ARTICLE 9
by
Stephen M. Phillips
Flat 4, Oakwood House, 117119 West Hill Road. Bournemouth. Dorset BH2
5PH. England.
Website: http://smphillips.mysite.com
Abstract
Previous articles showed that the superstring parameters 248
and 168 are embodied as numbers in the outer and inner forms of the Tree of
Life, as well as in various sets of regular polygons belonging to the latter
whose properties are determined by the number values of the Godnames of the ten
Sephiroth. This article will prove that the prescription by the Godnames of
'sacred geometry' encoding such numbers of universal (therefore scientific)
meaning applies also to single, regular polygon, particularly the square. The 25
yods of a square divided into four tetractyses symbolise the spatial dimensions
predicted by the quantum mechanics of spinless strings, their geometrical
differentiation corresponding to the distinction between bosonic and superstring
dimensions of space. There are 248 yods other than corners of tetractyses in a
square whose sectors are transformed into the next higherorder tetractys. 168
is the number of yods surround the centre of a square whose sectors are divided
into three triangles, the latter then being each divided into three tetractyses.
As further illustration of the Tetrad Principle proposed in Article 1, the
square, symbol of the number 4, or Pythagorean Tetrad, also represents in an
arithmetic way the properties of these transformed squares, whilst the hexagon,
the fourth regular polygon, embodies the dimension 248 of E_{8}, the
superstring gauge symmetry group. Finally, equivalence (in the sense of sharing
similar properties) is demonstrated between the square and the seven, enfolded,
regular polygons constituting each half of the inner form of the Tree of
Life.

1.
Introduction
The Pythagorean mathematician Nichomachus of Gerasa said that the Pythagoreans called the
number 4, or Tetrad, "the greatest miracle," "a God after another manner," "a manifold
divinity," and "holding the key of Nature." Why did he (and his precursors) value this number
so highly? Lacking insight into the mystical wisdom of ancient sages like Pythagoras and
possessing only a few surviving written fragments of his teachings — themselves only elementary
and distorted summaries of a now lost knowledge taught by word of mouth — modern scholarship
has no satisfactory answer to this question. Historians of Greek mathematics realise that
the
1
early Pythagoreans placed special emphasis on the number 4 because their
socalled 'perfect number' 10 is the fourth triangular number, otherwise known as the
'tetractys' (from the Greek word 'tetras,' meaning 'four'):
They knew also that the numbers 1, 2, 3 & 4 contain the symphonic ratios
that make possible the musical scale, i.e. 1:2, the octave; 2:3, the perfect fifth; and 3:4,
the perfect fourth. But the true reason why the Pythagoreans valued the tetrad so highly was
far more profound. They intuitively recognised that a fourfold pattern or structure existed in
the mathematics of the cosmos, generating not only the number 10 as the measure of divine
perfection and the tetractys as its symbol of wholeness but also numbers that quantify physical
phenomena. This rule was formulated as the ‘Tetrad Principle’ in Article 1, which showed how parameters of superstring theory have
a natural expression in terms of the numbers 1, 2, 3 & 4 — even the number 4 itself.
In this article it will be explained how the square, the geometrical symbol
of the number 4, embodies certain numbers of relevance to superstring theory (and beyond), thus
amply justifying the assignment to the Tetrad by the Pythagoreans of the title “holding the key
of Nature.”
2. The square encodes 25 spatial
dimensions & dimension 248 of E_{8}
When its sectors are tetractyses, a square contains 25 yods:
They symbolise the 25 dimensions of space predicted by the quantum mechanics
of spinless (bosonic) strings. The number 16 (4th square number) is the number of purely
bosonic string dimensions of the 26dimensional spacetime whose
compactification, according to the author’s theory outlined in Article 2, generates
superstrings with 9 remaining spatial dimensions (9 = 4th odd integer after 1). The central yod
() symbolises the longitudinal dimension of a bosonic string,
the surrounding 24 (=1×2×3×4=4!) yods represent the 24 transverse dimensions of a string
and the surrounding 8 yods () denote the 8 transverse, purely superstring dimensions (8 =
4th even integer). 25 is the arithmetic mean of the cubes of the first 4 integers:
25 = (1^{3} + 2^{3} + 3^{3} +
4^{3})/4.
When the cubes of the integers 1, 2, 3 & 4 are assigned to the positions
of the 25 yods in a square, their sum:
is the dimension of the unified, superstring gauge symmetry group
E_{8}. In other words, the square not only encodes the number of spatial dimensions of
spacetime but also embodies the number of quantum states
2
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. Where they appear in the text, they are written in
boldface.

