ARTICLE 52
by
Stephen M. Phillips
Flat 3, 32 Surrey Road South. Bournemouth.
Dorset BH4 9BP. England.
Website: http://smphillips.8m.com
Abstract
A systematic analysis of
the angles subtended by pairs of simple roots of all Lie groups is carried
out. The number value 21 of EHYEH, the Godname of Kether,
is the number of pairs of orthogonal roots of E_{8} and the
number of angles between the seven simple roots of E_{7}, its
largest exceptional subgroup. The number value 15 of YAH,
the older version of YAHWEH, the Godname of Chokmah, is the number of pairs
of orthogonal roots of E_{7} and the number of angles between the
six simple roots of E_{6}, which is the next largest exceptional
subgroup of E_{8}. None of the exceptional Lie groups has a
dimension that is a Godname number. Allowing dimensions of groups to be
10×Godname number, the Godname numbers 21, 15 & 36
define dimensions of subgroups of E_{8}. Only 21 &
15 define groups of less rank than E_{8}, all of which are
its subgroups. Only 21 & 15 define groups of rank less
than or equal to that of E_{8} whose numbers of orthogonal pairs of
simple roots are Godname numbers. The letter values of EYHEH & YAH
denote the numbers of right angles between sets of simple root vectors of
E_{6}, E_{7} & E_{8}. The integers 115
forming a cross pattée array sum to 4960. This is the number of spacetime
components of the 496 gauge fields of E_{8}×E_{8}
& SO(32), which are the gauge symmetry groups of the heterotic
superstring. A cross pattée array of integers 226 add up to 24800,
showing how YAHWEH determines the dimension 248 of E_{8}. The
Godnames ELOHIM, EL, YAHWEH SABAOTH & ELOHIM SABAOTH also determine this
number. ELOHIM with number value 50 prescribes 496 because
this number is the arithmetic mean of the first 50 triangular numbers
after 3.

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.

(All numbers in this table referred to
in the article are written in boldface).
2
1.
Numbers of EHYEH & YAH prescribe E_{8}
The gematria number values of the ancient Hebrew Godnames
have hitherto been found to prescribe the dimensions of the Lie groups
E_{8} and E_{8}×E_{8} or SO(32) either
arithmetically or in the geometrical context of the yod population of overlapping
trees or the yod/corner populations of enfolded polygons generated by the Tree of
Life. In this article, we shall demonstrate how the Godname numbers of Kether and
Chokmah select E_{8} and its two exceptional subgroups
E_{7} and E_{6} from the infinite number of Lie groups.
Because of its importance in proving rigorously that Godnames prescribe superstring
physics, the following discussion must be technical. It is therefore intended
primarily for mathematicians and physicists. For this reason, readers unfamiliar with
group theory are advised to turn to the simplified summary on page
10.
The mathematician E. Cartan showed in 1894 that there are
four* infinite series of simple Lie algebras:
D_{n}, generates SO_{2n}
A_{n}, generates SU_{n+1}
n = 1, 2,
3, 4
_{
} C_{n}, generates Sp_{2n }
B_{n},
generates
SO_{2n+2 }
and five "exceptional" Lie
algebras:
The structure of a
simple Lie algebra or group G is defined completely by a set of (rank G)dimensional
vectors called "simple roots," which span the "root space" of G, a (rank G)dimensional
Euclidean space. The "Dynkin diagram" specifies the set of simple roots
α_{i} (i = 1, 2, 3... rank G) of G. Each simple root is denoted by a dot
in 2d space. In the case of a G having α_{i}'s with two different lengths
α_{i},
the longer roots are denoted by open dots " š
" and the shorter roots by filledin dots "●" (no simple
group has roots with three or more different lengths). There
are four possible angles θ_{ij} between any pair
of simple roots α_{i} and α_{j} in root
space:
150°
90°
135°
120°
The angle
between a pair of simple roots is denoted in the Dynkin diagram of G by lines connecting
corresponding dots. The following convention is used:
* This is an example of the Tetrad
Principle discussed in Article 1.
3
Shown below is an exhaustive list of
the Dynkin diagrams of all simple Lie algebras G:
Suppose that G has N simple roots.
