ARTICLE 57
by
Stephen M. Phillips
Flat 4,
Oakwood House, 117119 West Hill Road. Bournemouth. Dorset BH2 5PH.
England.
Website: http://smphillips.mysite.com
Abstract
According to the Tetrad Principle formulated in
Article 1, the first four regular polygons constitute a holistic system.
Confirmation of this is provided by the fact that the properties of this
geometrical system are analogous to those of wellknown sacred
geometries revered in various religions. As such, they are prescribed by
the gematria number values of the Godnames assigned in Jewish Kabbalah
to the ten Sephiroth of the Tree of Life. One of the various parameters
characteristic of holistic systems is the number 496, which was found in
1984 to determine the two possible symmetry groups of the forces between
superstrings, namely, SO(32) and
E_{8}×E_{8}. The following question
arises: given that the first four enfolded polygons must embody this
number in some way, do they embody it in a manner that is compatible
with only one of the two groups? If so, it would imply that sacred
geometries pick out only one type of superstring as the actual, basic
particles making up the physical universe. Other articles have
accumulated considerable evidence indicating that sacred geometries
favour E_{8}×E_{8} over SO(32).
This article confirms this conclusion by showing that, constructed from
tetractyses, the first four enfolded polygons have 17 rows of yods with
496 permutations, of which 248 belong solely to the first three enfolded
polygons and 248 are shared by all four polygons, where 248 is the
dimension of E_{8}.

1
Table 1. Gematria 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 Yesodoth
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 in this table referred to in the article will be written in
boldface).
2
1. 496 permutations of rows
in tetractyses of 1st 4 enfolded polygons
The four rows of 1, 2,
3 & 4 yods in a tetractys have, respectively, (1! = 1) permutations, (2! = 2)
permutations, (3! = 6) permutations & (4! = 24) permutations (Fig. 1). The three
lowest rows have (2+6+24=32) permutations (see also here). This means that 10 objects
can be arranged in (1!×2!×3!×4! = 288) different tetractyses, where 288 =
1^{1} + 2^{2} + 3^{3} +
4^{4}. When their 17 sectors are
tetractyses, the first four enfolded polygons have 94 yods, that is, 90 yods lie outside
their shared root edge (Fig. 2). The individual yod populations of these polygons
conform to the pattern of integers in the Lambda Tetractys [1], so that they
constitute a holistic system.
An ngon with n
tetractys sectors has 6n yods surrounding its centre. They are arranged in 3n rows that
have 32n permutations. The n sides of the ngon have 24n permutations and its 2n second
& third rows have 8n permutations. Tabulated below are the numbers of permutations
of yods in the root edge and in the 48 rows of the 17 tetractyses in
the first four enfolded polygons:
Table 2. Number of permutations of rows in
the 17 tetractyses of the first four enfolded polygons.
Polygon