3
of the particle transmitting the unified force between superstrings!
Superstring theory, of course, predicts not 25 but nine spatial dimensions of spacetime.
Unless one is willing to dismiss as coincidence the appearance of the number
248 along with the number 25 and its breakdown into the numbers 9 &
16, it indicates that superstrings and their gauge symmetry group E_{8} have their
origin in 26d, not 10d, spacetime.
That the number 26 should be connected to the number
248 is less surprising when it realised that it is the number value of YAHWEH, the
Godname of Chokmah (Table 1).^{1} The Godnames of the Sephiroth of the Tree of Life were shown
in previous articles to prescribe the sacred geometries of its outer and inner forms, which
were also shown to embody the number 248. That the number
26 should have fundamental scientific significance is indicated by the
Tetrad Principle formulated in Article 1 (1), for it is the sum of the first
four Mersenne numbers (2), that is, the sum of the
combinations of 10 objects arranged in the four rows of a tetractys:
A
B C
D E F
G H I J

Number of combinations
2^{1} – 1 = 1
2^{2} – 1 = 3
2^{3} – 1 = 7
2^{4} – 1 = 15 TOTAL =
26

There is also a remarkable arithmetic relationship between the two numbers.
Using the identity:
6200 = 248×25 = 2^{2} + 3^{2} +
4^{2} + ... + 26^{2},
where the first term, 2^{2} = 4, is the Tetrad, we see that the
square displays the following remarkable property of the number 248:
Starting with the Tetrad, 2^{2} = 4, at the centre of the
square and assigning the 24 (=1×2×3×4) squares 3^{2} , 4^{2} , ...
26^{2} to the surrounding 24 yods of the four tetractyses, with squares of
even integers inside the square and squares of odd integers on its boundary, the sum of these
25 squares is that resulting from assigning the number 248 to each yod in the
square. This demonstrates the unique significance of the superstring parameter
248 visàvis the square.
The dimension 496 of the superstring gauge symmetry groups
E_{8}×E_{8} and SO(32) that apply to the five superstring theories is the
sum of the cubes of the first four odd integers:
This means that it can also be represented by a 4×4 square array of the squares
of 1, 3, 5 & 7. It is an illustration par excellence of the Tetrad
Principle.
The square encodes the number 248 geometrically as well
as arithmetically, as now explained. Starting
___________________________
^{1} Henceforth, all numbers in this table are written in
boldface.
4
from the point, or Pythagorean Monad, the tetractys has ten points
(symbolised below by circles), each of which is a potential tetractys of ten points, and so
on.
Let us call the point the 0thorder tetractys, the tetractys known to
mathematicians as the fourth triangular number the 1storder tetractys, the next member of the
infinite sequence the 2ndorder tetractys, etc. The number of yods in the 2ndorder tetractys
shown above is:
85 = 4^{0} + 4^{1} +4^{2} +
4^{3}
This is a property of the Pythagorean representation of holistic systems
that illustrates the Tetrad Principle. The number of white yods (o) at corners of the ten
tetractyses is 15, which is both the number of combinations of
four objects and the fourth Mersenne number

Figure 1. The 40 tetractyses in the square
when its sectors are 2ndorder tetractyses contain
248 coloured yods other than the 41 corners. These
hexagonal yods, which symbolise the Sephiroth of Construction, denote
the 248 gauge fields transmitting the unified force between
superstrings.