There are = ½N(N1) different
pairs of root vectors (α_{i},α_{j}) enclosing between
them the angle θ_{ij}. Therefore, the
mutual orientation between the N root vectors in root space is specified
completely by angles. This
number is the sum of the number of different pairs of orthogonal roots unjoined by
a line in the
Dynkin diagram of G and the number of pairs joined by one or more lines, which
enclose any of the angles 120º, 135º or 150º. In the Dynkin diagram of every
Lie algebra,
there are no isolated roots: every simple root is joined to at least one
other.
4
This means that there are
=½(N1) (N2)
right angles and = (N1) angles
which are either 120º, 135º, 150º or mixtures thereof [N.B. =
+
]. Tabulated
below is the number of angles enclosed by all pairs of simple roots defining
all possible Lie algebras:
The Godname number
21 of Kether is the number of pairs of orthogonal roots of the
superstring symmetry group E_{8} and the number
of angles between the seven simple roots of E_{7}. The Godname
number 15 of
Chokmah is the number of pairs of orthogonal roots of
E_{7} and the number of angles between the six simple roots
of E_{6}, a subgroup of E_{8} favoured
by many string theorists as the probable product of symmetry breaking of
E_{8}.
Turning to the nonexceptional groups:
1. for n = 6, number of angles between the simple
roots of SU_{7},
SO_{12},
SO_{13} & Sp_{12} = 15;
2. for n = 7, number of right angles between the
simple roots of SU_{8},
SO_{15},
SO_{14} & Sp_{14} = 15;
3. for n = 8, number of right angles between the
simple roots of SU_{9},
SO_{17}, SO_{16} &
Sp_{16} = 21.
Since the superstring symmetry group is
E_{8}, any other groups picked out by the Godnames of Kether or
Chokmah must be subgroups of E_{8}. These are:
SU_{9}, SO_{16} (n = 8),
SU_{8} (n = 7) & SU_{7},
SO_{13}, SO_{12} (n =
6).
5
The dimension
d_{G} of G is the number of its generators. For what Lie
algebra is d_{G} =
N_{GOD}, where N_{GOD} is
a Godname number? The Lie algebras have the following
dimensions:
None of the exceptional Lie
algebras has a dimension that is a Godname number. Below are tabulated all
values of n covering the range 15≤N_{GOD}≤543,
where 15 is the
smallest and 543 is the
largest Godname number:
6
n = 16 determines the
anomalyfree group SO_{32} with
dimension 496.
It is amusing that the dimension of
SU_{n+1} exceeds the largest Godname
number 543 for n>22, the number of Paths in the
Tree of Life. B_{3},
(SO_{7})
and C_{3} (Sp_{6})
have dimension 21, A_{3} (SU_{4})
and D_{3} (SO_{6})
have dimension 15, B_{4} (SO_{9})
and C_{4} (Sp_{8})
have dimension 36 and D_{9} (SO_{18})
has dimension 153. Allowing the further kind of
prescription:
N_{GOD}
N_{GOD}
N
_{GOD}
d_{G} =
N_{GOD}
N_{GOD}
N_{GOD}
_{
} N_{
GOD}
N_{GOD}
N_{GOD}
N_{GOD}
.
i.e., 15 ≤d_{G}≤ 5430, then B_{10} (SO_{21})
and
C_{10}
(Sp_{20}) have the dimension:
21
21
21
d_{G} =
21
21
21
21
21
21
21
and SU_{19} has the dimension:
36
36
36
d_{G} =
36
36
36
36
36
36
36
(there are no other possibilities). The
groups with rank less than or equal to the rank 8 of the superstring
group E_{8} are
SO_{7} and Sp_{6} (with
dimension 21), SU_{4} and
SO_{6} (with dimension 15), and
SO_{9} and
Sp_{8} (with dimension
36). Since
E_{8}⊃SO_{16}⊃SO_{9}⊃SO_{7}⊃SU_{4},
E_{8}⊃E_{6}⊃Sp_{6}⊃SU_{2}×Sp_{6}, and
SU_{4}≈SO_{6},
the Godname
numbers 21, 15 and 36 define dimensions of subgroups of
E_{8}.
