ARTICLE 10
by
Stephen M. Phillips
Flat 4, Oakwood House, 117119 West Hill Road. Bournemouth. Dorset BH2
5PH. England.
Website: http://smphillips.mysite.com
Abstract
The dodecagon has Pythagorean significance visàvis the
perfect number 10 because it is the tenth regular polygon. It
is also the last of the seven regular polygons constituting the inner form of
the Tree of Life. This article confirms the special status of the dodecagon by
showing how it geometrically embodies in a natural way the numbers 168, 336
& 1680 characterising the 3dimensional form of the superstring constituent
of up and down quarks and the number 248 characterising their unified dynamics.
The number values of the Godnames of the ten Sephiroth are shown to prescribe
properties of a single dodecagon and a pair of joined dodecagons, thus
indicating that they constitute ‘sacred geometry’ and must embody numbers of
universal significance, such as these defining parameters of superstrings. This
conclusion is confirmed by the simple and beautiful way in which the Pythagorean
Tetrad expresses their properties. As the last member of the set of seven
regular polygons, the dodecagon corresponds to Malkuth, the seventh and last
Sephirah of Construction, because it embodies information about the ‘Malkuth’ or
objective aspect of the microscopic manifestation of the Tree of Life in
spacetime. Being the polygonal counterpart of the Tree of Life, it has
analogous properties, some of which will be explored in later articles. Both the
yod population of the pair of joined dodecagons and the geometrical composition
of the separate pair manifest the 84:84 division that characteristic of holistic
systems. It is realised in the superstrings making up the quarks in atomic
nuclei as the 840 circularly polarised oscillations in the outer or inner halves
of each of their ten standing waves.