(see earlier). The number of coloured yods is
70 =

7
7 7
7 7 7
7 7 7 7 ,

where 7 is the fourth odd integer. 70 is the fourth,
4dimensional, tetrahedral number after 1, i.e., it is the number of balls that can be piled up
in 4dimensional space into the fourth tetrahedron.
Consider now a square divided into its four sectors, each of which is the
2ndorder tetractys (Fig. 1). The number of tetractyses
is
as is the number of corners surrounding the centre of the square. The number
of yods surrounding the centre of the square is
288 = 1^{1} + 2^{2} + 3^{3} +
4^{4}.
Notice how the Tetrad expresses the properties of both the 2ndorder
tetractys and the square when the latter is constructed from the former. There are
72 yods per sector, of which 10 are corners and
62 are coloured yods (what in earlier articles were referred to as
“hexagonal yods,” socalled because they are located at the centres and corners of hexagons) .
72 is the number of Chesed and 62 is the number
value of
5
Tzadkiel , the Archangel of Chesed. There are
48 yods lining the square, where 48 is the number
value of Kokab, the Mundane Chakra of Hod. 80 corners &
centres of tetractyses surround the centre of the square, where 80 is the
number value of Yesod. 49 yods line its diagonals, where
49 is the number value of EL ChAI, the Godname of Yesod.
64 hexagonal yods line the edges of the 2ndorder tetractyses, where
64 is the number value of Nogah, the Mundane Chakra of Netzach.
Fig. 1 shows that there are
248 hexagonal yods (yods other than corners of tetractyses).
248 is the number value of Raziel, the Archangel of Chokmah.
Symbolising Sephiroth of Construction, they denote the 248 gauge
fields of the superstring gauge symmetry group E_{8} transmitting the unified
force between superstrings. It is a remarkable demonstration of the way in which the Tetrad
determines numbers of scientific significance, in this case the dimension of E_{8}.
The Godname EHYEH prescribes it because the 36 tetractyses have
21 corners.
There are 32 hexagonal yods lying on the sides of the square that are
unshared with adjacent tetractyses. This leaves (248–32=216)
hexagonal yods inside the square. There are 46 internal hexagonal yods per sector that are
unshared. The number of unshared hexagonal yods in the square = 32 + 4×46 =
216. It is amusing that this is the number of years that, according to
Androcydes, the Pythagorean writer of the work On Symbols, and others was thought in
antiquity to elapse between successive lives of Pythagoras. 216 is the
number value of Geburah, the fifth Sephirah (see Table 1). It is also the notorious
number mentioned by Plato in a passage in his Republic, which is thought (3) to refer to its property as the
sum of the cubes of the numbers 3, 4 & 5 that measure the sides of
Figure 2. Symbol of the Pythagorean Tetrad, the
square can be formed from 36 tetractyses with
21 corners and with 168 yods surrounding its
centre.
Pythagoras’ famous rightangled triangle.
Of the 48 yods on the boundary of the square whose
sectors are divided into the next higherorder tetractys, four are its corners, leaving 44
other yods, once again illustrating the influence of the Tetrad in determining or expressing
the properties of objects with sacred geometry.
6
3. Square encodes
superstring structural parameter 168
In previous articles, it was explained how the outer and inner forms of the Tree of Life
encode the number 168, which is the number value of
Cholem Yesodeth, the Mundane Chakra assigned to Malkuth (physical
universe). It is the number of circularly polarised waves making up half a revolution of
each of the ten whorls of the superstring constituent of up and down quarks. We have
already seen that the square encodes its dynamical parameter 248. It will
now be shown how it embodies its structural parameter 168.
Instead of turning each sector of a square into a 2ndorder tetractys, consider it as a
triangles divided into three sectors, each of which is then split into three more triangles
turned into tetractyses (Fig. 2). Both a single square
constructed in this way and the pair of squares enfolded in the inner form of the Tree
of Life will be considered. Their properties are analysed below. Table 2 depicts the
numbers in Table 1 that appear in
the analysis presented in this article.
Table 2
(Numbers in coloured boxes appear in the discussion below)
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