For what Lie algebras is the total number of angles
between pairs of root vectors equal to N_{GOD}, and what are their dimensions? Below are tabulated values
of n determining N_{GOD} and the
corresponding dimension of the Lie algebra:
7
Godnames numbers of four
Sephiroth each define four, nonexceptional Lie
algebras:
SO_{14} (91)
SU_{8}(63)
n = 7:
=
21
Sp_{14}(105)
SO _{15} (105)
SO_{12} (66)
SU_{7}(
48)
n = 6:
=
15
Sp_{12}(78)
SO_{13}(78)
(number in brackets is dimension of Lie algebra)
SO_{18}(153)
SU_{10}(99)
n = 9:
=
36
Sp_{18}(171)
SO_{19}(171)
SO_{36}(630)
SU_{19}(360)
n = 18:
=
153
Sp_{36}(666)
SO_{37} (666)
21 also
defines E_{7}
and 15 defines E_{6}. The Lie
algebras corresponding to 153 have
rank 18, which exceeds the rank 8 of E_{8} and
the rank 16 of E_{8}×E_{8} and
SO_{32}. They are therefore disallowed. The groups
corresponding to 36 are also forbidden because they have rank 9. Only
the Godname numbers 21 and 15 define groups of lesser rank than
E_{8}, all of which are its
subgroups.
For what Lie algebras of rank N is the number of
right angles between pairs of simple roots =
N_{GOD}, and what are their dimensions? Below are tabulated
values of n
8
determining
N_{GOD}
and the corresponding dimension
d_{G} of the Lie algebra:
The Godname numbers of
four Sephiroth (the same as above) define four nonexceptional Lie
algebras:
SO_{16}(120)
SU_{9}(80)
n = 8:
= 21
Sp_{16}(136)
SO_{17} (136)
SO_{14}(91)
SU_{8}(63)
n = 7:
= 15
Sp_{14}(105)
SO_{15} (105)
(number in brackets is dimension of Lie algebra)
SO_{20}(190)
SU_{11}(120)
n = 10:
= 36
Sp_{20}(210)
SO_{21} (210)
SO_{38}(703)
SU_{20}(399)
n = 19:
= 153
Sp_{38}(741)
SO_{39}(741)
Comments
1. 21 also
defines E_{8} and 15 defines
E_{7}, as found earlier. SO_{17} is
excluded because SO_{16} is a
maximal subgroup of E_{8}. Since
E_{8} contains the others as subgroups, the group
selected by 21 with the largest dimension is
E_{8}.
2. 15 selects
the rank7 groups SU_{8},
SO_{15},
Sp_{14},
SO_{14} and E_{7}.
3. 36 selects only rank10 groups, which are forbidden,
having higher rank than E_{8}.
9
Similarly, the rank19
groups selected by 153 are
disallowed.
It is concluded that only
21 and 15 define
groups of rank less than or equal to that of
E_{8} whose numbers of orthogonal pairs of simple
roots are equal to Godname numbers.
We shall now prove that the Godnames EHYEH
and YAH have letter values that denote the numbers of right angles between
sets of simple root vectors of E_{6},
E_{7} and E_{8}. Below are tabulated the Dynkin diagrams of these groups,
their simple roots and the labelled simple roots that are perpendicular to
them:
Notice
that:
1. The number
15 of YAH (Hebrew: YH) is the number of right
angles between the seven simple roots of E_{7}. The
value 10 of the letter Y is the number of right angles between the
root vectors of E_{6} and the value 5 of letter H is the extra
number of right angles between the simple root vectors of
E_{7}.
2. The number
21 of EHYEH (Hebrew: AHIH) is the number of
right angles between the eight simple root vectors of
E_{8}. The value 1 of the letter A denotes the right
angle between roots 8 & 7, the value 5 of the first letter H is
the number of right angles between simple root 8 and simple roots 1,
2, 4, 5 & 6, and the value 5 of the second letter H is the number
of right angles between simple root 7 and roots 1, 2, 3, 4 &
5. Comparing AHIH with YH and remembering that the
Hebrew letters for I and Y are the same, we see that AHIH specifies
the superstring symmetry group E_{8}, YH
specifies E_{7} and I (or Y) specifies
E_{6}.
According to the Dynkin diagram of
E_{8}, any simple root vector is orientated at an angle of
120º to one, two or three other simple root vectors. Compare the eight
simple root
10
vectors of
E_{8}
with one of the sets of eight
generators of the (7+7) polygons enfolded in the Tree of
Life:
Each generator lies on
one, two or three circles as the endpoint of a vertical or horizontal
diameter. The simple root vector 3 is unique in being orientated at
120º to three other simple root vectors. Similarly, the generators
located at Daath or Tiphareth in the Tree of Life are unique in being
the points of intersection of three circles. The following
correspondence emerges between the (7+1) simple root vectors of
E_{8} and
the seven Sephiroth + Daath of the Tree of Life:
ROOT
SEPHIRAH
8
→
(Daath)
1
→
Chesed
2
→ Geburah
3
→
Tiphareth
4
→ Netzach
5
→
Hod
6
→ Yesod
7
→ Malkuth
Notice that the unique root vector 3
corresponds to Tiphareth, which uniquely occupies the centre of the
Tree of Life both in a geometrical and in a metaphysical sense. This
analogy between the generators of the polygons enfolded in the Tree of
Life and the Dynkin diagram of the superstring group
E_{8} should
come as no surprise because, as we have already seen, the group
mathematics of superstrings reflects the geometrical properties
of the Divine Image, the cosmic paradigm of
Creation.
