ARTICLE 24
by
Stephen M. Phillips
Flat 4, Oakwood House, 117119 West Hill Road. Bournemouth.
Dorset BH2 5PH. England.
Website: http://smphillips.mysite.com
Abstract
Article 22 presented evidence that the disdyakis triacontahedron
represents the 3dimensional manifestation of the exterior aspect of the
2dimensional, inner form of the Tree of Life — the geometrical representation
of “Adam Kadmon,” or Divine Man. Its 62 vertices define 28 regular and
semiregular solids. Constructed from Pythagorean tetractyses, the template of
sacred geometry, they are made up of 3360 hexagonal yods. This number is the
number of yods in the seven enfolded, regular polygons forming half of the inner
Tree of Life. It is further confirmation that the disdyakis triacontahedron is
the 3dimensional realisation of the inner Tree of Life. Its manifestation in
superstring spacetime are the 3360 circularly polarised oscillations made
during one complete revolution by the ten component closed curves of the
E_{8}×E_{8} heterotic superstring, as described 109 years ago by
the Theosophists Annie Besant and C.W. Leadbeater.

1. Introduction The author has discovered that
what is known to Kabbalists as the ‘Tree of Life’ is actually only the
outer form of the complete, sacred geometrical system embodying the divine
paradigm. It has an inner form, and Articles 22 and 23 showed that the Catalan solid
called the ‘disdyakis triacontahedron’ is its polyhedral manifestation. Through the
equivalence between the (6n+1) yods of an nsided, regular polygon with its sectors
turned into tetractyses and the (6n+1) Sephirothic emanations up to Chesed of the highest
tree in n overlapping Trees of Life, this inner form can be shown^{1} to encode the replication of its outer form to map all
levels of reality, both physical spacetime and
1
Table 1. 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.

(All numbers from this table that are referred to in the article are
written in boldface).
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superphysical realms. The seven enfolded, regular polygons constituting one
half of the inner Tree of Life map what are known in Theosophy as the 49
subplanes of the seven planes of consciousness: physical, astral, mental, buddhic, atmic,
anupadaka (or Monadic) and adi (or Divine), each subplane being represented by its own Tree of
Life. The 120 yods on the boundaries of the seven polygons signify the 120
faces of the disdyakis triacontahedron and the 144 yods inside them denote the 144 faces of
another polyhedron (Fig. 1). This ‘144 Polyhedron’ was identified in Article 23 as generated from
the disdyakis dodecahedron (one of the 13 Catalan solids), which has 48
faces, 26 vertices & 72 edges. This is the only
semiregular polyhedron that has 48 faces, which means that it, alone, can
generate a polyhedron with 144 faces when tetrahedra are stuck on its faces.
2. Polyhedral content of the disdyakis
triacontahedron The 62 vertices of the disdyakis
triacontahedron define the vertices of 28 regular and semiregular solids: ten
tetrahedra, five cubes, five octahedra, one icosahedron, one dodecahedron, five rhombic
dodecahedra and one rhombic triacontahedron. A rhombic dodecahedron (Fig. 2) is an Archimedean solid with 14 vertices, 24 edges and 12 rhombic
faces. Constructed from four tetractyses, each of the 12 rhombic faces has 12 internal
hexagonal yods, whilst two hexagonal yods lie on each of its 24 edges. The number of
hexagonal yods in the rhombic dodecahedron is therefore 2×24 + 12×12 = 192 (see Article
3
18 for the significance of this parameter of holistic systems visàvis the
I Ching table and the Bode numbers of the planets). When the five, separate rhombic dodecahedra
group, their (5×14=70) separate vertices become the 62 vertices and raised
centres of the 30 Golden Rhombic faces of the disdyakis triacontahedron, their 30 unshared
vertices coinciding with the centres of these faces and their 32 shared vertices coinciding
with their vertices (Fig. 3). Each rhombic dodecahedron has six unshared vertices and eight shared
vertices that are vertices of these
faces. Each face vertex coincides with two vertices of different
dodecahedra:
A_{1}B_{1}
A_{2}C_{1}
A_{3}D_{1}
A_{4}E_{1}
A_{5}B_{2}
A_{6}C_{2}
A_{7}D_{2}
A_{8}E_{2} 
B_{3}C_{3}
B_{4}D_{3}
B_{5}E_{3}
B_{6}C_{4}
B_{7}D_{4}
B_{8}E_{4} 
C_{5}D_{5}
C_{6}E_{5}
C_{7}D_{6}
C_{8}E_{6} 
D_{7}E_{7}
D_{8}E_{8} 
where the letters label the five dodecahedra and the numbers indicate their vertices.
Let us now construct the 28 solids from tetractyses — the template of sacred
geometry — and then work out their populations of hexagonal yods. The significance of the
latter is that the seven hexagonal yods of the tetractys symbolise the seven Sephiroth of
Construction, the formative degrees of freedom expressing the 'objective' aspects of God). The
hexagonal yod populations of the 28 polyhedra in the disdyakis triacontahedron
4
are:^{2}
tetrahedron: 
10×48 =

