.
11
of this central 12sided polygon generates the number 168
as their sum. This is an example of the beautiful coherence between number and sacred geometry.
It also illustrates the Tetrad Principle because the fourfold sequence of 8sided, 10sided,
6sided and 12sided polygons ends with one that embodies a number of universal significance,
namely, the very number 168 that defines the form of an
E_{8}×E_{8} heterotic superstring because each of its 10 curves makes
168 circularly polarized oscillations during one half of a revolution around
its spin axis. It manifests in the disdyakis triacontahedron as its 168 edges
above and below its equator that give it a 3dimensional shape.
4. The seven
polygons encode the roots of E_{8}
The seven polygons have 240 geometrical elements surrounding their centres, of which
72 belong to the uppermost two layers and 168 belong to the
five lowest layers. This 72:168 division appears in the
structure of the nonzero roots of the superstring gauge symmetry group E_{8}. The
roots of the E_{8} algebra can be described in terms of eight orthonormal unit vectors
{u_{i}}. Eight zero roots correspond to points at the centre of the root diagram and
240 nonzero roots all have length √2. They are given by:
±u_{i} ±u_{j} (i, j = 1, 2, …, 8)
and
½(±u_{1}, ±u_{2}, …±u_{8})
(even number of +’s)
Their explicit forms as 8tuples and their numbers are listed below:
The 240 nonzero roots of E_{8} comprise 168
nonzero roots made up of four sets of 28 and one set of 56 and 72 nonzero
roots comprising one set of 70 and two single ones. Remarkably, surrounding the seven centres
of the polygons are as many geometrical elements (240) as there are nonzero roots of
E_{8}, the first two polygons having as many elements (72) as there
are nonzero roots of E_{6}, an exceptional subgroup of E_{8} of much interest
to physicists seeking to extend the Standard Model of particle physics using superstring
theory. This provides highly convincing evidence that superstring physics is encoded in the
pattern of vertices belonging to the seven layers of the disdyakis triacontahedron.
The 2:5 division of layers generating the
72:168 division of geometrical elements surrounding centres
of polygons reflects the distinction in the Tree of Life between the two Sephiroth of
Construction — Chesed and Geburah, which lie outside its Lower Face, and the remaining five
Sephiroth of Construction that span it. The eight zero roots of E_{8} consist of seven
zero roots of one kind and one zero root of another kind. The former are denoted by the seven
centres of the polygons and the latter is denoted by the topmost A vertex of the disdyakis
triacontahedron (see Fig. 10).
The 72:168 division was encountered in
Article 267 in discussion of the construction of the disdyakis triacontahedron from
tetractyses. When its 180 internal triangles formed by its centre and each edge of the
polyhedron are divided into three tetractyses and its 120 triangular faces are turned into
single tetractyses, there are 1680 (=168×10) geometrical elements surrounding
the axis that joins two diametrically opposite A vertices. When each of its 120 faces is
constructed from three tetractyses, there are 2400 (=240×10) such elements, i.e., an extra 720
(=72×10) elements are added. What is being represented geometrically here are
the root structures of the superstring gauge symmetry group E_{8} and its exceptional
subgroup E_{6}, for E_{8} has 168 nonzero roots that do not
belong to E_{6}, which has 72 such roots.
12
Every simple root of E_{8} is associated with a YangMills gauge
charge — the source of a 10dimensional YangMills gauge field. The factor of 10 in these
properties therefore encodes the information that each gauge field has ten spacetime
components. The layers of the disdyakis triacontahedron encode the number of 10dimensional
gauge fields defining the unified superstring force, whilst the various numbers of geometrical
elements making up the interior and exterior of this polyhedron are the numbers of
components of these fields. They represent the number of bits of information needed to
define the gauge fields of the superstring in 10dimensional spacetime. In the context of the
disdyakis triacontahedron, the additional 720 elements correspond to the (240×3=720) components
of these fields in 3dimensional space.
The 72:168 division of the nonzero roots
of E_{8} appears in the E_{8}×E_{8} heterotic superstring as the
distinction between the three major whorls and the seven minor whorls in the object described
over a century ago by Annie Besant and C.W. Leadbeater^{8} as being
the basic unit of matter (Fig. 14). As the manifestation in the spacetime continuum of the Tree of
Life blueprint, the major whorls correspond to the Kether, Supernal Triad of Chokmah and
Binah, whilst the minor whorls
correspond to the seven Sephiroth of Construction. Besant & Leadbeater
connected them to the three “solar Logoi” and to the seven “Planetary Logoi:”
“The Anu is a sun in miniature in its own universe of the inconceivably
minute. Each of the seven whorls is connected with one of the Planetary Logoi, so that each
Planetary Logos has a direct influence playing on the very matter of which all things are
constructed. It may be supposed that the three conveying electricity, a differentiation of
Fohat, are related to the Solar Logos.”^{9}
Although each whorl is a separate closed curve that never intersects itself
or touches one another, this separateness is an illusion, existing only in ordinary
3dimensional space. The curves are the 3dimensional crosssections of a hyperdimensional
object embedded in a higher, 15dimensional space with a topology such that
its crosssection is a set of ten closed curves topologically equivalent to circles, each one
twisting 2½ times in an outer circuit and 2½ times in a tighter revolution around the spin axis
of the spin½ superstring. A 1dimensional curve embedded in 26dimensional
spacetime has 24 transverse vibrational degrees of freedom. These generate generalised 24
KaluzaKlein gauge fields that, on compactification of the 16 dimensions beyond superstring
spacetime, become 24 YangMills gauge fields of E_{8} existing in 10dimensional
spacetime. Their charges are spread along each curve, 24 to each one. The three major whorls
have 72 E_{8} gauge charges and the seven minor whorls have
168 charges. The former are the gauge charges of E_{6}, an exceptional
subgroup of E_{8}, and the latter are the remaining gauge charges of E_{8}.
They correspond in the seven polygons of vertices of the disdyakis triacontahedron to the
72 geometrical elements surrounding the centres of the two uppermost polygons
and to the 168 elements around the centres of the lowest five polygons. The
first of these polygons has 24 elements, the second has 24 edges and 24 vertices and triangles,
the third has 24 elements and the last two polygons have 72 elements. These
sets of 24 elements are the polygonal counterpart of the 24 gauge charges
13
spread along each of the ten curves of the superstring.
Of the 72 elements in the two uppermost polygons, 18
elements are vertices of the disdyakis triacontahedron (eight A, four B & six C vertices).
Hence, these polygons have (72–18=54) elements that are not polyhedral
vertices. Of the 168 elements in the lowest five polygons, 42 elements are
vertices of the disdyakis triacontahedron (20 A, eight B & 14 C vertices). This leaves
(168–42=126) elements in the lowest five polygons that are not polyhedral
vertices. Including their
five centres, there are (126+5=131) elements that are not
polyhedral vertices. 131 is the number value of Samael, the Archangel
of Geburah. All seven polygons have 180 edges and triangles. Including the seven centres of
polygons, there are 187 geometrical elements that are not polyhedral vertices.
This is the number value of Auphanim, the Order of Angels assigned to Chokmah.
The counterpart in the outer Tree of Life of the 240 geometrical elements
surrounding the centres of the seven polygons is the 240 extra yods needed to construct the
1tree when its 19 triangles are each divided into three tetractyses (Fig. 15). The counterpart of the centres of the seven polygons is the
locations of the seven Sephiroth of Construction.
Suppose now that the 60 sectors of the seven polygons are each divided into
three tetractyses instead of single ones. Table 4 lists the number of vertices, edges and
triangles in each polygon.
Table 4. Geometrical composition of polygons (three triangles as sectors).