1
Table 1. Number values of the ten Sephiroth in
the four Worlds.

SEPHIRAH

GODNAME

ARCHANGEL

ORDER OF ANGELS

MUNDANE CHAKRA

1 
Kether
(Crown)
620

EHYEH
(I am)
21

Metatron
(Angel of the
Presence)
314

Chaioth ha Qadesh
(Holy Living
Creatures)
833

Rashith ha Gilgalim
First Swirlings.
(Primum Mobile)
636

2 
Chokmah
(Wisdom)
73

YAHWEH, YAH
(The Lord)
26,
15

Raziel
(Herald of the
Deity)
248

Auphanim
(Wheels)
187

Masloth
(The Sphere of
the Zodiac)
140

3 
Binah
(Understanding)
67

ELOHIM
(God in multiplicity)
50

Tzaphkiel
(Contemplation
of God)
311

Aralim
(Thrones)
282

Shabathai
Rest.
(Saturn)
317


Daath
(Knowledge)
474





4 
Chesed
(Mercy)
72

EL
(God)
31

Tzadkiel
(Benevolence of God)
62

Chasmalim
(Shining Ones)
428

Tzadekh
Righteousness.
(Jupiter)
194

5 
Geburah
(Severity)
216

ELOHA
(The Almighty)
36

Samael
(Severity of God)
131

Seraphim
(Fiery Serpents)
630

Madim
Vehement
Strength.
(Mars)
95

6 
Tiphareth
(Beauty)
1081

YAHWEH ELOHIM
(God the Creator)
76

Michael
(Like unto God)
101

Malachim
(Kings)
140

Shemesh
The Solar Light.
(Sun)
640

7 
Netzach
(Victory)
148

YAHWEH SABAOTH
(Lord of Hosts)
129

Haniel
(Grace of God)
97

Tarshishim or
Elohim
1260

Nogah
Glittering
Splendour.
(Venus)
64

8 
Hod
(Glory)
15

ELOHIM SABAOTH
(God of Hosts)
153

Raphael
(Divine
Physician)
311

Beni Elohim
(Sons of God)
112

Kokab
The Stellar Light.
(Mercury)
48

9 
Yesod
(Foundation)
80

SHADDAI EL CHAI
(Almighty Living God)
49,
363

Gabriel
(Strong Man of
God)
246

Cherubim
(The Strong)
272

Levanah
The Lunar Flame.
(Moon)
87

10 
Malkuth
(Kingdom)
496

ADONAI MELEKH
(The Lord and King)
65,
155

Sandalphon
(Manifest
Messiah)
280

Ashim
(Souls of Fire)
351

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

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

2
1.
Introduction
In earlier articles, analysis of the sacred geometry of the Tree of Life
uncovered properties quantified by numbers that are the gematria number values of the ten Sephiroth, their Godnames, Archangels,
Orders of Angels & Mundane Chakras (Table 1). It was shown in Article 9 (1) that the square embodies the structural parameter 168^{1} and the dynamical parameter 248 of
the superstring constituent of up and down quarks (the latter embodiment was also discussed
in Article 1 (2)). Although the relevance of these numbers to the physics of the universe
was, of course, unknown to the early Pythagoreans, it illustrates in a remarkable way their
profound intuition about the fundamental importance of the number 4 to the study of the
natural world. This principle, which the author has called the “Tetrad Principle,” was
formally postulated in Article 1. But the number 10 was also central to Pythagorean
mathematics because it was symbolised as the fourth triangular number by the
tetractys:
The dots will be called “yods,” after the name (yod) of the tenth letter (י)
of the Hebrew alphabet, which is somewhat shaped like a dot or point. The inner form of the
Tree of Life (Fig. 1) comprises seven enfolded, regular
polygons: triangle, square, pentagon, hexagon, octagon, decagon & dodecagon. The last of
these is the tenth regular polygon, counting from the simplest one — the
triangle. Given the titles “All Perfect,” “God,” & “Kosmos” given by the Pythagoreans to
the number 10, which they regarded as the perfect completion
of number, it should come as no surprise that the dodecagon, too, embodies
numbers of universal (and therefore scientific) significance. This article discusses how the
dodecagon encodes the numbers 168, 248, 336 and 1680 as parameters of the structure and
dynamics of superstrings. Of these numbers, only the second — the dimension of the rank8,
exceptional Lie group E_{8} used in superstring theory — has as yet been
recognised by particle physics. According to Table 1, it is the number value of
Raziel, the Archangel of Chokmah.
2. Properties of
the dodecagon
With its 12 sectors turned into 12 tetractyses, the dodecagon is made up of
73 yods (Fig. 2), of which
36 yods are on the boundary and 72 yods surround
its centre. 73 is the number value of Chokmah, the second member of
the Supernal Triad at the head of the Tree of Life, 36 is the Godname
number of Geburah, the fifth Sephirah from the top, and 72 is the
number value of Chesed, the fourth Sephirah from the top (see Table 2 below). The fact that a
Godname — ELOHA — prescribes the shape of the dodecagon by quantifying how
many yods are needed to mark out its boundary is the first sign that the dodecagon
constitutes ‘sacred geometry.’ A dodecagon whose sectors are triangles or tetractyses
(Fig. 2) will be called ‘Type A.’ A
dodecagon whose sectors are divided into three triangles or tetractyses (Fig. 3) will be called ‘Type B.’ This
type contains 181 yods (3). Corresponding nomenclature will apply
to all other polygons.
______________________
^{1} All numbers belonging to this table will be written in
boldface.
3
Rather than give tedious calculations, the properties of both types of
dodecagon are listed below for later discussion. For the sake of reference, the number values
of the Sephirothic titles, their Godnames, Archangelic Names, Angelic Names and Mundane Chakras
are shown in the following table. Numbers in coloured cells either have been already referred
to or will appear in later discussion.
Table 2. Gematria number values.
Sephirah