Single square Square has four sectors, each with (3×3=9)
triangles/tetractyses having 12 edges inside it.
Number of corners of triangles = 1 + 4 + 4×4 = 21 (19
outside root edge).
Number of sides of triangles = 4 + 4 + 4×12 = 56 (55 outside root edge,
48 not sides of sectors).
Number of triangles = 4×9 = 36.
Number of corners & sides of triangles = 21 + 56 = 77.
Number of corners & triangles = 21 + 36 = 57
(55 outside root edge).
Number of sides & triangles = 56 + 36 = 92 (91 outside root
edge).
Number of corners, sides & triangles = 77 + 36 = 113 (113 – 3 = 110
outside root edge, 113 – 8 = 105 inside square).
Number of yods = 21 + 36 + 56×2 = 169 (169 – 4 =
165 outside root edge, 168 surrounding centre).
Number of yods at corners of tetractyses = 21.
Number of hexagonal yods = 169 – 21 = 148.
Number of yods on edges of tetractyses = 169 – 36 = 133 (133 – 4 =
129 yods outside root edge or generated from square with four
corners).
Number of hexagonal yods on edges of tetractyses = 133 – 21 =
112.
Number of yods on boundary of square = 12.
Number of yods inside square = 169 – 12 = 157 (17 corners and
140 hexagonal yods).
Number of yods generated from square with 4 yods at corners = 169 – 4 = 165.
Two joined squares
Two squares have 8 sectors, each with (3×3=9) triangles/tetractyses with 12 edges inside
it.
7
Number of corners of triangles = 2 + 2×19 = 40.
Number of sides of triangles = 1 + 2×55 = 111 (110 outside root edge).
Number of triangles = 2×36 = 72.
Number of corners & sides of triangles = 40 + 111 = 151.
Number of corners & triangles = 40 + 72 = 112.
Number of sides & triangles = 111 + 72 = 183.
Number of corners, sides & triangles = 151 + 72 = 223 (223 – 3 = 220
outside root edge, 223 – 13 = 210 inside squares).
Number of yods = 4 + 2×165 = 334 (334 – 4 = 330 outside root edge).
Number of yods at corners of tetractyses = 2 + 2×19 = 40.
Number of hexagonal yods = 334 – 40 = 294 (292 outside root edge).
Number of yods on edges of tetractyses = 334 – 72 = 262 (262 – 4 = 258
outside root edge).
Number of hexagonal yods on edges of tetractyses = 262 – 40 = 222 (220 outside root edge).
Number of yods on boundary of squares = 4 + 2×(12–4) = 20.
Number of yods inside squares = 334 – 20 = 314 (34 corners and
280 hexagonal yods, 140 in each square).
Number of yods generated from two joined squares with 6 corners = 334 – 6 = 328.
Listed below are the ways in which the number values of the Godnames of the 10
Sephiroth prescribe the properties of the square and the pair of joined squares:
Kether: 21 
36 triangles in square have
21 corners. also, 210 (=21×10) corners,
sides & triangles inside squares. 
Chokmah: 15 
Number of corners & sides of triangles = 151 = 150th integer
after 1, where 150 = 15×10. 
26 
36 triangles in square have 56 sides, of
which 52 are generated from square, where 52 = 26th even
integer. 
Binah: 50 
Number of corners, sides & triangles in two squares separated
by root edge = 3 + 2×113 = 229 = 50th prime number. 
Chesed: 31 
Number of new yods outside root edge on edges of tetractyses in
square = 129 – 2 = 127 = 31st prime
number. Also, number of yods outside root edge of two joined squares other than
corners = 330 – 19 = 311 = 310th integer after 1, where 310 =
31×10. 
Geburah: 36 
Square has 36 triangles. also, two joined
squares have 151 corners & sides, where 151 = 36th prime
number. 
Tiphareth: 76 
Square has 77 corners & sides, where 77 =
76th integer after 1. 
Netzach: 129 
Number of yods in two joined squares outside root edge on sides of
tetractyses = 258 = 129th even integer. Also, the number of
yods generated from square with four corners. 
Hod: 153 
Number of corners & sides of two separate squares = 2×77 = 154
= 153rd integer after 1. 
Yesod: 49 
72 triangles in two joined squares have 111
sides, of which 14 are sides of sectors other than root edge. number of sides
other than sector edges = 111 – 14 = 97 = 49th odd
integer. 
Malkuth: 65 
129 yods outside root edge on sides of two
squares, where 129 = 65th odd
integer. 
155 
Number of yods inside square surrounding centre = 157 – 1 = 156 =
155th integer after 1. 
The most important property listed above is that
168 yods surround the centre of a square whose sectors are constructed
from three triangles, themselves constructed from three tetractyses. The number values of the
Godnames of the ten Sephiroth therefore prescribe this basic structural parameter of the
superstring.
The Pythagorean Tetrad determines the following properties of the square and
the pair of joined squares present in the inner form of the Tree of Life:
1. Number of corners of tetractyses in two joined squares = 40
2. Number of sides outside the root edge of the square = 55
8
Figure 3. The fourth type of square has
168 yods
surrounding its centre.
1
2 3
= 4 5 6
7 8 9 10.