SUMMARY
Physicists use the mathematical language
of group theory to describe the symmetries displayed by the four forces
known to act between subatomic particles. Postulating that these
symmetries exist at every point in
spacetime requires the existence of socalled 'gauge bosons.' These
are quantum particles, the exchange of which between particles
generates the force whose symmetry is described by the group in
question. The number of gauge bosons associated with a gauge symmetry
group describing a given kind of force is its dimension. Superstring
theory predicts that the gauge symmetry group that accounts for all the
forces (other than gravity) acting between superstrings must have a
dimension of 496. The groups having this dimension are SO(32) and
E_{8}×E_{8}, where E_{8} is the
socalled 'exceptional group' with the largest
dimension (248). A group is defined by its
roots, the number of which is equal to its dimension. Each root can be
expressed in terms of a set of socalled ‘simple roots,' which are
depicted schematically by the Dynkin diagram of the group. The simple
roots characterizing a group can be represented by finite straight
lines that point in various directions in a mathematical space. The
Dynkin diagram specifies both the lengths of these lines and the angles
between them. Only four angles are possible: 90º, 120º, 135º &
150º. The number of right angles between the lines in this space
representing the simple roots of E_{8} is 21, as
11
is the total
number of angles between the lines representing the simple roots of
E_{7}
(a group whose symmetries are
part of the larger set of symmetries of
E_{8}). The number of
right angles between the lines representing the simple roots of
E_{7}
is
15
, as is the total number of
angles between the simple roots of E_{6} (a group that belongs to
E_{8} as well). Hence, the Godname numbers of
Kether (21) and Chokmah (15)
prescribe the very gauge symmetry
group (and two of its subgroups) that accounts for the
unified force between superstrings.
2. YAH & YAHWEH prescribe 248 &
496 The
dimension 248 of E_{8} and
the dimension 496 of E_{8}×E_{8}
and SO(32) are determined in a
purely arithmetic way by the two Godname numbers of Chokmah:
YAH = 15 and YAHWEH = 26.
It is instructive to
analyse these arithmetic prescriptions in some detail because
their existence is no fortuitous coincidence but, instead,
arises from and reflects the beautiful, geometrical
properties of the inner form of the Tree of
Life.
The sum of the
squares of the first 15 integers is
1240
= 1^{2} + 2^{2} + 3^{2} + ... +
15^{2}.
This is a
triangular array of integers 1–15.
Therefore, 2480 (=2×1240) is
the sum of a 16×16 square array of these integers shown in Figure 1
with a
diagonal of zeros
separating the two triangular arrays. It contains 256
(=4^{4}) integers
comprising a diagonal row of 16 (=4^{2})
zeros and
24
24
24
4^{4}
–
4^{2}
=
240 =
24 24
24
(24 = 1×2×3×4)
24 24
24 24
nonzero integers
1, 2, 3... 15, where
2^{0}
2
^{1}
15
=
2^{3}
2 ^{2}
12
is the number
of YAH. Remarkably, the square representation of the dimension
of E_{8} also encodes as nonzero integers its
number of nonzero roots. The Pythagorean Tetrad prescribes
this arithmetic representation of the superstring
parameter 248.
The identity:
4960 = 2×2480 = 4(1^{2} +
2^{2} + 3^{2} + ... +
15^{2}).
is represented in Figure 2 by a
cross pattée array of integers 1, 2, 3... 15.
The cross is made up of
480 integers, of which
integers form
its boundary. Once again, the number
168
of the Mundane Chakra
of
Figure 2.
Integers 1–15 in a cross pattée array sum to
4960.
13
Figure 3.
Integers 1–15 in a St Andrews cross array sum to
4960.