Title

Godname

Archangel

Order of
Angels

Mundane
Chakra

Kether

620

21

314

833

636

Chokmah

73

15, 26

248

187

140

Binah

67

50

311

282

317

Chesed

72

31

62

428

194

Geburah

216

36

131

630

95

Tiphareth

1081

76

101

140

640

Netzach

148

129

97

1260

64

Hod

15

153

311

112

48

Yesod

80

49

246

272

87

Malkuth

496

65, 155

280

351

168

Properties of: dodecagon; two separate dodecagons; joined dodecagons (4)
(Nonbracketed numbers refer to the Type A dodecagon; bracketed numbers refer to the Type B
dodecagon)
1. Number of corners of dodecagon = 12 (12) ; 24 (24); 22 (22).
2. Number of sides of dodecagon = 12 (12) ; 24 (24); 23 (23).
3. Number of corners & sides of dodecagon = 24 (24); 48
(48); 45 (45).
4. Number of triangles = 12 (36); 24 (72); 24
(72).
5. Number of corners of triangles = 13 (25); 26 (50); 24
(48).
6. Number of sides of triangles = 24 (60); 48 (120); 47 (119).
7. Number of corners & sides of t riangles = 37 (85); 74 (170); 71 (167).
8. Number of sides & triangles = 36 (96); 72 (192); 71
(191).
9. Number of corners & triangles = 25 (61); 50 (122); 48
(120).
10. Number of corners, sides & triangles = 49 (121); 98 (242);
95 (239).
11. Number of corners, sides & triangles outside root edge = 46 (118); 95
(239); 92 (236).
12. Number of yods = 73 (181), 146 (362), 142 (358). Number of yods other than
centres = 72 (180); 144 (360); 140 (356).
13. Number of yods outside root edge = 69 (177) ; 138 (354); 138 (354).
14. Number of hexagonal yods = 60 (156); 120 (312); 118 (310).
15. Number of hexagonal yods outside root edge = 58 (154); 116 (308); 116 (308).
16. Number of yods on boundaries of dodecagon = 36 (36); 72
(72); 68 (68). Number of boundary yods outside root edge = 32 (32) ;
68 (68); 64 (64).
17. Number of internal yods = 37 (145); 74 (290); 74 (290).
18. Number of yods on sides of tetractyses = 61 (145); 122 (290); 118 (286). Number of yods on
sides of tetractyses outside root edge = 57 (141); 118 (286); 114 (282).
19. Number of yods on sides of tetractyses other than corners & centre of dodecagon =
48 (132); 96 (264);
94 (262). Number of such yods outside root edge = 46 (130); 92 (260); 92 (260).
20. Number of yods other than corners & centre of dodecagon = 60 (168);
120 (336); 118 (334).
21. Number of yods other than corners of dodecagon & centres of its sectors =
49 (157); 98 (314); 96 (312). Number of such yods outside root edge = 47
(155); 94 (310); 94 (310).
22. Number of yods other than corners of dodecagon = 61 (169); 122 (338); 120 (336).
Set out below are the ways in which the Godname numbers prescribe these
properties of the dodecagon and two separate or joined dodecagons:
Kether: 21 
21 corners & sides of dodecagon outside the root edge.
The Type A dodecagon has 73 yods, where
73 = 21st prime number. Also, 121
corners, sides & triangles in the Type B dodecagon, where

4

121 = 11^{2} = 1 + 3 + 5 + … +
21
is the sum of the first ten odd integers after 1. The
Pythagorean measure of perfection — the number 10 — therefore defines the
geometrical composition of the tenth regular polygon. The Decad
determines not only the number value 73 of Chokmah but also
the number value 67 of Binah because the tenth integer after 1
is 11 and an undecagon constructed from tetractyses has 67 yods. These
two number values have a remarkable connection to
the geometry of what was called in earlier articles the
“1tree.” An ntree (n an integer) is defined as the n lowest trees of any set
of N overlapping Trees of Life (n<N). Below Binah in the 1tree constructed
from 19 tetractyses are 67 yods (Fig. 4). There are 73 yods up to the Path joining
Binah and Chokmah. Far from being arbitrary appellations, the Kabbalistic
titles of the Sephiroth have a geometrical basis visàvis the Tree of Life
and any equivalent geometrical object that embodies the divine, mathematical
paradigm.

Chokmah: 15
26 
47 sides of Type A dodecagon, where 47 = 15th
prime number. This is the number of sectors of the seven enfolded polygons of the
inner Tree of Life (Fig. 1).
Two separate, Type A dodecagons have 24 sectors with 26
corners. Also, number of yods outside the root edge on sides of
72 tetractyses in two
joined, Type B dodecagons which are not corners or centres of dodecagons = 260
= 26×10.

Binah: 50 
Two separate, Type B dodecagons have 72 triangles
with 50 corners.

Chesed: 31 
Number of hexagonal yods in two joined, Type B dodecagons = 310 =
31×10. This is also the number of yods outside the root edge of
two joined,
Type B dodecagons other than the corners and centres of their sectors.

Geburah: 36 
Number of yods on boundary of dodecagon. Also, 36
is the number of triangles in the Type B dodecagon.

Tiphareth: 76 
Number of hexagonal yods outside root edge of Type B dodecagon =
154 = 77th even

5

integer. 77 = 76th integer after 1. 
Netzach: 129 
Number of yods outside root edge on sides of 36
tetractyses other than corners or centre of Type B dodecagon = 130 =
129th integer after 1. 
Hod: 153 
Number of hexagonal yods outside root edge of Type B dodecagon =
154 = 153rd integer after 1. 
Yesod: 49 
Number of corners, sides & triangles of Type A dodecagon =
49. 
Malkuth: 65 
Number of yods outside root edge on sides of tetractyses other
than corners and centre of Type B dodecagon = 130 = 65th even
integer. 
155