n

Number of vertices
2n+1

Number of edges
5n

Number of triangles
3n

Total
10n+1

1 

8

16+1

40

24

80+1

2 

10

20+1

50

30

100+1


Subtotal 
18

36+2

90

54

180+2

3 

6

12+1

30

18

60+1

4 

12

24+1

60

36

120+1


Subtotal 
36

72+4

180

108

360+4

5 

6

12+1

30

18

60+1

6 

10

20+1

50

30

100+1

7 

8

16+1

40

24

80+1


Total 
60

120+7

300

180

600+7

The seven polygons have 120 vertices surrounding their centres. This is a
beautiful property because it indicates that the disdyakis triacontahedron has the same number
of triangular faces as there are vertices in the 180 triangles making up the polygons in its
seven layers of vertices. 60 of them are polyhedral ones, leaving 67
nonpolyhedral vertices, where 67 is the number value of Binah. The numbers
120 and 60 conform to the Tetrad Principle^{10} because
120 = 2^{2} + 4^{2} + 6^{2} + 8^{2} =
2^{3} + 2^{4} + 2^{5} + 2^{6}
14
and
60 = 2^{2} + 2^{3} + 2^{4} + 2^{5}.
The 8sided polygon at the base of the disdyakis triacontahedron has
80 vertices, edges & triangles surrounding its centre. 80
is the number value of Yesod, the penultimate Sephirah. Its meaning — ‘foundation’ — is
appropriate, given that this is the lowest polygon. The first two polygons have
36 vertices, 90 edges and 54 triangles surrounding their centres, which total
180 geometrical elements. The last five polygons have 84 vertices, 210 edges & 126
triangles surrounding their five centres, totalling 420 elements. With single triangles as
sectors, the last five polygons have 84 edges. Once again, the heterotic superstring structural
parameter 84 appears. The number of edges & triangles in the lowest five polygons = 210 +
126 = 336. This, too, is a structural parameter of the superstring, being the number of
circularly polarized oscillations in one revolution of each whorl of the UPA/superstring. Both
numbers are consistent with the Tetrad Principle governing Tree of Life parameters because
84 = 1^{2} + 3^{2} + 5^{2} + 7^{2}
and
336 = 4×84 = 2^{2}×84 = 2^{2}(1^{2} + 3^{2} +
5^{2} + 7^{2})
= 2^{2} + 6^{2} + 10^{2} +
14^{2}.
As
10 = 1 + 2 + 3 + 4
and
30 = 1^{2} + 2^{2} + 3^{2} + 4^{2},
the number of edges of the 180 triangles is
300 = 10×30 =
(1+2+3+4)(1^{2}+2^{2}+3^{2}+4^{2}).
The number of edges and triangles is
480 = 2×240 = 2(2^{4} + 2^{5} + 2^{6} +
2^{7}) = 2^{5} + 2^{6} + 2^{7} + 2^{8}.
= 4×120 =
2^{2}(2^{2} + 4^{2} + 6^{2} + 8^{2}) = 4^{2} +
8^{2} + 12^{2} + 16^{2}.
This property again illustrates the Tetrad Principle.
The number of geometrical elements surrounding the seven centres is 600,
which has the tetractys representation:
Remarkably, this is the number of geometrical elements surrounding the axis
of the disdyakis triacontahedron, for it has 60 vertices surrounding this axis, each joined
internally to its centre as an edge of an internal triangle, 180 edges, each the edge of an
internal triangle, and 120 triangles — a total of (60+60+180+180+120=600) elements.
The Divine Name EL assigned to Chesed with number value 31
prescribes the seven layers of vertices because the total number of vertices is 127, which is
the 31st prime number. In view of the fact that there are seven layers, it is
fitting that 127 is the seventh Mersenne prime number! EL prescribes the disdyakis
triacontahedron also because it consists of 31 vertices and their mirror
images. The Divine Name YAH assigned to Chokmah with number value 15
prescribes the vertex composition of the polygons because 120 vertices surround their centres
and 120 is the 15th triangular number:
120 = 1 + 2 + 3 + … + 15.
Including the apex and nadir of the disdyakis triacontahedron, there are
129 vertices. This is the number value of YAHWEH SABAOTH, the Godname of
Netzach, showing how the Godname of the seventh Sephirah prescribes the seven
layers of vertices in the disdyakis triacontahedron.
We saw earlier that the 8sided polygon at the base of the disdyakis
triacontahedron comprises 80 vertices, edges & triangles when its sectors
are divided into three triangles. With these triangles regarded as tetractyses, the polygon has
120 yods around its centre. In other words, it has as many yods around
15
its centre as all seven polygons have vertices surrounding their centres!
When each sector is regarded as a tetractys, the 8sided polygon has 49 yods.
This shows how EL CHAI, Godname of Yesod with number value 49, prescribes the
seven polygons, which begin and end with this one.
Table 5. Yod composition of the seven polygons (single tetractys).