480

cube: 
5×96 =

480

octahedron: 
5×96 =

480

icosahedron: 
1×240 =

240

dodecahedron: 
1×240 =

240

rhombic dodecahedron: 
5×192 =

960

rhombic triacontahedron: 
1×480 =

480


Total =

3360

This result is truly astounding for two complementary reasons:
1. Divine Unity symbolised by the Pythagorean Monad, or mathematical point
("0thorder tetractys"), differentiates, firstly, into the familiar tetractys ("1storder
tetractys") with 10 yods (three corners, seven hexagonal yods), secondly, into the “2ndorder
tetractys” with 85 yods (15 corners, 70 hexagonal yods), and so on:
3360 is the number of yods in the seven enfolded, regular polygons
constituting the inner form of the Tree of Life when their 47 sectors are each turned into the
2ndorder tetractys (Fig. 4).^{3} We now see that the seven types of solids terminating in the
disdyakis triacontahedron contain 1680 hexagonal yods (as shown on page 14 of Article
5
22^{4} ), whilst the actual numbers for the seven types of solids in the
disdyakis triacontahedron total 3360 hexagonal yods, where 3360 = 2×1680. This is the total
number of yods making up the polygonal, inner form of the Tree of Life constructed from the
template of the 2ndorder tetractys. The number 3360 expresses a holistic structure
both in the 2dimensional space of the polygons and in the 3dimensional space of the
disdyakis triacontahedron. This marvellous, beautiful property of the polyhedron clearly
demonstrating its Tree of Life basis!
2. 3360 is the number of circularly polarised standing wave oscillations
made during each of the five revolutions of the ten closed curves making up the
E_{8}×E_{8} heterotic superstring, as described by Annie Besant and C.W.
Leadbeater in 1908, when they used anima to magnify subatomic particles (Fig. 5), whilst 1680 is the number of such oscillations in each curve. What
this means is that, as the completion of the sevenfold sequence of regular and
semiregular polyhedra, the disdyakis triacontahedron has to be made up of the
same number of formative degrees of freedom (hexagonal yods) as there are yods needed
to represent its 2dimensional counterpart, namely, the seven enfolded, regular polygons.
Far from being a coincidence, the presence of the same number in two superficially
different contexts reveals in an unambiguous way the beautiful, mathematical design of a
transcendental, creative Intelligence. To discover this mathematical harmony, we need to
understand 'sacred geometry' — not the distorted version found in many books, which lack
understanding of the fundamental principles — but the only geometry worthy of being called
'sacred,’ namely, that of the Tree of Life.
Each of the 42 sides of the seven enfolded polygons has 11 yods between
their ends, which number 36. The number of yods forming the boundaries of the polygons = 11×42
+ 36 = 498. In other words, 496 yods form the sides of the polygons
between the two endpoints of the root edge that generates them. This is the number value
of the Hebrew word ‘Malkuth’ signifying the last Sephirah of the Tree of Life. It is yet
another confirmation that the disdyakis triacontahedron is the outer (or Malkuth) aspect
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of the inner Tree of Life. As discovered by physicists Michael Green and
John Schwarz^{5} in 1984, 496 is the dimension of the nonabelian gauge
symmetry group defining superstring interactions that are free of quantum anomalies. We
therefore encounter the following amazing property of the polygonal form of the inner Tree
of Life blueprint: it encodes not only the oscillatory pattern of the
E_{8}×E_{8} heterotic superstring but also the number of gauge bosons that
transmit its unified force — the first as its yod population and the second as the number of
yods forming its boundary between the endpoints of its generative root edge.
3. The rhombic
dodecahedron and rhombic triacontahedron
The rhombic dodecahedron has 12 rhombic faces (Fig. 6). The longer diagonal of each face (shown as a red line in Fig. 6) is the edge of a cube and the shorter diagonal (shown as a blue line)
is the edge of an octahedron. The ratio of the lengths of the longer and shorter diagonals
is √2 = 1.414... . These Platonic solids are dual to each other. The two other Platonic
solids that are dual to one another — the icosahedron and the dodecahedron — share an
analogous property in that their edges are, respectively, the longer and shorter diagonals
of the faces of the rhombic triacontahedron. Their ratio is the Golden Ratio φ = 1.618…
.These rhomboids are generated in a simple way by the geometry of the Tree of Life.
Fig. 6 indicates how the ten Sephiroth are the centres or points of
intersection of a column of white circles. Let us take their radii as one unit. The central
Pillar of Equilibrium intersects the path joining Chesed and Geburah at a point A that is
one unit away from the vertical righthand tangent BC to these circles. ABCD is a square
with sides of length 1. Therefore, its diagonal AC = √(1^{2} + 1^{2}) = √2.
With A as centre, draw a circle passing through C of radius √2 (shown as a dashed line in
Fig. 6). It intersects the line drawn along AB at E. E' is the corresponding
point on the other side of the central pillar. EE' = 2√2. The central pillar intersects the
path joining Netzach and Geburah at D'. DD' = 2. Therefore, EE'/DD' = 2√2/2 = √2. We find
that the rhombus DED'E' has the same shape as the rhombic face of the rhombic
dodecahedron.
Extend the tangent at B to the point F, where BC = CF = 1. Then, BF = 2 and
AF = √(2^{2} + 1^{2}) = √5. The line AF intersects CD at G, where AG = GF =
√5/2. With G as centre, draw a circle of radius ½. It intersects AF at H, where AH = √5/2 + ½ =
(√5+1)/2 = φ, the Golden Ratio. With A as centre, draw a circle of radius AH. It intersects the
extension of AB at I, where AI = φ. I' is its counterpart on the other side of the central
pillar. II' = 2φ. Therefore, II'/DD' = 2φ/2 = φ. The rhombus DI D'I' has the same shape as the
rhombic face of the rhombic triacontahedron. What manifests finally as the fruit of the Tree of
Life, namely, the disdyakis triacontahedron with golden rhombic faces, was within it as their
seed shape from the very beginning!
References
^{1} Phillips, Stephen M.The Mathematical Connection between Religion and
Science (Antony Rowe Publishing, England, 2009).
^{2} The hexagonal yod populations of these solids are taken from
Article 22: “The disdyakis triacontahedron as the 3dimensional counterpart of the Inner
Tree of Life,” Stephen M. Phillips, (WEB, PDF), p. 6.
^{3} Proof: The 2ndorder tetractys has 85 yods, of which 13 yods
line each of its sides. When each of the n triangular sectors of an nsided, regular polygon
are turned into a 2ndorder tetractys, there are (85–13=72) independent
yods per sector of the polygon. Its yod population = 72n + 1, where “1” denotes the
yod at the centre of the polygon. The polygonal form of the inner Tree of Life consists of a
triangle, square, pentagon, hexagon, octagon, decagon and dodecagon. They are enfolded in
one another and share the same base, or what the author has called the “root edge,” as they
should be thought of as
7
growing out of this fundamental line joining Daath and Tiphareth in the Tree
of Life. When the seven separate polygons are superposed on one another in their enfolded
state, corresponding members of the set of 13 yods forming what becomes their shared side
coincide and therefore must not be counted separately in a calculation of their yod population.
Below are listed the yod populations of each polygon and (except for the triangle) their
numbers of yods outside the root edge:
Polygon 
n