155 hexagonal yods associated with each joined,
Type B dodecagon. Also, the number of yods outside the root edge of the Type B
dodecagon other than its corners and centres of its sectors. 
3. Encoding of superstring structural parameter
168 The Type A dodecagon has 60 yods other than corners surrounding
its centre (Fig. 5), whilst the Type B dodecagon has 168 yods other
than corners surrounding its centre (Fig. 6). In other words, 168 new yods are needed to
transform its sectors into tetractyses. Compare this with what was found for the square in
Article 9 (5): the Type B square with three tetractyses as each sector has 60 yods
surrounding its centre, whilst the Type C square with nine tetractyses as each sector has 168 yods surrounding its centre. Polygons of
Type A, B, C, etc represent successive levels of complexity in their construction from
tetractyses. What is so remarkable and significant in the context of the special emphasis
given by the Pythagoreans to the Decad and to the Tetrad symbolised by the square is that
both the square and the tenth regular polygon embody the same pair of
numbers, although differently. 168 is just the number of extra yods
required to turn the twelve sectors of a dodecagon into tetractyses. In the case of the
outer form of the Tree of Life, there are 60 extra yods needed to construct it from
tetractyses. The dodecagon bears to the first six polygons the same relation as Malkuth
bears to the six higher Sephiroth of Construction. This is suggested by the fact that it
contains as many hexagonal yods as the Tree of Life — only its skeletal (Malkuth) boundary
is different. It is confirmed by the fact that there are
155 hexagonal yods associated with each of the two joined, Type B
dodecagons (Fig. 7), whilst it has 168 yods other than corners of
its sectors, where 155 is the number value of ADONAI MELEKH, the
Godname of Malkuth, and 168 is the number value of Cholem
Yesodeth, the Mundane Chakra of this Sephirah. EL, the Godname of Chesed, prescribes
the pair of joined dodecagons because they contain
(155+155=310=31×10) hexagonal yods, where
31 is its number value.
Further remarkable confirmation that the dodecagon constitutes sacred
geometry because its properties are prescribed by Godnames is the fact that, outside their root
edge, the two joined, Type B dodecagons
6
contain 260 (=26×10) yods on the sides of their
72 tetractyses that are not their 48 corners or centres,
where 26 is the number value of YAHWEH, the Godname of Chokmah. Compare this
with the fact that the seven enfolded, regular polygons contain 260 yods outside their shared
root edge (Fig. 8). In the first case, this number is that needed to delineate the edges
of their tetractyses outside the root edge, given their corners and centres; in the second
case, it is the number of yods required to construct the seven enfolded polygons, starting
with the root edge. The ways in which the generative Godname YAHWEH prescribes both
geometrical objects are analogous.
It is not coincidental that the two objects possess properties that are
quantified by the same sets of numbers listed in Table 2. The dodecagon is the polygonal form of the Tree of Life and will —
like any other holistic structure — embody the numbers listed in this table.
4. Encoding of
1680 in pair of joined dodecagons
It was shown in Article 9 that, when the yods in a square constructed from tetractyses are
themselves replaced by tetractyses (Fig. 9):
there result 248 yods other than corners of tetractyses,
that is, yods symbolising the seven Sephiroth of Construction. These symbolise the
248 quantum states of the particle transmitting the unified superstring force
described by the gauge symmetry group E_{8}. A dodecagon with its 12 sectors turned
into such higherorder tetractyses contains 120 tetractyses, where
showing how the Tetrad determines this number. The number of yods in each
sector is
7
8
again illustrating the role of the Tetrad. Taking into account that 12 yods
on each internal edge of a sector apart from the centre of the dodecagon are shared with
adjoining sectors, there are (84–12=72) yods per sector, where
72 is the number of Chesed, the fourth Sephirah from the top of the
Tree of Life. Of these, 10 (=1+2+3+4) are corners of tetractyses symbolising Kether, Chokmah
and Binah and 62 are hexagonal yods symbolising Sephiroth of
Construction, where 62 is the number value of Tzadkiel, the
Archangel corresponding to Chesed. Therefore, the number of yods in the 120 tetractyses of a
dodecagon = 12×72 + 1 = 865. 13 yods lie along the shared edge of the
pair of dodecagons, leaving (865–13=852) yods outside it. The number of yods in the
dodecagon outside
the root edge that surround the centres of each sector = 852 – 12 =
840, where
and
yet again illustrating the basic role of the Tetrad in defining properties
of sacred geometry with universal significance (as will become evident shortly). A pair of
joined dodecagons therefore has (840+840=1680) yods outside their shared edge that surround the