n

Number of vertices
n+1

Number of yods
6n+1

Number of hexagonal yods
5n

1 

8

8+1

48+1

40

2 

10

10+1

60+1

50


Subtotal 
18

18+2

108+2

90

3 

6

6+1

36+1

30

4 

12

12+1

72+1

60


Subtotal 
36

36+4

216+4

180

5 

6

6+1

36+1

30

6 

10

10+1

60+1

50

7 

8

8+1

48+1

40


Total 
60

60+7

360+7

300

Table 4 indicates that the four polygons in each half of the disdyakis
triacontahedron have 360 geometrical elements surrounding their centres (including
36 polyhedral vertices), the fourth polygon having 36
triangles. This demonstrates how the Godname ELOHA assigned to Geburah with number value
36 prescribes the polygons formed by the 60 vertices of the disdyakis
triacontahedron. They have 72 vertices (36 polyhedral
vertices, 36 nonpolyhedral vertices).
Including their four centres, the first four polygons have
76 vertices. This is the number value of YAHWEH ELOHIM, the Godname of
Tiphareth, showing how this Godname prescribes each half of the disdyakis triacontahedron
The first four polygons have 36 internal vertices
surrounding their centres and 180 edges in their tetractyses, that is, 216
vertices & edges. This is the number value of the Sephirah Geburah, whose Godname has
the number value of 36.
5. The yod composition of the seven
polygons Table 5 lists the yod populations of the seven polygons when their sectors
are turned into tetractyses. Figure 16 displays the yods in the first four polygons. The
polygon in the second layer has 10 vertices and therefore is 10sided even though its
boundary has six straight lines. Similarly, the polygon in the fourth layer is 12sided
because it has 12 vertices, even though it is bounded by eight straight lines. The seven
polygons have 67 yods at the corners of their 60 tetractyses, the first two
polygons have 20 yods at corners of their 18 tetractyses and the last five polygons have 47
yods at the corners of their 42 tetractyses. As 47 is the 15th prime
number, this shows how the Godname YAH with number value 15 prescribes the
last five polygons. They have 210 (=21×10) hexagonal yods, showing how the
Godname EHYEH with number value 21 prescribes the last five polygons. The
first four polygons have 36 yods at corners of their 36
tetractyses surrounding their centres, showing how the Godname ELOHA of Geburah with number
36 prescribes these polygons. They have 216 yods
surrounding their centres, where 216 is
16
the number value of Geburah. All seven polygons have 360
(=36×10) yods surrounding their centres, yet again demonstrating how ELOHA
prescribes the seven polygons. The number of yods in the fourth polygon is 73,
which is the number value of Chokmah. It has 72 yods surrounding its centre.
This is the number of Chesed, the Sephirah next below Chokmah on the righthand Pillar of
Mercy. The number values of Chokmah, Binah, Chesed and Geburah measure in a natural way
different groups of yods in the seven polygons. EHYEH also prescribes the central layer of the
disdyakis triacontahedron because 73 is the 21st prime
number.
There are 307 yods on the edges of the 60 tetractyses. ELOHIM SABAOTH with
number value 153 prescribes the shapes of the polygons because 307 is the
153rd odd integer after 1. Formally including the two A vertices above and
below the sheets, 309 yods shape the nine layers of vertices.
ADONAI MELEKH with number value 155 prescribes them as 309
is the 155th odd integer after 1. 247 new yods are needed to delineate the
edges of the tetractyses (240 hexagonal yods on their 120 edges and seven centres). We found
earlier that they are composed of 247 geometrical elements, seven of which are their centres
(see Table 1). Remarkably, the 240:7 division exists for both yods and
geometrical elements.
Now consider the last four regular polygons enfolded in the inner Tree of
Life that are the counterparts of the polygons in onehalf of the disdyakis triacontahedron.
Tabulated below is the number of yods in each polygon outside the root edge, the number of
centres of their tetractyses and the number of yods on the edges of tetractyses:
Table 6. Yod composition of the last four enfolded, regular polygons.
There are 168 yods in the four polygons outside the root
edge on the edges of their 36 tetractyses (Fig. 17). The number value of Cholem Yesodeth, the Mundane Chakra
of Malkuth, defines the shapes of all the tetractyses that form the four polygons having
counterparts in the disdyakis triacontahedron. It demonstrates the meaning of this Mundane
Chakra: it expresses the form that the Tree of Life takes in the physical world. This form
is the heterotic superstring and 168 is its structural parameter, as many
previous articles have discussed. The first two polygons have the same number of yods (84)
as the last two. Thus, there is the same 84:84 division as that which appears in the 84
edges and 84 vertices and triangles of the tetractyses in the last five polygons. This
distinction manifests in the superstring as the
17
840 circularly polarised waves in the outer half of each of its 10 curves
and the 840 oscillations in its inner half (Fig. 18).
We saw earlier that, associated with the first two regular polygons — the
hexagon and the octagon — are 80 yods symbolizing the 80
bones of the axial skeleton, whilst the decagon and dodecagon has 126 yods symbolizing the 126
bones of the appendicular skeleton. Now we find that two pairs of polygons in a different order
display an 84:84 division of yods on the boundaries of tetractyses that has significance for
superstrings.
According to Table 6, the first four polygons of vertices in the disdyakis triacontahedron
have 184 yods on the edges of their tetractys sectors and the
last three polygons have 123 such yods. The seven polygons therefore have
307 yods lining their tetractyses. 307 is the 153rd odd integer after 1. This
shows how the Godname ELOHIM SABAOTH of Hod with number value 153 prescribes
the seven polygons of vertices.
Table 7 lists the yod populations of the seven polygons when their sectors
are turned into three tetractyses:
Table 7. Yod populations of the seven polygons (three tetractyses).


n

Number of vertices
2n+1

Number of yods
15n+1

Number of hexagonal yods
13n

1 

8

16+1

12+1

104

2 

10

20+1

150+1

130


Subtotal 
18

36+2

270+2

234

3 

6

12+1

90+1

78

4 

12

24+1

180+1

156


Subtotal 
36

72+4

540+4

468

5 

6

12+1

90+1

78

6 

10

200+1

150+1

130

7 

8

16+1

120+1

104


Total 
60

120+7

900+7

780

120 yods are vertices surrounding their seven centres, where 120 is the
15th triangular number. The Godname YAH with number 15 also
prescribes the number of yods per sector surrounding their centres. Including the seven centres
and the A vertices at the apex and nadir of the disdyakis triacontahedron, there are
(120+7+2=129) vertices altogether. The Godname YAHWEH SABAOTH with number
value 129 therefore prescribes the disdyakis triacontahedron.
The seven polygons have 900 yods surrounding their seven centres. Article
26^{11} showed that, when both the internal and external triangles
formed by the edges of the disdyakis triacontahedron are divided into three tetractyses, it
has 900 triangles. The yod population of the seven polygons is equal to the number of
triangles needed to build the disdyakis triacontahedron.
An nsided polygon has (15+1) yods when its sectors are
constructed from three tetractyses. Three yods
18
19
in each sector are at the centres of tetractyses, so that (12n+1) yods lie
on the boundaries of the 3n tetractyses. The seven polygons have (12×60+7=727) boundary yods.
727 is the 129th prime number, showing the remarkable way in which YAHWEH
SABAOTH, the Godname of Netzach (the seventh Sephirah) with number value
129, prescribes the seven layers of vertices in the disdyakis triacontahedron.
720 boundary yods surround the seven centres. This constitutes a Tree of Life pattern because
the seven separate, regular polygons of the inner Tree of Life have 720 yods surrounding their
seven centres when their sectors are constructed from three tetractyses. The two sets of
polygons are compared in Fig. 19 (for the sake of clarity, the regular counterparts of the polygons
in the seven layers are shown). It is yet more proof that the disdyakis triacontahedron is
the polyhedral form of the Tree of Life.
The first four polygons have (12×36+4=436) yods on the
edges of tetractyses. Of these, 76 are vertices, where 76 is
the number value of YAHWEH ELOHIM, Godname of Tiphareth, leaving 360 (=36×10)
yods. YAHWEH ELOHIM prescribes the first four polygons in terms of their vertices and the
Godname ELOHA with number 36 prescribes those yods that are not vertices.
The first or last three polygons with 24 vertices have (12×24+3=291) yods
lying on edges of tetractyses. 288 yods surround their three centres, where 288 = 1^{1}
+ 2^{2} + 3^{3} + 4^{4}, showing how the integers 1, 2, 3 & 4
symbolized by the tetractys express properties of Tree of Life patterns. They have 240
geometrical elements surrounding their centres. As the seven separate polygons of the inner
Tree of Life have 288 yods surrounding their centres and 240 hexagonal yods, this indicates
that each set of three polygons constitute a Tree of Life pattern, just as the first four
polygons do.
Table 7 indicates that the seven polygons need 907 yods to build them out of
tetractyses. 907 is the 155th prime number, where 155 is
the number value of ADONAI MELEKH, the Godname of Malkuth. This is a truly sublime property,
for it powerfully demonstrates how the Godname of the most material Sephirah — Malkuth —
arithmetically prescribes the number of yods needed to construct all seven polygons from
tetractyses. ADONAI MELEKH also prescribes the fourth polygon because it has 156 hexagonal
yods, where 156 is the 155th integer after 1. Article 26 showed^{12} that the number of yods other than polyhedral vertices needed
to construct each of its 120 faces from three tetractyses is 1560 (= 156×10), so that ADONAI
MELEKH prescribes the shape of the surface of the disdyakis triacontahedron as well as its
interior polygons.
The last three polygons have 363 yods. This is the number
value of SHADDAI EL CHAI, the complete Godname of Yesod. It differentiates between the first
four layers forming half of the polyhedron and its remaining three layers. The latter have
48 vertices surrounding their centres, that is, they have 51 vertices. 51 is
the 50th integer after 1, showing how ELOHIM, the Godname of Binah with number
value 50, differentiates the first four layers of vertices from the last three
layers. The first two polygons have 272 yods. This is the number value of
Cherubim, the Order of Angels assigned to Yesod.
The seven polygons have 187 yods that lie on internal edges
of tetractyses. This is the number value of Auphanim, the Order of Angels assigned to
Chokmah. The first four polygons have 112 such yods. This is the number value
of Beni Elohim, the Order of Angels assigned to Hod. 36 of these are
vertices, leaving 76 radial yods that are not vertices of the disdyakis
triacontahedron. YAHWEH ELOHIM therefore prescribes the four layers of each half of the
disdyakis triacontahedron. The number of yods other than polyhedral vertices that lie on the
300 edges of the 180 tetractyses in the first four polygons is 400 = (2+4+6+8)^{2}.
This is another beautiful illustration of the Tetrad Principle because 400 is the square of the
sum of the first four even integers.
6. Encoding of
the superstring structural parameter 336
The disdyakis triacontahedron has 60 vertices, 180 edges and 120 triangular faces (360
geometrical elements) surrounding any axis passing through two diametrically opposite vertices.
The fourth polygon of vertices forming its equator has 12 vertices and 12 edges, totalling 24
geometrical elements. (360– 24=336) elements lie above and below the equator,
168 on either side. As the polyhedron has 180 edges, there are
(180–12=168) edges above and below its equator, 84 on either side. There are
48 vertices above and below the equator, 24 vertices on either side.
Furthermore, there are 60 triangles on each side of the equator. Therefore, the
168 geometrical elements on each side comprise 24 vertices, 84 edges and 60
triangles, that is, 84 edges and (24+60=84) vertices and triangles. Once again, we find the
84:84 division in the superstring structural parameter 168. It is created by
the mirror symmetry of the disdyakis triacontahedron whereby every geometrical element has its
mirror image, both 84’s referring to the same upper or lower half of the polyhedron. Its
inversion symmetry divides the 336 elements into two sets of 168 geometrical
elements, one the mirror image of the other.
20
21
22
The number 336 has the beautiful property:
336 = 4×84 = 2^{2}×(1^{2} + 3^{2} + 5^{2} + 7^{2}) =
2^{2} + 6^{2} + 10^{2} + 14^{2}.
It is an illustration of the Tetrad Principle, which determines parameters
of any system designed according to the blueprint of the Tree of Life.
The 12sided polygon in the equator has 24 vertices and edges on its
boundary and 24 edges and triangles in its sectors (12 edges & 12 triangles). Using the
results obtained above, the pattern of elements in the disdyakis triacontahedron is set out
below:

vertices

internal edges

external edges

internal triangles

external triangles

Total

First 3 layers: 
24

24

84

24

60

216

Equator: 
6+6

6+6

6+6

6+6

0

48

Last 3 layers: 
24

24

84

24

60

216

Total =

60

60

180

60

120

480

The first three layers have 216 internal and external
geometrical elements, where 216 is the number value of Geburah. Similarly for
the last three layers. The fourth polygon has 48 elements (24 internal, 24
external) surrounding its centre, totalling 49 elements. This shows how the
Godname EL CHAI of Yesod with number value 49 prescribes the central polygon.
48 is the number value of Kokab, the Mundane Chakra of Hod. The seven
layers have 480 geometrical elements made up of 240 elements and their 240 mirror images. Each
set of 240 elements consists of 168 external ones (24 vertices, 60 triangles
& 84 edges) belonging to the three layers and 72 elements comprising six
vertices and six external edges of the fourth polygon, 30 internal edges and 30 internal
triangles. This 72:168 division corresponds to the division
of nonzero roots of the superstring gauge symmetry group E_{8}. The
72 elements correspond to the 72 nonzero roots of its
subgroup E_{6} and the 168 elements correspond to the
168 roots of E_{8} that are not also roots of E_{6}. The
geometry of the disdyakis triacontahedron therefore encodes the 480 nonzero roots of the
E_{8}×E_{8} heterotic superstring, the direct product of the two identical
groups reflecting the inversion symmetry of the polyhedron, not the distinction between its
exterior and interior, which creates a 120:360 division. Onehalf of the polyhedron encodes the
forces between superstrings of ordinary matter, its mirror image half encodes the forces
between superstrings of shadow matter.
The 168:24:24:168 pattern of yods in the
(6+6) polygons of the inner Tree of Life appears in the I Ching table (Fig. 20). The 24 corners outside the root edge of the first six polygons on
one side correspond to
23
24
the 24 lines in the upper trigrams of the eight diagonal hexagrams and the
24 corners on the other side correspond to the 24 lines in the lower trigrams. The
168 yods associated with each set of polygons that are not corners correspond
to the 168 lines and broken lines in the 28 hexagrams on either side of the
diagonal. The fourth polygon in the disdyakis triacontahedron has (6+6) polyhedral edges, (6+6)
polyhedral vertices, (6+6) internal edges and (6+6) internal triangles, that is, eight groups
of six geometrical elements. They correspond to the eight diagonal hexagrams, each with six
lines/broken lines. The disdyakis triacontahedron is the polyhedral counterpart of the ancient
I Ching table of 64 hexagrams, each line or broken line denoting a geometrical
element of its faces and internal polygons. The 12 lines and 12 broken lines in the lower
trigrams of the diagonal hexagrams correspond to the 12 vertices and 12 edges of the fourth
polygon. The 168 lines and broken lines in the 28 offdiagonal hexagrams in
the lower half of the table correspond to the 168 geometrical elements in the
faces either above or below the equator. The 12 lines and 12 broken lines in the upper trigrams
of the diagonal hexagrams correspond to the 12 internal edges and 12 triangular sectors of the
12sided polygon, whilst the 168 lines and broken lines in the hexagrams above
the diagonal correspond to the 168 elements that are the mirror images of the
other set. There are (48+168+168=384)
elements in the faces of the disdyakis triacontahedron and in its central polygon, i.e., lines
and broken lines in the I Ching Table. This is the 383rd integer after 1. 383 is the
76th prime number, showing how YAHWEH ELOHIM, Godname of Tiphareth, prescribes
the disdyakis triacontahedron.
The seven polygons in the disdyakis triacontahedron have 60 internal edges
and 60 internal triangles (30 edges and 30 triangles per half). Including the
168 external elements in the first three layers and similarly for the last
three layers, there are (30+30+168=228) elements and their mirror images other
than the 12 vertices and 12 edges in the equator of the disdyakis triacontahedron. Including
the top and bottom A vertices, there are 229 elements in each half of the disdyakis
triacontahedron other than its equator. 229 is the 50th prime number, showing
how ELOHIM with number value 50 prescribes the disdyakis triacontahedron.
Figure 21 depicts the inner Tree of Life counterpart of this. There are 228
yods other than external corners of polygons
associated with each set of seven polygons (456 yods in total). Outside the
root edge are 227 yods in each set of polygons other than their corners. The Divine Name EL
CHAI of Yesod with number value 49 prescribes this set of yods symbolizing the
geometrical elements of the seven polygons because 227 is the 49th prime
number and 228 is the 227th integer after 1. Associated with either set of the first six
enfolded polygons are 168 yods symbolizing the 168 elements
in the three polygons above or below the equator. 60 yods symbolizing the 60 elements inside
each half of the disdyakis triacontahedron are associated with each dodecagon. The first six
polygons in the Tree of Life and their mirror images, reflected across the root edge, are the
counterpart in the disdyakis triacontahedron of the first three layers of polygons and their
mirror images, inverted through the centre of the polyhedron.
The (6+6) enfolded polygons constitute a Tree of Life pattern in themselves
because they are prescribed by the ten Godnames. The simplest example of this is the fact that
they have 50 corners, where 50 is the number value of ELOHIM,
Godname of Binah (the complete prescription is given in Article 4).^{13} As before, the 168 yods associated with each
set of the first six polygons symbolize the 168 geometrical elements in
each set of three polygons, whilst the 24 corners on each side of the root edge denote the
24 vertices and edges on the equator and the 24 edges and triangles inside the polyhedron
(Fig. 22). These degrees
25
of freedom mark out the boundary of the new Tree of Life pattern just as the
48 elements create the polygon that completes the sequence of polygons forming
half of the disdyakis triacontahedron — a Tree of Life pattern in itself as the remainder of
the polyhedron is merely its mirror image.
Figure 23 displays the equivalence between the seven layers of polygons with
60 vertices and the first (6+6) regular polygons of the inner Tree of Life. The six vertices
of the fourth polygon and their mirror images correspond to the five centres and one end of
the root edge on either side of the root edge. The eight vertices of the first polygon
correspond to the eight external corners of the decagon. The 10 vertices of the second
polygon correspond to the 10 external corners of the first four regular polygons. The six
vertices of the third polygon correspond to the six external corners of the octagon. The
mirror symmetry of the inner Tree of Life corresponds to the invariance of the disdyakis
triacontahedron under the operation of inversion.
Figure 18 shows that each curve of the E_{8}×E_{8} heterotic
superstring makes 1680 circularly polarised oscillations as it twists five times around its
axis. 336 oscillations occur during one complete revolution, 168 in each
half revolution. They correspond to the 168 geometrical elements defining
the shape of each set of three polygons. The 24 polyhedral vertices and edges on the equator
correspond to the 24 E_{8} gauge charges that are spread along each curve.
E_{8}×E_{8} has 336 nonzero roots that do not belong to the
subgroup E_{6} in each factor. Their associated gauge charges are sources of
10dimensional vector gauge fields, which therefore have 3360 components. As discussed in
Article 31,^{14} there are 28 regular and semiregular solids defined by the
vertices of the disdyakis triacontahedron. When their faces are constructed from
tetractyses, they have 3360 hexagonal yods that symbolize the seven Sephiroth of
Construction. This
amazing property shows how the superstring parameter 3360 is present in the
geometrical potential of the disdyakis triacontahedron as a number characterizing the forms of
all the possible polyhedra within it. It is a Tree of Life parameter because, when the seven
enfolded polygons of the inner Tree of Life are constructed not from the tetractys but from its
next higher order — the 2ndorder tetractys — it is found to have exactly 3360 yods