(10=1+2+3+4)

This is also the number of corners and triangles in the square outside the root
edge.
3. Number of corners, sides & triangles in two joined squares outside the root edge =
220
9
10
11
4
8 12
= 16 20 24
28 32 36 40.
This is also the number of hexagonal yods outside the root edge in two joined
squares.
1. Number of yods in the square = 169 = 13^{2}
168 yods surround its centre, which is denoted by the
number 1. 168 is the sum of the 12 odd integers after 1 on the square’s
boundary, four to a side.
2. Number of yods in the square generated by its construction from tetractyses (i.e., yods
other than its corners) = 169 – 4 = 165
Again, the central number, 1^{2} = 1, denotes the centre of the
square.
3. Number of yods outside the root edge of two joined squares = 330
The central number, 6^{2} = 36, denotes the
36 hexagonal yods at the centres of the
36 tetractyses. The Tetrad also determines the superstring structural
parameter 168 because it is the number of yods surrounding the centre of
the fourth member of the sequence of tetractysgenerated squares, starting with the
square itself (Fig. 3). It is a remarkable
illustration of the Tetrad Principle discussed in Article 1.
4. Equivalence of the square
& the 7 enfolded polygons
According to the properties of the square listed in Section 2, the pair of squares has 262 yods
on the boundaries of their 72 tetractyses. The seven regular polygons
enfolded in the Tree of Life have 264 yods, that is, 262 yods are associated with either set of
polygons. The number of yods defining the shapes of the tetractyses making up the pair of
squares is that needed to fill up each half of the inner form of the Tree of
Figure 6. The number of hexagonal (i.e.,
new) yods in the Tree of Life constructed from tetractyses is the
number of yods needed to create a square, starting from
its centre, with three tetractyses as each sector.
12
Life (Fig.4). Furthermore, there are 222
hexagonal yods on the boundaries of the 72 tetractyses. These yods
correspond to the Sephiroth of Construction above Malkuth, the lowest Sephirah of
Construction. 222 is the number of hexagonal yods associated with each set of seven enfolded
polygons (Fig. 5). These two remarkable
similarities demonstrate that the square possesses an implicit or hidden equivalence to the
inner form of the Tree of Life. As there are 444 hexagonal yods in the (7+7) enfolded
polygons (4), the Tetrad expresses the
hexagonal yod population of the inner form of the Tree of Life.
The square is also equivalent in the same sense to the outer form of the
Tree of Life. The reason for this is as follows: when the 16 triangles of the Tree of Life are
turned into tetractyses, they contain 70 yods made up of the 10 corners of tetractyses at the
positions of the Sephiroth and 60 hexagonal yods (Fig. 6). When each sector of a
square is divided into three triangles which are then turned into tetractyses, the square is
found to contain 61 yods. 60 yods are needed to construct such a transformed square from the
point at its centre, just as 60 additional yods are required to construct the outer form of
the Tree of Life from 16 tetractyses.
The hexagon is the fourth of the regular polygons.
36 yods surround its centre when its sectors are tetractyses. When the
latter are of the same type as that discussed in Section 3 for the embodiment of the number
168 by the square, the hexagon contains 252 yods
Figure
7. The (248
+
248 )
yods outside the root edge surrounding the centres
of the two joined hexagons symbolise the
(248
+
248 )
gauge fields transmitting the unified
force between E_{8}×E_{8} heterotic superstrings. The
78 green yods in each hexagon that are either
centres or corners of tetractyses denote the 78
gauge fields of E_{6}.
surrounding its centre. Hence, there are 248 yods
outside the root edge that surround its centre. Outside the root edge of the pair of joined
hexagons (Fig. 7) are
(248+248=496) yods surrounding their
centres. They symbolise the (248+248) gauge fields of the
E_{8}×E_{8} heterotic superstring. Once again, we see how the Tetrad
Principle picks out a mathematical object that embodies a number of fundamental
significance. The regular polygon with four corners embodies the structural parameter
168 of superstrings and the fourth regular polygon embodies its
dynamical parameters 248 & 496. ELOHA, the
Godname of Geburah with number value 36, prescribes this representation
because each hexagon has 36 yods surrounding its centre when its
sectors are tetractyses. ELOHIM, Godname of Binah with number value 50,
prescribes it because, including the four yods of the root edge, 500
(=50×10) yods surround the centres of the two hexagons. YAHWEH, the Godname
of Chokmah with number value 26, prescribes this representation of the
number 496 because 26 yods on edges of the