centres of their 24 sectors, where 24 = 1×2×3×4 (Fig. 10). This is the number of turns in each of the ten helical whorls
(Fig. 11) of the ‘ultimate physical atom,’ or UPA (Fig. 12), observed over 100 years ago by the two Theosophists Annie Besant
and C.W. Leadbeater, using a siddhi, or psychic ability, known to Indian yoga. Each whorl
makes 2½ outer revolutions about the vertical axis of spin of the UPA and 2½ inner
revolutions, spiralling 840 times in circles in each half. We see that each dodecagon
containing 840 yods distributed outside the root edge about the centres of its sectors
encodes the number of coils in half a whorl; the two identical dodecagons correspond to its
inner and outer halves. The ‘Malkuth’ level of the microscopic Tree of Life, that is, each whorl of the superstring, is encoded in the tenth regular
polygon and in the last of those constituting the inner form of the Tree of Life. Each one
of the 1680 yods both shaping the pair of dodecagons and surrounding the centres of their
24 sectors denotes a circularly polarised oscillation or wave in a whorl. These yods
represent the ‘material’ manifestation of the 240 tetractyses of the 24 higherorder
tetractyses making up the pair of dodecagons. The question arises: what do these
higherorder tetractyses denote? Twentyfour of them are associated with each whorl, that
is, 240 higherorder tetractyses are associated with the UPA itself. The gauge symmetry
group E_{8} describing the unified superstring force has 240 socalled
‘generators’ corresponding to the 240 socalled ‘nonzero roots of its Lie algebra.’ To
each generator corresponds a kind of charge analogous to the electric charge of a
particle. Each charge is the source of a gauge field, i.e., a particular kind of force.
Each higher order tetractys represents one of the 240 gauge charges, and 24 such charges
are spread along each whorl, making a total of (10×24=240) for the superstring itself. As
1680 = 24×70 and the Tree of Life comprises 70 yods when its 16 triangles are turned into
tetractyses (see Figure 5), this number is the number of yods in 24 separate Trees of
Life. This reflects the fact that the 24 gauge charges manifesting in each whorl are all
independent and ‘smeared’ along its length in a way analogous to that proposed in the
E_{8}×E_{8} heterotic model of the superstring. 70 is also the number of
yods corresponding to Sephiroth of Construction in the higher order tetractys making up
each sector of the dodecagons, showing again that the gauge charges are wholes —
complete Tree of Life entities in themselves.
9
The last statement should answer the following question that may have
arisen in the reader’s mind during the discussion above of how the number 1680 was embodied
in the pair of dodecagons: what, if any, is the significance of the seemingly arbitrary way
in which the 840 yods in each dodecagon were selected — namely, picking out the 840 yods
that surround centres of sectors? The yod at the centre of a tetractys denotes Malkuth, the
material manifestation of the whole symbolised by the tetractys. The six yods surrounding
it at the corners of a hexagon denote the six Sephiroth of Construction above Malkuth.
There are 84 yods surrounding the centre of the next higher order tetractys (see
Figure 9). On the cosmic level, these correspond to the 42 subplanes of
the six superphysical planes of consciousness and the 42 subplanes of
their cosmic counterpart (see Article 5 for more details). On the microcosmic level,
they denote the number of circularly polarised waves in a quarter of a revolution about
the axis of the UPA, i.e. , a 90° turn in space. In conformity with its tenfold nature
— both in ordinary space and in 10dimensional spacetime — each whorl makes
ten halfrevolutions, five in an outer twisting and five in a more tightly
knit, double helical twist. This 5:5 split corresponds to the division in the Tree of
Life between the five uppermost Sephiroth, which span its Upper Face, and the five
lowest Sephirah forming its Lower Face. The yod at the centre of a higherorder
tetractys denotes the Malkuth level of manifestation of a Tree of Life system and so
does not enter the count of the yods symbolising differentiations of Sephiroth
beyond Malkuth. Each of the 24 gauge charges spread out along each whorl
is that manifestation. What appears at first sight to be merely an
ad hoc choice of yods contrived to generate the number 840 in each dodecagon
is in fact a selection dictated by the proper, physical interpretation of their
higherorder tetractys sectors.
Another similarity between the powers of the square and dodecagon to
embody various superstring parameters like 248 and
168 is the fact that the latter number is the sum of the first 12 odd
integers after 1 (Fig. 13) and that the shapes of both polygons are defined by the number
12 because a square divided into tetractyses has 12 yods along its boundary, whilst a
dodecagon is delineated by its 12 corners. As the template for constructing objects