(Fig. 24). This is the number of circularly polarized oscillations made by
all ten curves of the E_{8}×E_{8} heterotic superstring during one complete
revolution. Following through with the analogy with the 168 elements in the
60 faces of the disdyakis triacontahedron above or below its equator, it would be more
accurate to associate each set with the inner or outer half of the superstring curves, so
that the 168 elements in one set correspond to 168
oscillations in an outer halfrevolution and the 168 elements in the other
set correspond to the 168 oscillations made as the curve winds in a half
revolution in the core of the
26
superstring. It is truly remarkable that the same number
168 should be encoded not only in the faces above or below the equator but
also in the fourth polygon forming the equatorial plane as the extra number of yods needed to
construct its 12 sectors from three tetractyses!
6. The Platonic
Lambda
In his Timaeus, Plato described how the Demiurge measured the World Soul, or substance of the
spiritual universe, as a strip divided according to the simple proportions of the first three
squares of 2 and 3. This is traditionally represented by his ‘Lambda,’ socalled because of its
resemblance to the Greek letter Λ (Fig. 25). These numbers line two sides of a tetractys array of ten numbers
from whose relative proportions the scientists and musicians of ancient Greece worked out
the frequencies of the notes of the now defunct Pythagorean musical scale.^{15} The numbers missing from the Lambda are shown in red in
Fig. 25. The sum of the 10 integers is 90 and the sum of the integers 1, 8
and 27 at the corners of the tetractys is 36. The seven integers at the
centre and corners of the hexagon shown in Fig. 25 with dashed edges add up to 54 and the sum of the integers at the
corners of the hexagon is 48.
The archetypal nature of this arrangement of integers was demonstrated in
previous articles. It is known today for how it generates the tone ratios of the notes of the
Pythagorean musical scale, the ancient father of the modern equaltempered scale. What is not
realised by historians of music and mathematics is that it has connections to the I Ching
table,^{16} superstrings,^{17} the seven musical scales^{18} and the nature of the spiritual cosmos.^{19} This is because it is the arithmetic representation of the universal
Tree of Life pattern underlying these examples of holistic systems. The first four enfolded
polygons in the inner form of the Tree of Life have 90 yods outside their shared edge
(Fig. 26). This is the sum of the ten numbers of the Lambda Tetractys. This
cannot be coincidental because the yod
populations of the individual polygons are also the sums of
diagonal rows of integers in the Lambda! Indeed, the individual numbers in each
diagonal row can be seen to denote different sets of yods in each polygon (Fig. 27)! They are listed below:
Triangle
1 centre;
2 2 hexagonal yods on horizontal edge of tetractys;
4 4 hexagonal on 2 internal edges;
8 8 yods on edges of triangle & at centres of 3 tetractyses.
Square
3 centre & 2 corners;
6 6 hexagonal yods on sides of square;
12 12 internal hexagonal yods.
Pentagon
9 9 yods at corners & centres of 5 tetractyses;
18 18 hexagonal yods.
Hexagon
27 27 yods.
Article 26^{20} showed how the Lambda Tetractys matches the tetractys
composition of the disdyakis
27
triacontahedron. Constructed from tetractyses, its 120 triangular faces
divide into (120×3=360) tetractyses. Each of its 180 edges is the side of an internal triangle
formed by two vertices and its centre. They are constructed from (180×3=540) tetractyses. The
total number of tetractyses forming the disdyakis triacontahedron = 360 + 540 = 900. The
following correspondences appear:
The numbers 1^{3}, 2^{3} and 3^{3} at the
shapedefining corners of the Lambda Tetractys add up to 36, which is the
number of tetractyses whose yod population is the number of tetractyses creating the
shape of the disdyakis triacontahedron. The seven numbers located at the centre and
corners of a hexagon add up to 54 — the number of tetractyses whose yod population is the
number of tetractyses inside the disdyakis triacontahedron. Notice that the number
36 — the number value of the Godname ELOHA — defines the external shape in
both cases and that the number 54 in both cases defines the interior. Moreover, the largest of
the seven integers in the hexagon is 18, whilst the number of internal tetractyses creating the
180 edges is 180 = 18×10. Lastly, of the 900 tetractyses in the disdyakis triacontahedron, 60
(=6×10) are tetractyses having edges that are edges of the rhombic triacontahedron, leaving 840
(=84×10) tetractyses other than these. The integer 6 at the centre of the Lambda Tetractys
therefore corresponds to the 60 internal tetractyses of the disdyakis triacontahedron sharing
edges with the rhombic triacontahedron, whilst the sum (84) of the integers on the boundary of
the Lambda Tetractys correspond to the 840 additional tetractyses needed to construct the
disdyakis triacontahedron. Six arithmetic properties of the Lambda Tetractys correlate
precisely with the geometry of the disdyakis triacontahedron. This cannot be coincidental.
Instead, it demonstrates that the Lambda Tetractys and the disdyakis triacontahedron are,
respectively, arithmetic and geometric expressions of the same universal archetype
expressing the holistic nature of God.
Table 7 indicates that the seven polygons formed by the vertices of the
disdyakis triacontahedron have 900 yods surrounding their centres, that is, the number of
yods in 90 tetractyses. The first four polygons in half the disdyakis triacontahedron have
540 yods surrounding their centres. This is the number of yods in 54 tetractyses. The
remaining three polygons have 360 yods surrounding their centres — the number of yods in
36 tetractyses. This 54:36 division is identical to that
displayed by the Lambda Tetractys. 60 yods of the fourth (12sided) polygon other than its
centre lie on its sides and on radial edges of tetractyses — the number of yods in six
tetractyses. They correspond to the number 6 at the centre of the Lambda Tetractys. There
remain 480 yods surrounding the centres of the first four polygons — the number of yods in
48 tetractyses. They correspond to the sum (48) of the
integers at the corners of the hexagon in the Lambda Tetractys. There are (900–60=840) yods
in the seven polygons not lying on the sides of the fourth polygon and radial edges of
tetractyses. They correspond to the sum of the integers surrounding the integer 6 at the
centre of the Lambda Tetractys.
As
29^{2} – 1 = 840 = 3 + 5 + 7 + 9 + … + 57,
the second perfect number 28 defines the number 840 because the latter is
the sum of the first 28 odd integers after 1. The third perfect number 496 is
the dimension of the two possible gauge symmetry groups of superstrings (SO(32) and
E_{8}×E_{8}). As discussed in Section 5, 840 is also the number of circularly
polarised oscillations made during 2½ revolutions in an outer or inner half of each of the ten
closed curves making up the heterotic superstring (see Fig. 18). As the seven polygons have 900 yods around their centres, of which
60 are vertices of the disdyakis triacontahedron, 840 yods surround their centres that are
not polyhedral vertices. In other words, 840 extra yods are needed to construct the
seven polygons from tetractyses. The appearance of the superstring structural parameter
840 in the polygons formed by the vertices of the disdyakis triacontahedron is astounding
and demonstrates the formcreating character of this number. It was encountered earlier in
the analysis of the disdyakis triacontahedron, when it was found to have 840 tetractyses not
sharing edges with the 60 edges of the rhombic triacontahedron formed by B and C vertices.
The seven polygons therefore collectively encode both the
28
form of the superstring and its gauge symmetry group E_{8}. As we
saw earlier, the central polygon needs 168 extra yods to construct it from
tetractyses. It, too, embodies a superstring structural parameter, for each closed curve of the
heterotic superstring makes 168 oscillations during half of a complete
revolution around its spin axis.
To summarise, the following remarkable correspondences exist between the
Lambda Tetractys, the disdyakis triacontahedron and the seven polygons formed by its
vertices:
Lambda Tetractys