Malkuth
appears in the context of the representation of the
dimension of the superstring symmetry groups
E_{8}×E_{8} and SO(32) with dimension
496. This cross pattée representation of
4960 — the number of spacetime components of the
496 gauge particles predicted by
superstring theory — encodes the number 480 of nonzero
roots of this group as the number of nonzero integers
summing to 4960. Alternatively, a
31×31
quare array of
integers 1–15
in the form of a St.
Andrews cross represents the number 4960 (Fig. 3).
This demonstrates how the Godname EL with number
value 31 prescribes the number of components
of the gauge fields of superstrings.
The number of YAHWEH defines
the dimension 248 of E_{8}
in the
following way: using the identity
6201 = 1^{2} +
2^{2} + 3^{2} +… +
26^{2},
then
24800 = 4×6200 =
4(2^{2} + 3^{2} +… +
26^{2}).
Figure 4 shows the cross
pattée representation of this number. 1400 integers are
present. The sum of the 296 integers forming the boundary
of the cross is
14
Figure
4. Integers 2–26
in a cross pattée
array sum to 24800.
15
331
331
331
331
331
331
331
331
5246
=
331
331
331
331
331
331
331
331,
where
331 is the 67th prime number,
67 being the number value of
Binah. Observe how, through this 4×4 square array
representation, the Pythagorean Tetrad reveals that the
number of Binah — the Sephirah embodying the most
abstract archetypes of form — defines the shape of an
archetypal pattern of numbers whose sum characterizes
the physics of superstrings! Since 24800 is the sum of
1400 integers, 49600 is the sum of 2800 integers,
where
280
280
280
2800
=
280
280
280
280
280
280
280
,
thus relating the second
perfect number 28 to the third perfect number
496 as well as the number
496 of Malkuth to the number
value 280 of Sandalphon,
its Archangel. All these
representations of 248 and 496 in terms of the two Godname numbers
of Chokmah have a fourfold, rotational symmetry: their
appearance is unaltered by a rotation of 90º. They
illustrate the fundamental importance of the
Pythagorean Tetrad in expressing parameters of the Tree
of Life.
3.
Godnames of Hod & Netzach prescribe 248 &
496
It should not be supposed
that only two of the ten Godname numbers prescribe the
dimensions of the superstring gauge symmetry groups
E_{8}, SO(32)
and E_{8}×E_{8}. The Godnames of
all the Sephiroth participate in this prescription,
their specificity increasing, the lower the position of
the Sephirah in the Tree of Life. As an example, we
shall now consider how the numbers of the Godnames
YAHWEH SABAOTH of Netzach and ELOHIM SABAOTH of Hod
conspire to determine the numbers
248 and 496.
The identity
_{
26}
24800
=
Σ
(2n)
^{2} =
4^{2}
+
6^{2}
+
8^{2}
+ ... +
52^{2}.
^{
n=2}
can be
written as follows in terms of the odd integers
composing each square number in this
summation:
4
^{
2}
= 1
+
3
+
5
+
7.
6
^{
2}
=
1
+
3
+
5
+
7
+
9
+
11.
8
^{
2}
=
1
+
3
+
5
+
7
+
9
+
11
+
13
+
15
.
●
●
52
^{
2}
= 1 +
3
+
5
+
7
+
9
+
11
+
13
+
15
+...+
101
+
103
.
Adding,
24800 = 25(1+3+5+7) + 24(9+11) +
23(13+15)
+ ... +
1(101+103).
The number 24800 is the
sum of a stack of
26
pairs of odd
integers, the base of which comprises 25 ‘1's and
the apex of which is the number
103. Figure 5 shows two
similar
16
Figure
5
17
stacks placed side by
side to represent the number 49600. The reason why
they are depicted as inverted will be given
shortly. The base of the pair of stacks consists of
50 ‘1's. The height and width of the
pair of stacks is prescribed by the numbers of,
respectively, YAHWEH and ELOHIM. The range of
integers is specified by the number 103,
which is the number of SABAOTH ("Hosts"). Each
stack is prescribed by YAHWEH SABAOTH, whilst both
stacks are prescribed by ELOHIM SABAOTH. This is
how these two Godnames define patterns of integers
adding up to 24800 and 49600. Ignoring the factor
of 100, which merely reflects the 10‑dimensional
nature of superstring spacetime, we see that the
number 496 naturally splits up into two
248s, reproducing what mathematicians call
the "direct product" of two
similar E_{8} groups with dimension
248. These Godnames prescribe in an
arithmetic way the E_{8}×E_{8}
heterotic superstring.