possessing sacred geometry, the tetractys unveils a
beautiful harmony between geometry and arithmetic that exists only
in such objects.
5. Encoding of 336
in the pair of dodecagons
Two joined dodecagons have 22 corners, where
Its 24 (=1×2×3×4) sectors have 24 corners. This illustrates once more
how the integers 1, 2, 3, & 4 express properties of the dodecagon. As there are 22
compactified dimensions in 26dimensional spacetime, each corner of a
dodecagon can be regarded as symbolising the higher dimensions of space. The ten
corners outside the root edge of one dodecagon symbolise the ten curledup dimensions
generating the ten string like components of the superstring (see Article 2). The twelve
corners of the other dodecagon denote the twelve remaining compactified dimensions. These
consist of the five dimensions that define a compactified space whose symmetry generates
the superstring gauge group E_{8} and the seven curledup dimensions predicted by
supergravity theory. The centres of the two dodecagons symbolise the two transverse
dimensions of 26dimensional strings.
Notice that the division:
22 = 3 + 7 + 12
of the 22 letters of the Hebrew alphabet into the three mothers: aleph,
mem & shin, the seven double
10
consonants: beth, gimel, daleth, caph, pe, resh, & tau, and the
twelve simple consonants has a remarkable geometrical counterpart in the 22 corners of the
pair of joined dodecagons. This is because the three mother letters correspond to three
corners symbolising the curledup dimensions beyond supergravity spacetime that generate
the three major whorls of the UPA, the seven double consonants correspond to seven corners
that denote the curledup dimensions generating its seven minor whorls
and the twelve simple consonants correspond to the corners of the other
dodecagon symbolising the five E_{8}generating dimensions and the seven curledup,
supergravity dimensions.
Property number 22 in the list given in Section 2 states that the number
of yods in two joined dodecagons other than their 22 corners is 336 (Fig. 14), where
Starting with the Tetrad, 2^{2}= 4, the sum of the
squares of the four integers 2, 6, 10 & 14 spaced four units apart is
336. It was stated in the last section that the inner and outer halves of a whorl makes 2½
revolutions. This means that its 1680 turns are spread over five revolutions, 336 turns per
revolution. Each turn in a revolution of a whorl is symbolised by a yod in the two joined,
Type B dodecagons other than their 22 corners and two centres, that is, new yods
generated by their construction from
tetractyses. The 168 such yods in each
dodecagon denote the number of circularly polarised oscillations made during the traverse
of either half of one revolution of a whorl.
With their sectors turned into the next higherorder tetractys after the
Pythagorean tetractys, each dodecagon was found earlier to contain 840 yods outside their
root edge surrounding their centres. Each dodecagon represents half of a whorl made up of
840 coils (Fig. 15). Enfolded in each Tree of Life belonging to CTOL are the two
sets of seven regular polygons. The lowest ten Trees of Life have 140
enfolded polygons, where 140 is the number value of Masloth,
the Mundane Chakra of Chokmah. Their 20 dodecagons contain (10×1680=16,800) yods outside
their root edges surrounding the centres of their 240 sectors representing the 240 gauge
charges of E_{8}. This Tree of Life representation of the superstring shows that
the Godname ADONAI prescribes the number 16,800 because its number value
65 is the number of SLs in the lowest ten trees of CTOL. The number of
corners of the 70 polygons enfolded on either side of these trees is
351, which is the number value of Ashim, the Order of Angels
assigned to Malkuth. 351 is also the sum of the first
26 integers, showing how the Godname YAHWEH with number value
26 prescribes the ten overlapping Trees of Life representing the ten
whorls of the superstring. Each dodecagon has ten
11
corners outside its root edge. The ten dodecagons enfolded in the lowest
ten trees have (10×10=100) external corners. This means that the 60 polygons enfolded on
either side of the ten trees that are not dodecagons have (351–100=251)
corners. Article 5 (6) discussed the significance of the number 251 in relation to the
superstring. The concurrence in the same context (the lowest ten trees) of
this structural parameter encoded in the first six types of polygons with the number
16,800 encoded in the seventh type is remarkable evidence for the Kabbalistic basis of
superstring theory and the author’s identification of the UPA as a superstring. Notice
that the proportion of the dodecagons to the first six types enfolded in ten trees,
namely, 10:60, corresponds in the tetractystransformed Tree of Life to the 10:60
pattern of yods created by the ten Sephiroth and the 60 hexagonal yods. Indeed, the
inner form of the Tree of Life has the same pattern, namely, the ten
corners of a dodecagon outside the root edge that it shares with the other 13 polygons