Disdyakis triacontahedron


7 Layers

Sum of 10 integers = 90 

900 (=90×10) tetractyses 

900 (=90×10) yods surround 7
centres; 
Sum of integers at corners = 36 

360 (=36×10) tetractyses in 120
faces 

360 (=36×10) yods in lowest 3
polygons surround 7 centres; 
Sum of integers at centre and
corners of hexagon = 54 

540 (=54×10) tetractyses inside
polyhedron 

540 (=54×10) yods in first 4
polygons surround 7 centres; 
Integer at centre = 6 

60 (=6×10) internal tetractyses with
edges of rhombic triacontahedron 

60 (=6×10) yods on edges of
sectors of 4th polygon surround
7 centres; 
Sum of integers at corners of
hexagon = 48 

480 (=48×10) internal tetractyses
do not share edges of rhombic
triacontahedron 

480 (=48×10) other yods in first
4 polygons surround 7 centres; 
Sum of integers surrounding
centre = 84 

840 (84×10) tetractyses do not
share edges of rhombic
triacontahedron 

840 (=84×10) yods surround
centres other than those on
edges of 4th polygon 
The amazing parallels demonstrate the holistic nature of the disdyakis
triacontahedron because the construction of both its external shape and its seven interior
polygons from tetractyses conforms to the archetypal Lambda Tetractys array of integers, whose
ratios determine the tone ratios of the notes of the Pythagorean musical scale.
The corners of a tetractys symbolize the Supernal Triad and its seven
hexagonal yods denote the seven Sephiroth of Construction. The former correspond to the three
lowest layers and the latter correspond to the first four layers in half the polyhedron. The
yod at the centre of a tetractys symbolizes Malkuth, the lowest Sephirah signifying the
material aspect of the Tree of Life, e.g., the physical universe. Its counterpart in the seven
layers of the disdyakis triacontahedron is the formgenerating boundary of
the 12 sectors of the polygon in the central layer of
vertices. This is delineated by 60 yods (36 yods on the edges of
the
polygon and 24 yods inside it). Division of these sectors into three
tetractyses requires 168 more yods (see Fig. 13). This is the number value of Cholem Yesodeth, the Mundane
Chakra of Malkuth, confirming the correspondence with Malkuth of the number 6 at the centre
of the Lambda Tetractys and the 12sided polygon as the middle layer of the disdyakis
triacontahedron. The 12sided polygon has 156 hexagonal yods, where 156 is the
155th integer after 1, showing how ADONAI MELEKH, the Godname of Malkuth,
prescribes this polygon. 155 is also the number of hexagonal yods
associated with each
29
dodecagon in the inner Tree of Life when each of its 12 sectors is divided
into three tetractyses (Fig. 28). The number 156 is also the sum of the first 12 even integers that
can be assigned to the vertices of a 12sided polygon (Fig. 29). Remarkably, it is the sum of the letter values of all possible
combinations of the Divine Name YHVH assigned to Chokmah with number value
26:


Y = 10, H = 5, V = 6

Y
+ H + V

= 
21

YH + HV + YV + HH 
= 
52

YHV + HVH + HYH 
= 
57

YHVH 
= 
26

TOTAL 
= 
156

This illustrates the mathematical power in a Godname, for the disdyakis
triacontahedron is the 26th of the family of 13 Archimedean solids and their
duals — the 13 Catalan solids, and so it is chosen by YAHWEH as the mother shape encoding the
archetypal pattern of the form of the Tree of Life.
According to Table 7, the number of hexagonal yods in the seven polygons is 780. This is
the sum of the first 12 multiples of 10 that can be assigned to its 12 corners:
780 = 10 + 20 + 30 + … + 120.
The 12sided dodecagon is the tenth regular polygon and 120 is the sum of
the first ten odd integers:
120 = 3 + 5 + 7 + … + 21.
The number 10 or Decad was called “All Perfect” by the ancient Pythagoreans.
We see that it defines 1. the 120 vertices surrounding the centres of the seven polygons when
they are constructed from tetractyses, 2. the 780 hexagonal yods that these tetractyses contain
and 3. — through the Lambda Tetractys, when its numbers are multiplied by 10 — the 900 yods that
surround their centres. Truly, 10 is the measure of divine perfection, whose polyhedral
manifestation is the
disdyakis triacontahedron.
7. Encoding of the superstring parameters 840 &
1680 Considered as tetractyses, the 60 sectors of the seven polygons
are made up of 360 yods surrounding their seven centres. Four of their 60 edges connect
pairs of A vertices (see the 10sided polygon in Figures 3 & 4). These are not edges of the disdyakis triacontahedron. The seven polygons
share 56 sides with it, leaving (180–56=124) unshared edges. Each one has two hexagonal
yods. The 124 unshared edges have (124×2=248) hexagonal yods. This is the
dimension of the superstring gauge symmetry group E_{8}. There are nine hexagonal
yods inside each of the 120 faces of the disdyakis triacontahedron when divided into three
tetractyses, i.e., (120×9=1080) hexagonal yods, and 120 vertices, totalling 1200 yods. The
polygons have four sides with eight hexagonal yods unshared with the faces of the disdyakis
triacontahedron. They have (360–8=352) yods that are either inside them or lying on edges
shared with the polyhedron. The faces and internal polygons of the disdyakis triacontahedron
have (248+1080+120+352=1800) such yods. Of these, (1800–120=1680) are yods
that are not vertices inside faces. Other than the vertices of the polyhedron, they are all
hexagonal yods. Its mirror symmetry means that every one of the 840 yods in one half of the
polyhedron has a yod that is its mirror image. This 840:840 division manifests in the
E_{8}×E_{8} heterotic superstring as the 840 circularly polarized waves
in its outer half and the 840 waves in its inner half (see Fig. 18). Divided into three tetractyses, a face of the disdyakis
triacontahedron has three internal edges, three triangles and a vertex, making seven
geometrical elements. The 120 faces therefore have (7×120=840) elements inside them. The
appearance again of the number 840 is further evidence that the disdyakis triacontahedron
encodes the oscillatory form of the E_{8}×E_{8} heterotic
superstring.
30
A polygon with n sides has (15n+1) yods, that is, 14n yods
other than its internal corners of tetractyses and centre. The first three polygons with 24
edges have (14×24=336) such yods, as have the last three polygons. The fourth polygon with 12
sides has (14×12=168) yods other than internal corners and centres. All seven
polygons have (336+168+336=840) such yods (60 vertices of polygons and 780
hexagonal yods). Once again, we see this structural parameter of the heterotic superstring
encoded in the seven polygons. The 240 vertices and edges of the disdyakis triacontahedron
create 840 geometrical elements making up
the interior of the faces of the disdyakis triacontahedron and 840 yods
other than its vertices making up its seven layers of polygons. Figure 30 shows the counterpart in the inner Tree of Life of this
representation of the inner and outer halves of the heterotic superstring. When the 24
sectors of the pair of dodecagons are changed into 2ndorder tetractyses, there are 1680
yods outside their root edge surrounding their centres, 840 yods in each polygon. The two
dodecagons correspond to the two halves of the disdyakis triacontahedron. The (12+12=24)
sectors signify the 24 gauge bosons of E_{8} whose charges are spread along each
curve as 1680 circularly polarised waves. They correspond in the disdyakis triacontahedron
to the 12 vertices and 12 edges on its equator.
The 840 yods in the seven polygons include the eight hexagonal yods of the
four AA edges of the 10sided polygon on the second level that are not shared with the
polyhedron and which are excluded from the 1680 hexagonal yods and polyhedral vertices
calculated above. The seven polygons therefore have (840–8=832) yods made up of 60 polyhedral
vertices and 772 hexagonal yods other than the unshared hexagonal yods on the four edges, that
is, 833 yods when the topmost A vertex is included. 833 is
the number value of Chaioth ha Qadesh (“Holy Living Creatures”), the Order of Angels
assigned to Kether. Starting from a point, 833 more yods are needed to
construct from tetractyses the seven sheets of vertices and the lowest A vertex. Kether is the
seed point from which the Tree of Life grows.
31
The 56 edges of the polygons shared with the disdyakis triacontahedron
define 56 tetractyses with (6×56=336) yods surrounding their centres. 168 yods
are in the 28 tetractyses in each half of the polyhedron. 84 yods lie on their edges and 84
yods are inside them. The 168:168 pattern is the same as that
displayed by the edges of the disdyakis triacontahedron above and below its equator. The four
shared edges not included in the counting have 24 yods, just as the 12 edges in the 12sided
polygon in the equator have 24 hexagonal yods. In fact, the pair of dodecagons in the inner
Tree of Life has 336 yods other than their corners, 168 in each one
(Fig. 31).
The 56 sectors of the polygons whose edges are shared with the disdyakis
triacontahedron have 840 yods surrounding their centres when they are divided into three
tetractyses. This is the same as the number of yods other than corners in all 60 sectors of the
seven polygons. In the former case, we have 56×15 = 840, where
15 is the number of yods per sector; in the latter, we have 60×14=840, where
14 is the number of yods per sector other than corners. Amazingly, both cases encode the same
superstring parameter 840. However, as each sector has 10 geometrical elements (two vertices,
five edges and three triangles), the 56 sectors have (56×10=560) elements. Each set of 28
sectors has 280 elements. 280 is the number value of
Sandalphon, Archangel of Malkuth. This is a remarkable conjunction in the same context
of the number values of an Archangel and Mundane Chakra of the same Sephirah, for we have just
seen that the 28 sectors have 168 yods surrounding the centres of their
polygons and 168 is the number value of the Mundane Chakra of Malkuth.
The 56 shared edges have (2×56=112) hexagonal yods.
112 is the number value of Beni Elohim, the Order of Angels assigned
to Hod. There are also 112 hexagonal yods on the 56 radial edges of the 56
sectors. Including the seven centres, 119 yods line the sectors. The disdyakis triacontahedron
has 180 edges and 62 vertices, so that (180×2+62=422) yods
lie on its edges. (119+422=541) yods line the edges of the polyhedron and the 56 sectors of its
polygons. Two of these are the top and bottom A vertices. (541–2=539) yods other than these
vertices therefore line the (180+56=236) edges. Amazingly, 539 is the sum of the Godname
numbers of the seven Sephiroth of Construction:
31 + 36 + 76 +
129 + 153 + 49 + 65 =
539.
This demonstrates the constructive power of the Divine Names.
The faces of the disdyakis triacontahedron have the top and bottom A
vertices, 124 edges and 120 triangles that it does not share with the seven polygons, that is,
246 geometrical elements. This is the number value of Gabriel, the
Archangel of Yesod.
There are 127 yods on the internal edges of all 60 sectors of the polygons.
There are (422+127=549) yods lining both polyhedral edges and polygonal sectors. Excluding the
centres of the three polygons on each side of the equatorial plane, there are
543 boundary yods. This is the number value of EHYEH ASHER EHYER (“I am that I
am”), the complete Godname assigned to Kether.
With its faces constructed from three tetractyses, the disdyakis
triacontahedron has seven yods lying on edges of tetractyses inside each face.
32
There are (120×7=840) such yods. Together with its 62
vertices and 360 hexagonal yods lying on its edges, the polyhedron has
(840+62+360= 1262) boundary yods, that is, 1260 yods other
than its two opposite A vertices. This is the number value of Tarshishim, the Order of
Angels assigned to Netzach.
Of the 541 yods lining the edges of the polyhedron and the 56 sectors with
edges shared with it, 67 are vertices of the polygons.
(541–67=474) yods other than vertices lines these edges,
creating their shapes. 474 is the number value of Daath (“knowledge”), the
nonSephirah. The number 539 is the sum of 474 and 65, the
number value of ADONAI, Godname of Malkuth. It measures “knowledge of the Lord”.
8. The two sets
of three polygons
The seven Sephiroth of Construction of the Tree of Life are grouped above Malkuth into two
triads: Chesed–Geburah–Tiphareth and Netzach–Hod–Yesod (Fig. 32). In terms of human psychology, they represent, respectively, the
transpersonal levels of human consciousness and the incarnated personality (loosely,
“Spirit” and “Soul,” respectively). The two sets of three polygons above and below the
equatorial plane of the disdyakis triacontahedron correspond to this 3:3 pattern, with the
Malkuth aspect represented by the 12sided polygon, the last of the enfolded polygons of the
inner Tree of Life, which we saw earlier embodies the number value of Cholem
Yesodeth, the Mundane Chakra of Malkuth (see Fig. 13). A polygon and its mirror image are the counterpart of two
Sephiroth of Construction on the same pillar.
9. The Godnames prescribe the seven layers of
polygons Below are listed the ways in which the Godnames of the ten
Sephiroth mathematically define properties of the seven layers of vertices in the
disdyakis triacontahedron between two diametrically opposite A vertices:
Kether 
EHYEH: 21 
disdyakis triacontahedron has 21 layers in 3
perpendicular sets of 7;
21 A and B vertices in first 4 polygons and their
21 mirror images;
41 vertices of sectors in first 4 polygons (including top A vertex). 41 =
21st odd integer after 1;
5 lowest polygons with 168 geometrical elements have 42 sectors.
42 = 21st even integer;
210 (=21×10) hexagonal yods in last 5 polygons;
73 yods in 4th polygon. 73 =
21st prime number; 