Number of yods = 72n + 1

Number of yods outside root edge


triangle 
3

217

217

square 
4

289

289 – 13 = 276

pentagon 
5

361

361 – 13 = 348

hexagon 
6

433

433 – 13 = 420

octagon 
8

577

577 – 13 = 564

decagon 
10

721

721 – 13 = 708

dodecagon 
12

865

865 – 13 = 852




Total
= 3385

Inspection of Fig. 4 reveals that the tip of the triangle viewed with the root edge as its
base is also the centre of the hexagon (the triangle is simply a triangular sector of the
hexagon). Similarly, the tip of the pentagon is the centre of the decagon.
With 2ndorder tetractyses as their sectors, the centroid of the triangle where corners of
its three 2ndorder tetractyses meet is also the central yod of the tetractys at the
centre of the 2ndorder tetractys constituting a sector of the hexagon (see diagram
above). The 11 yods between corners on each of the two sides of the triangle outside its
shared base coincide with yods on the sides of this sector of the hexagon. There are (1 +
1 + 1 + 2×11 = 25) yods in the total population calculated above that coincide with yods
belonging to other polygons (these are the only yods occupying the same positions). In
determining the yod population when the separate polygons are superposed, these yods must
be subtracted in order to avoid doublecounting Therefore, the yod population of the seven
enfolded polygons constructed from 2ndorder tetractyses = 3385 – 25 = 3360.
^{4} Ref. 2, p. 14.
^{5} Green, M.B. & Schwarz, J.H. “Anomaly cancellations in
supersymmetric d = 10 gauge theory and superstring theory.” Physics Letters, B149, 117.
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