with 60 corners. Just as the points in space where the ten Sephiroth are located define
the basic geometrical aspect of the Tree of Life — its Malkuth level — so their
counterpart in its inner form — the pair of dodecagons — quantitatively embodies the
physical nature of the Tree of Life in the subatomic world as measured by the 1680 coils
in each helical whorl that makes up the UPA/superstring. One can only affirm the
hermetic axiom: “as above, so below.”
6. Encoding of 248
& 168 in Type A & Type B dodecagons
Up till now, both dodecagons have been regarded as the same type. Suppose that one of the
dodecagons is Type A and that the other is Type B. The former contains
73 yods and the latter has 181 yods. The pair of joined dodecagons
has 250 yods. We saw above that the first six types of polygons enfolded in ten overlapping
trees of Life have 251 corners. The topmost corner of the hexagon enfolded in the tenth
tree coincides with the lowest corner of the hexagon enfolded in the 11th tree. 250 corners
are therefore intrinsic to these polygons. They correspond to the 250 yods in a Type A and
Type B dodecagon, thereby further demonstrating the holistic nature of the pair of
dodecagons. As Articles 5 and 6 discuss in more detail, the UPA is formed from a dbrane
embedded in 26dimensional spacetime that wraps itself around ten
circular dimensions to generate the ten independent whorls of the superstring. A point on
each whorl is specified by 25 spatial coordinates, so that the ten whorls have (10×25=250)
such variables.
We see that the two types of dodecagon embody the number of variables defining the
positions of ten points in 26dimensional spacetime which can never
coincide, i.e., they belong to ten curves that never touch or intersect. This is highly
significant, for it is evidence for the tenfold nature of the superstring.
248 yods surround the centres of the joined dodecagons, where
248 is the number value of Raziel, the Archangel of Chokmah
(Fig. 16). They symbolise the 248 gauge bosons of
E_{8} that transmit the unified superstring force.
246 yods are outside the root edge, where
246 is the number value of Gabriel, the Archangel of
Yesod. According to the properties listed on pages 3 and 4, the Type A dodecagon has
49 geometrical elements, where 49 is the
number of EL ChAI, Godname of Yesod, and the Type B dodecagon has 121 geometrical
elements. The pair of separate dodecagons has 170 geometrical elements, i.e., they have
168 geometrical elements surrounding their centres. In other
words, 168 geometrical elements are needed to construct their
48 tetractyses, starting from their two centres. The pair of joined dodecagons
embodies the superstring dynamical parameter 248 because
248 yods are needed to construct them, starting from their two
centres, and the pair of separate dodecagons embodies its structural parameter
168 because 168 geometrical elements are
needed, as well as 168 yods, starting from a dodecagon divided
into its sectors. This is more remarkable evidence of how the Malkuth aspect of the
microscopic manifestation of the Tree of Life is encoded in the last of the regular
polygons constituting its inner form. 48 is the number value of
Kokab, the Mundane
12
13
Chakra of Hod, the Sephirah that signifies mental activity and
communication. Previous articles discussed how this number is a parameter of the Tree of
Life, being the number of corners of the seven separate, regular polygons making up its
inner form and the number of corners, edges & triangles making up its outer form. Its
superstring interpretation is as follows: as discussed earlier, each of the 24 gauge
charges carried by a string component of the superstring/UPA manifests as a circularly
polarised standing wave. Each such wave has two orthogonal, plane wave components that are
90º out of phase. Each whorl therefore consists of (2×24=48) independent
standing plane waves. This 24:24 division manifests geometrically in the Type A dodecagon
as the 24 vertices & edges on its boundary and the 24 edges and triangles inside it.
The pattern appears in the first (6+6) enfolded polygons as the 24 corners intrinsic to
each set. As future articles will demonstrate, it is a characteristic of any
holistic system possessing sacred geometry. The ten whorls of the UPA/superstring comprise
(10×48=480) plane waves. The encoding of these in the inner form of the
Tree of Life is the set of 240 hexagonal yods in either set of seven separate regular
polygons, i.e., their 480 hexagonal yods (Fig. 17). In the case of the first (6+6) enfolded polygons enfolded in
ten Trees of Life, the two sets of 240 plane waves are the counterpart of the (240+240)
corners of the (60+60) polygons of the first six types. Every hexagonal yod or corner in
one set is the mirror image of its counterpart in the other set. Every such pair denotes
the two orthogonal plane waves making up each of the 24 circularly polarised
oscillations in each of the ten whorls of the superstring
constituent of up and down quarks paranormally described over a century