EHYEH ASHER EHYEH: 543 
543 yods other than centres of 1st three polygons
on each side of equator lie on the edges of the disdyakis triacontahedron and
sectors of 7 polygons. 



Chokmah 
YAH: 15 
15 Golden Rectangles in 3 perpendicular sets of
5;
15 sets of seven layers created by 15 opposite
pairs of A vertices;
15 A vertices in half of disdyakis triacontahedron and their
15 mirror images;
47 vertices of sectors of lowest 5 polygons. 47 = 15th prime
number;
120 vertices of 180 tetractyses in 7 polygons surround their centres. 120 = 1 + 2 +
3 + … + 15; 

YAHWEH: 26 
26 A and B vertices (or A and C vertices) in
first 4 polygons;
51 vertices in first (or last) 3 polygons. 51 = 26th odd
integer. 



Binah 
ELOHIM: 50 
229 geometrical elements in each half of polyhedron (including top
or bottom A vertex). 229 = 50th prime number;
51 vertices in first (or last) 3 polygons. 51 = 50th integer after
1. 



Chesed 
EL: 31 
31 vertices in each half of polyhedron;
127 vertices of sectors of 7 polygons. 127 = 31st prime
number 



Geburah 
ELOHA: 36 
360 (=36×10) geometrical elements in faces
surround axis;
36 vertices of first 4 polygons;
36 yods along perimeter of 4th polygon;
360 (=36×10) geometrical elements surround centres of first 4
polygons with 36 vertices/sectors;
360 (=36×10) yods surround centres of 7 polygons;
360 (=36×10) hexagonal yods on boundaries of 108 tetractyses in
first 4 polygons. 



Tiphareth 
YAHWEH ELOHIM: 76 
76 vertices of tetractyses in first 4
polygons;
384 elements in faces of disdyakis triacontahedron and in central, 12sided
polygon. 384 = 383rd integer after 1. 383 = 76th prime
number. 
33
Netzach 
YAHWEH SABAOTH: 129 
129 vertices in 180 tetractyses of 7 polygons,
including the top
and bottom A vertices;
727 yods on boundaries of 180 tetractyses in 7 polygons. 727 =
129th prime number. 



Hod 
ELOHIM SABAOTH: 153 
307 yods on boundary of 60 sectors of 7 polygons. 307 =
153rd
integer after 1. 



Yesod 
EL CHAI: 49 
49 geometrical elements in 4th polygon;
49 yods in first (or last) polygon. 

SHADDAI EL CHAI: 363 
363 yods in first (or last) 3 polygons 



Malkuth 
ADONAI: 65
ADONAI MELEKH:155

129 vertices in 180 tetractyses of 7 polygons.
129 = 65th odd integer;
907 yods in seven polygons. 907 = 155th prime number. 
The last property is perhaps the most remarkable of these demonstrations of
the power of Godnames to define properties of a Tree of Life system such as the disdyakis
triacontahedron. Very aptly, the number value of the Godname of the Sephirah Malkuth signifying
the outer form of the Tree of Life defines a prime number that is the very number of yods
creating the complete form of the seven polygons!
References
^{1} Phillips, Stephen M. Article 32: "Derivation of the Bones & Classical
Acupoint Compositions of the Human Body and Their Relationship to the Seven Musical Scales,”
(WEB, PDF).
^{2} Phillips, Stephen M. Article 33: "The Human Axial
Skeleton is the Trunk of the Tree of Life,” (WEB, PDF).
^{3} Phillips, Stephen M. Article 15: “The Mathematical
Connection Between Superstrings and Their Micropsi Description: a Pointer Towards Mtheory,”
(WEB, PDF), p. 13.
^{4} Phillips, Stephen M. Article 18: “Encoding of Planetary
Distances and Superstring Structural Parameters in the I Ching Table," (WEB, PDF), p. 19.
^{5} Phillips, Stephen M. Article 21: “Isomorphism Between
the I Ching Table, the 3×3×3 Array of Cubes and the Klein Configuration,” (WEB, PDF), pp. 3–4.
^{6} Phillips, Stephen M. Article 16: “The Tone Intervals of
the Seven Octave Species and Their Correspondence With Octonion Algebra and Superstrings,”
(WEB, PDF).
^{7} Phillips, Stephen M. Article 26: “How the Seven Musical
Scales Relate to the Disdyakis Triacontahedron,” (WEB, PDF), p. 15.
^{8} Besant, A. and Leadbeater, C.W. Occult
Chemistry, 3rd ed., Theosophical Publishing House, Adyar, Chennai, India, 1951.
^{9} Ibid, p. 14.
^{10} Phillips, Stephen M. Article 1: “The Pythagorean
Nature of Superstring and Bosonic String Theories,” (WEB, PDF), p. 5.
^{11} Ref. 7, p. 20.
^{12} Ibid, p. 16.
^{13} Phillips, Stephen M. Article 4: “Godnames Prescribe
Inner Tree of Life,” (WEB, PDF), pp. 2–5.
^{14} Phillips, Stephen M. Article 31: “The Musical Nature
of the Polyhedral Tree of Life,” (WEB, PDF), pp. 30–31.
^{15} Phillips, Stephen M. Article 11: “Plato’s Lambda — Its
Meaning, Generalisation and Connection to the Tree of Life,” (WEB, PDF).
^{16} Phillips, Stephen M. Article 20: “Algebraic,
Arithmetic and Geometric Interpretations of the I Ching Table,” (WEB, PDF), pp.18–22.
34
^{17} Ref. 14, p. 7.
^{18} Ibid, pp. 2–3, and Article 12, “New Pythagorean
Aspects of Music and Their Connection to Superstrings,” (WEB, PDF), pp. 2–4.
^{19} Ibid, pp. 3–7.
^{20} Ref. 7, pp. 28–29.
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