Number of permutations on
sides

Number of permutations of 2nd & 3rd
rows

Total

Root edge

24×1 = 24

−

24

Triangle

24×2 =
48

8×3 = 24

72

Square

24×3 =
72

8×4 = 32

104

Pentagon

24×4 = 96

8×5 = 40

136

Subtotal

240

96

336

Hexagon

24×5 =120

8×5 = 40

160

Total

360

136

496

Their 49 rows of yods
have 496 permutations. Remarkably, this is both the number value of
Malkuth, the last Sephirah of the Tree of Life:
and the dimension
of SO(32) and E_{8}×E_{8},
the two symmetry groups that generate interactions
between superstrings that are free of quantum anomalies [2]. As each permutation of two, three & four yods has its
mirrorimage counterpart, the 496 permutations consist of 248 permutations and their 248 mirrorimages, where 248 is the dimension of E_{8}. However,
the actual division of the 496 permutations into two sets
of 248
permutations may be more subtle than this, and we
must analyse the contribution of each polygon in more detail. The first four
enfolded polygons have 49 rows of yods with 496 permutations, showing how EL ChAI, the Godname of Yesod
with number value 49, prescribes this number at the heart of superstring theory. In
fact, EL ChAI prescribes the first (4+4) enfolded polygons because, when they are
all Type B polygons, they possess 490 (=49×10) yods (Fig. 4). The transformation Type
A → Type B adds 153 yods to each
set of four polygons, where 153 is the number
value of ELOHIM SABAOTH, the Godname of Hod.
Table
2 indicates that the first three enfolded polygons have 240 permutations of the yods on
their 10
3
sides (24 permutations per side) and 96
permutations of their 24 second & third rows — a total of 336 permutations. This
chancedefying conjunction of the superstring structural parameter 336 (the number of
circular turns in each revolution of a helical whorl of the UPA) and the superstring
dynamical parameters 240 & 496 is amazing. It
constitutes powerful evidence for the UPA described by Annie Besant & C.W.
Leadbeater being a superstring. This is
confirmed by the fact that the 14 sides of the first four enfolded polygons outside
their shared root edge with 24 permutations of its four yods also have 336 permutations,
so that:
24 + 336 + 136 = 496.
The seven sides of the
square & pentagon outside the root edge have (72+96=168) permutations and the seven sides
of the triangle & hexagon outside the root edge have (48+120= 168) permutations. The geometry of
the first four enfolded polygons naturally allows the division 336 =
168 +
168 that is
characteristic of holistic systems [3]. Its counterpart in the superstring is the 336
turns in one revolution of a helical whorl of the UPA and the 168 turns in a halfrevolution
(scroll down the lefthand column of the table in ref. 4). Alternatively, the three
sides of the triangle have 72 permutations and the seven
sides of the square & pentagon have 168 permutations. The
72:168 division of the 240
permutations of the sides
4
of the triangle, square & pentagon is
a division that is characteristic of the parameter 240 embodied in all holistic systems,
corresponding to the 72 roots of E_{6}, the rank6 exceptional
subgroup of E_{8}, and the 168 other roots of E_{8}. Given this remarkable
correspondence between the 496 roots of E_{8}×E_{8}
and the 496 permutations of the 49 rows of yods in the first four enfolded polygons, the
question arises: what eight permutations are the counterpart of the eight simple
roots of each E_{8}? If no natural counterparts exist in the geometry of the first
four enfolded polygons, this would be evidence favouring SO(32) over
E_{8}×E_{8}. However, analysis in
previous articles of other sacred geometries have certainly demonstrate their
embodiment of the latter symmetry group, and so we should expect the geometry of the
first four enfolded polygons to be able to generate these (8+8) permutations.
Perhaps, however, one set of polygons corresponds to E_{8}×E_{8} and its mirror
image in the other half of the inner Tree of Life corresponds to SO(32)? Given that
each side of a polygon generates 24 permutations of their four yods and that the
second & third rows of yods in each tetractys generate eight permutations, it
seems reasonable to infer that the eight simple roots in one
E_{8} correspond to the eight permutations in the second &
third rows in one of the 17 tetractyses and that the eight simple roots in the other
E_{8} correspond to the eight permutations in the second &
third rows in another tetractys. This leaves 120 permutations in the second &
third rows of the remaining 15 tetractyses (120=8×15), as well as the (5×24=120) permutations of the yods in the five
sides of the hexagon outside the root edge, making a total of 240 permutations. The
two tetractyses should, intuitively speaking, be expected to belong to the simplest
polygon — the triangle — because the simple roots of a Lie group are the most basic
ones. The former 120 permutations comprise the eight permutations in the second &