ago by the Theosophists Annie Besant and C.W. Leadbeater.
The numbers of corners, edges & triangles surrounding the centres of
the separate Type A and Type B dodecagons are:
There are 84 edges and 84 corners & triangles. This 84:84 division
of the 168 geometrical elements in the two types of dodecagons is
characteristic of holistic systems. Later articles will provide numerous examples. Its
remarkable consequence is that, if we consider the ten dodecagons enfolded on one side of
the central pillar of ten overlapping Trees of Life as Type A and their counterparts on the
other side as Type B, the ten pairs of separate dodecagons have 840 edges and 840 corners
and triangles. This is the same 840:840 division as was found for the yods
surrounding the centres of the 24 sectors of two enfolded polygons when each sector is a
higherorder tetractys. It manifests physically as the 840 circularly polarised waves in
each half of a whorl of the E_{8}×E_{8} heterotic superstring. Here is
clear evidence that the superstring’s oscillatory form, as described by Besant &
Leadbeater, conforms to the geometry of the pair of dodecagons in the inner Tree of Life.
The Type A dodecagon has 48 elements surrounding its centre and the Type B
has 120 elements surrounding its centre. The counterpart of this 48:120
division in a Type B dodecagon with 168 yods other than the 13 corners of
its 12 sectors is the 48 yods at corners & centres of tetractyses and
the 120 hexagonal yods on their 60 edges (Fig. 18). Its counterpart in the 168 yods outside
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the root edge on the sides of the first (6+6) enfolded polygons is their
48 corners and the 120 hexagonal yods on their 60 sides. Equivalent
holistic structures always display analogous patterns.
7.
Conclusion
The dodecagon and the pair of joined dodecagons in the inner form of the Tree of Life can
be transformed into two types, depending on whether their sectors are turned into single
tetractyses or three tetractyses. The ten Godname numbers prescribes their resulting
properties, suggesting that they embody numbers of cosmic significance. This is confirmed
by the way they encode the numbers 168 and 336, these being the
number of coils in, respectively, half and one revolution of a string component of the
superstring constituent of up and down quarks, proved by the author to have described
paranormally with a yogic siddhi over a hundred years ago by the two Theosophists Annie
Besant and C.W. Leadbeater. A pair of dodecagons is found to embody the number (1680) of
such coils in all five revolutions of a string when their sectors are constructed from the
next higherorder tetractys. As each of the ten strings of the superstring is the space
time manifestation of a Sephirah, it, too, can be represented by a Tree of Life. This means
that the superstring is modelled by ten overlapping trees in whose inner forms are enfolded
20 dodecagons containing 16,800 yods that are outside their root edges and surround the
centres of their 240 sectors. These correspond to the 16,800 coils in the superstring. They
denote circularly polarised oscillations in its ten strings generated by the 240 gauge
charges of the superstring symmetry group E_{8}, which are ‘smeared’ along each
whorl, 24 per whorl. These gauge charges are the physical meaning of the 24 higherorder
tetractys sectors in the pair of dodecagons enfolded in each overlapping Tree of Life as
the last of the regular polygons constituting its inner form. A Type A dodecagon and a Type
B dodecagon separately have 168 geometrical elements surrounding
their centres, whilst, joined together, they have 248 yods unshared
with the outer form of the Tree of Life or, alternatively, 248 yods
surrounding their centres. These yods symbolise the 248 gauge bosons
of E_{8}. Superstring physics has been reduced to sacred geometry and then to
number as its generating principle. Truly, as the Pythagoreans declared: “Number is form
and form is number.”
References 1. Phillips, Stephen M. Article 9: “How the square encodes the superstring
parameters 168 & 248,” (WEB, PDF), pp. 2–9.
2. Phillips, Stephen M. Article 1: “The Pythagorean nature of
superstring and bosonic string theories,” (WEB, PDF), p. 4.
3. The number of yods in a polygon with n corners is: N = 6n + 1 (Type
A); N = 15n + 1 (Type B). A Type A dodecagon (n=12) has
73 yods. A Type B dodecagon has 181 yods.
4. Formulae for a polygon with n corners:

Type A

Type B

Number of hexagonal yods in polygon with n corners =
Number of corners of triangles =
Number of sides of triangles =
Number of triangles =
Number of corners & sides =
Number of corners & triangles =
Number of sides & triangles =
Number of corners, sides & triangles =

5n
n + 1
2n
n
3n + 1
2n + 1
3n
4n + 1

13n
2n + 1
5n
3n
7n + 1
5n + 1
8n
10n + 1

5. Ref. 1.
6. Phillips, Stephen M. Article 5: “The superstring as microcosm of the
spiritual macrocosm,” WEB, PDF).
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