sectors of each hexagon surround its centre. Each sector has nine hexagonal yods at the
centres of tetractyses. There are (248 – 6×9 = 194)
yods other than centres. 194 is the number value of Tzadekh,
the Mundane Chakra of Chesed. The nine tetractyses in each sector have four corners inside
the boundary of the hexagon (48 for both hexagons), which has four
corners outside the root edge. Hence, there are (6×4 + 4 + 1 = 29) external corners and
(248–28=220) hexagonal yods, where 220 = 4×55
4
8 12
= 16 20 24
28 32 36 40,
13
showing how the Tetrad determines this number. The two joined hexagons have
(2×29 + 2 = 60) corners and (2×220 + 2 = 442) hexagonal yods, i.e., 221 hexagonal yods are
associated with each hexagon. This is the number of hexagonal yods associated with each set of
seven enfolded polygons that are outside the root edge (see Fig. 5). This demonstrates how the
pair of joined hexagons embodies a property of the inner Tree of Life. There are 13 corners
& centres of the nine tetractyses in each sector. The 248 yods
therefore comprise (6×13=78) corners & centres (shown in Fig. 7 as green yods). They
symbolise the 78 roots of E_{6}, an exceptional subgroup of E_{8} that
physicists have considered as a basis for the physics of the Standard Model (in other words,
the currently known properties of subatomic particles). Six of the corners are centres of
the sectors, leaving 72 other corners & centres. They correspond
to the six simple roots of E_{6} and its 72 other roots.
There are 156 green yods in the two hexagons, where 156 is the 155th
integer after 1 and 155 is the number value of ADONAI MELEKH, the
Godname of Malkuth. As this Sephirah is the universe of physical matter, it is an indication
that E_{6} does, indeed, describe part of its physics.
5.
Conclusion
The simple square symbolises the number 4, or Tetrad, so highly esteemed by the ancient
Pythagoreans. The four integers 1, 2, 3 & 4 add up to 10, which is the number of Sephiroth
of the Tree of Life and the number of points in the tetractys — an equivalent symbol of the
tenfold nature of divine unity and a representation of these integers. A square divided into
tetractyses has 25 yods symbolising the 25 spatial dimensions of the universe. A square with
each sector divided into three tetractyses has 60 yods surrounding its centre that correspond
to the 60 extra yods making up the triangles of the Tree of Life when they are turned into
tetractyses. A square with each sector divided into three triangles that are then divided into
three tetractyses contains 168 yods surrounding its centre. This number
characterises the manifestation of the Tree of Life in the subatomic world as the number of
circularly polarised waves made by each one of the ten whorls of a spinning superstring as it
completes half a revolution in space about its axis of spin, that is, onetenth of the five
complete revolutions of each whorl . The number 168 is defined
arithmetically by a square because it is the sum of the twelve, successive odd integers after 1
that can be spaced along the sides of a square, four to each side. A square with each sector
turned into the next higherorder tetractys above the tetractys contains
248 yods other than corners of tetractyses. These yods, which symbolise
the seven Sephiroth of Construction, signify the 248 quantum states of
the particle transmitting the unified force acting between superstrings of either ordinary or
shadow matter. A 4×4 square array of the squares of the first four odd integers has as its sum
the number (496) of such states responsible for the interactions of
superstrings of both kinds of matter. Successive transformations of the square generate numbers
corresponding to ever more concrete realisations of the cosmic blueprint. The fourth of the
regular polygons embodies the 248 gauge fields of the superstring
symmetry group E_{8}, the pair of these polygons embodying the dimension
496 of E_{8}×E_{8}. Truly, one must agree with the
Pythagoreans in their calling the number 4 “the greatest miracle” and “holding the key of
Nature."
References 1. Phillips, Stephen M. Article 1: “The Pythagorean nature of superstring
and bosonic string theories,” (WEB, PDF), p. 4.
2. Named after Father Marin Mersenne, a natural philosopher, theologian,
mathematician, musical theorist and friend of René Descartes, the nth Mersenne number is of
the form M_{n} = 2^{n} – 1, where n is an integer.
3. J. Adams, The Republic of Plato, Cambridge University Press,
Cambridge, 1929.
4. Proof: Of the 264 yods in the seven enfolded polygons,
36 are corners of polygons and five more are just their centres. There
are (264–36–5=223) hexagonal yods. Two of them lie on the root edge and 221
are external. Number of hexagonal yods in the (7+7) enfolded polygons = 2 + 2×221 = 444.
14