third rows of the third tetractys in the triangle and
the 112 permutations of the second & third rows in the square,
pentagon & hexagon (see table). The division: 240
= 112 + 8 +120 = 112 + 128 has its group counterpart in E_{8}, as now explained. The
E_{8} root system is a rank 8 root system containing 240 root
vectors spanning R^{8}. It is irreducible in
the sense that it cannot be built from root systems of smaller rank. All the root
vectors in E_{8} have the same length. It is convenient for a number of
purposes to normalize them to have length √2. In the
socalled even coordinate
system E_{8} is given as the
set of all vectors in R^{8} with length
squared equal to 2 such that coordinates are either all integers or all halfintegers
and the sum of the coordinates is even. Explicitly, there
are 112 roots with integer entries obtained from
(±1, ±1, 0, 0, 0, 0, 0,
0)
by taking an arbitrary combination of
signs and an arbitrary permutation of coordinates, and 128 roots with halfinteger
entries obtained from
(±½, ±½, ±½, ±½, ±½,
±½, ±½, ±½)
by taking an even number of minus signs
(or, equivalently, requiring that the sum of all the eight coordinates be even). There
are 240 roots in all.
The 28 red yods lining the 10 sides of the
triangle, square & pentagon with black yods at their centres form 10 sets of 24
permutations of four yods corresponding to the 240 roots of E_{8}. The tetractys sector
with the root edge as its base has a row of two red yods and a row of three red yods
(those yods shared with the two other sectors of the triangle are shown as half red and
half blue). These two rows have eight permutations that correspond to the eight simple
roots of E_{8}. Hence, 12 rows of red yods have 248 permutations. The five sides of the hexagon outside the root edge have
14 blue yods and two red yods. They form five sets of 24 permutations of four yods,
i.e., 120 permutations. One of the two remaining tetractys sectors in the triangle has
two rows with eight permutations; they correspond to the eight simple roots of the
second E_{8} group. The pair of rows in the other sector of the triangle, the
four pairs of rows in the square, the five pairs of rows in the pentagon and the five
pairs of rows in the hexagon constitute
15 pairs
of rows of yods with (15×8=120) permutations. The number of permutations of the
(5+2+2+8+10+10=37) rows of blue yods = 120 + 8 + 120 = 248 . This is the how the pattern of permutations in the rows of yods
in the first four enfolded polygons is analogous to the pattern of
(248+248) roots in the symmetry group
E_{8}×E_{8}.
2. Superstrings are either 10fold or
5fold
Four rows have 16
permutations corresponding to the 16 simple roots of the two similar
E_{8} groups and 45 rows have 480
permutations corresponding to their 480 roots. The 10 sides of the first three
enfolded polygons generate 240 permutations of red yods (24 per side). There remain
five sides of the hexagon and three sets of five pairs of rows, i.e., 35 rows of blue
yods, with 240 permutations. They create five sets of seven rows (one side with 24
permutations & three pairs of rows with (3×8=24) permutations. In other words,
the 240 roots in one E_{8} correspond to
10 sets of 24 permutations and the 240 roots in the other E_{8} correspond to five sets of
(24+24=48)
permutations. Article 53 [5] demonstrates the 10fold division of the holistic
parameter 240 embodied in five sacred geometries. It manifests in the subquark state
of the E_{8}×E_{8} heterotic superstring paranormally described over a
century ago by Annie Besant & C.W. Leadbeater as the 10 whorls of their UPA. 24
gauge charges of E_{8} are spread along
each whorl. It was shown in ref. 6 that the seven separate Type B polygons of the
inner Tree of Life are composed of
5
240 geometrical elements surrounding the
centres of the triangle, square, pentagon & dodecagon and 240 geometrical elements
surrounding the centres of the hexagon, octagon & decagon. The former polygons form
10 sets of 24 elements and the latter polygons form five sets of 48. It was proposed in ref. 6 that the former
correspond to the 10 whorls of the UPA and that the latter correspond to the five whorls
reported by another remoteviewer (private communication) to make up the superstring of
shadow matter whose forces are described by the second E_{8} group. We now find
support for this interpretation in the fivefold pattern of distribution of the 240
permutations of the 37 rows of blue yods. The 10 sides of the triangle, square &
pentagon determine the 10 whorls of the superstring of ordinary matter and the five
sides of the hexagon outside the root edge, the five sectors of the pentagon and the
five remaining sectors of the square & triangle determine the five whorls of the
superstring of shadow matter.
There are 360 permutations of the yods in
the 15 sides and 136 permutations of the 34 second & third
rows. 496 is the 31st
triangular number:
1 + 2 + 3 + ...
+ 31 = 496,
and 136 is the 16th triangular
number:
1 + 2 + 3 + ... + 16 = 136.
A pentagon whose five sectors are
tetractyses contains 31 yods. 30 yods surround its centre.
31
is the number value of EL (אל), the Godname of Chesed. The value 1 of the letter aleph
(א) denotes the
centre of the pentagon and the value 30 of lamed (ל) denotes the 30 yods. 15 black yods lie on the sides
of the pentagon, inside which are 16 red yods (Fig. 6).
Assigning the integers
1–31 to the 31 yods, starting with 1 assigned to the central yod, the first 16
integers inside the pentagon add up to 136 and the
next 15
integers 17–31 assigned to yods on its sides add up to 360. This division is
the counterpart of the 496 permutations (see the last row in Table 2). The interior of
the pentagon generates the number of permutations of the 34 second & third rows
and its boundary generates the number of permutations of
the 15 sides of the first four enfolded polygons. Displayed here is
a remarkable harmony of number and sacred geometry coexisting with an equally
remarkable conjunction of theoretical physics (namely, the
number 496 at the heart
of superstring theory) and the paranormal
observations of Besant & Leadbeater (namely, the number 336 — the number of
circular turns in each revolution of a whorl of the UPA revealed by their micropsi
vision).
The 90 yods outside the root edge of the
first four enfolded polygons comprise 13 corners of 17 tetractyses and 77 hexagonal
yods. The first (4+4) enfolded polygons have (17+17=34) tetractyses with 184 yods made
up of (13+13=26)
corners outside the root edge and 156 hexagonal
yods. 26
is the number value of YAHWEH, the Godname of Chokmah
(Hebrew: YHVH), and 156 is the sum of the values of all combinations of the Hebrew
letters of YHVH:
YHVH = 26.
Y = 10, H = 5, V = 6.
Y + H + V
= 21;
YH + YV + VH + HH = 52;
YHV + YHH + HVH = 57;
YHVH
= 26.
Total = 156.
(17+17=34) hexagonal yods are at the
centres of the 34 tetractyses and (184−34=150=15×10) yods line their 61 sides, where 15 is the number value
of YAH, the older Godname assigned to Chokmah. (150−4=146) yods line the 34 tetractyses
outside the root edge. 73 yods outside the root edge line the 17 tetractyses in each set
of four enfolded polygons. 73 is the number value of Chokmah. The first (4+4) enfolded Type A
polygons have 34 sectors with 61 sides and 28 corners, i.e., 123 geometrical elements.
This is the tenth Lucas number L_{10}, i.e., the tenth member
of the Lucas series after the first one, L_{0} = 2:
2, 1, 3, 4, 7, 11, 18, 29,
47, 76,
123, 199, 322, ...
6
so that it is determined by the Decad
(10). In terms of the Golden Ratio Φ = (1+√5)/2, the nth Lucas number
L_{n} = Φ^{n} +
(−Φ)^{−n}, so that:
123 = L_{10} =
Φ^{10} + Φ^{−10}.
This is
how the Golden Ratio and the Decad express the geometrical composition of the first
(4+4) enfolded Type A polygons, each half of which embodies the third perfect number
496:*
496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124
+ 248
at the
heart of superstring
theory! The second perfect number 28:
28 = 1 + 2 + 4 + 7 + 14
also appears in them as the number of
corners of their 34 sectors, whilst the first perfect number 6:
6 = 1 + 2 + 3
is the number of corners of the hexagon,
which is the fourth polygon.
The 3tree (the lowest three Trees of four
or more overlapping Trees of Life) also possesses 123 points, lines & triangles
(Fig. 7). Article 32 [7] explains the significance of the 3tree and its inner
form as the Tree of Life map of the human skeleton and its acupuncture
meridians.
120 geometrical elements are outside the
root edge of the first (4+4) enfolded Type A polygons, where
120 = 2^{2} +
4^{2} + 6^{2} +
8^{2}.
The Godname YAH with number value
15 prescribes not only the 15 sides of the first four
enfolded polygons and the 15 corners of their 17 sectors but also these
120 geometrical elements because 120 is the 15th triangular
number:
1 + 2 + 3 + 4 + ... + 15 = 120.
The Decad also determines this
number because it is the sum of the first ten odd integers after 1:
120 = 121 − 1 =
11^{2} − 1 = 3 + 5 + 7 + 9 + 11 + 13
+ 15 + 17 + 19 + 21.
See Article 8 [8] for a discussion of the
properties of the first four polygons — both separate and enfolded — and their
prescription
by the Godnames of the 10 Sephiroth. The holistic nature of the first (4+4) enfolded
polygons is proved in Article 48 [9].
References
1. For example, see: http://smphillips.mysite.com/plato%27slambda08.html.
2. http://smphillips.mysite.com/occultchemistry05.html.
3. http://smphillips.mysite.com/theholisticpattern.html.
4. http://smphillips.mysite.com/realgodparticle.html.
5. http://smphillips.mysite.com/article53.html.
6. http://smphillips.mysite.com/treeoflife36.html.
7. http://smphillips.mysite.com/article32.html.
8. http://smphillips.mysite.com/article08.htm.
9. http://smphillips.mysite.com/article48.htm.
* A perfect number is an integer that is the sum of its
divisors.
7