The number of yods in
the ntree (the lowest n Trees of Life) with all
their triangles turned into tetractyses is given
by
Y(n) =
50n
+ 30.
The
49tree represents what Theosophists
call the "cosmic physical plane," each Tree mapping one
of its 49 subplanes. It has
Y(49) = 2480 yods. This is the
number of spacetime components of the
248
10dimensional gauge
fields of E_{8}. It can be
pressed as the
sum
_{4}
_{4}
2480
=
Σ
t
_{n}
+
Σ
T
_{n}
,
^{n=0}
^{n=1}
where
4
^{n}
1^{n}
3
^{n} 3
^{n}
2^{n} 2
^{n}
t
_{n}
= 2
^{n} 2
^{n }
2^{n}
and T
_{n}
=
3
^{n}
3^{n} 3
^{n}
1^{n} 1
^{n }
1^{n}
1
^{n}
4
^{n} 4
^{n }
4^{n} 4
^{n}.
It
is remarkable that the number of integers in
each stack summing to 24800 is
_{
3}
700 =
Σ
(t
_{n} +
T_{n})
,
^{
n=1}
which
is the number of yods in ten separate Trees of Life
whose triangles are tetractyses. As well as
illustrating the basic designing role of the
Pythagorean Decad, this shows how yods can denote
things such as numbers when they collectively define
a Tree of Life parameter like the number
248.
The
column of numbers on the righthand side of Figure 5
is the running total of the integers in successive
rows of either stack, addition commencing from the
top row. Three partial sums are multiples of 100.
The first two rows sum to 100, the first four rows
sum to 400 and the first
36
rows add up to18800, after which a further
72 integers
in 16 rows
sum to 6000, making a total of 24800. The Godname
number 36 of Geburah specifies a point in
the summation of rows of integers yielding a
partial sum that is a multiple of 100. Being
specified by the number of a Godname, this stage
has significance visàvis the Tree of Life, as
we now explain. There are 103
stages of descent
of the Lightning Flash from Kether of the 25th
Tree, which, as the 155th
SL, prescribed by the Godname
ADONAI MELEKH of Malkuth. The first stage of
descent reaches Hod of the 26th
Tree, the
153rd SL specified by ELOHIM
SABAOTH with number value
153. By comparing stages of descent
of the Lightning Flash in the
26tree
18
shown on the left in Figure 5
with the integers 1, 3, 5 ...
103,
it will be
seen that, for example, the 9th and 11th
stages of descent occur at the Lower Face
of the 24th Tree, the 13th and
15th stages occur at the
Lower Face of the 23rd Tree, etc. In
other words, the integers in the paired
rows of the stacks are, simply, the
numbers of stages of descent of the
Lightning Flash in successive Trees. The
71st stage of descent, at which the
running sum of the first
36 rows
of
integers is 18800, reaches Chesed of the
eighth tree, which is the
49th SL in the
Cosmic Tree of
Life.
The Godname number of Yesod specifies the
row where the partial sum is a multiple
of 100 and therefore of possible physical
significance. This is why we have
considered an inverted stack of
integers.
The summation of
odd integers given above leading to the number
24800 has a simple interpretation visàvis the
26tree
that is the
counterpart of
26dimensional
spacetime.
Noting that: 1 + 3 = 2×2, 5 + 7 = 2×6, 9 + 11 =
2×10, etc., 24800 can be
written:
24800 = 50×2 +
50×6 + 48×10 + 46×14 +... + 4×98 +
2×102
_{
25}
= 50×2 + Σ 2n(106 – 4n).
^{
n=1}
This is the sum of the
numbers of SLs on the central pillar up to Yesod of each successive
Tree, each number being weighted with the number
of stages of descent of the Lightning Flash from the 155th SL
specified by ADONAI MELEKH to that Yesod. The
additional term 50×2 is t he
product of the number (50) of SLs on the central pillar up to Yesod of the 25th
Tree and the number (2) of stages of descent of
the Lightning Flash from the 155th SL to Yesod of the 26th (not the 25th) Tree. Its difference from the other terms correlates with
the fact that the 26th Tree is
the counterpart of the dimension of time, whereas lower Trees correspond to dimensions of space.
Yesod of the nth tree is the 2nth SL on the central
pillar. The Lightning Flash descends in (4n–3) stages from this SL. So the number of
stages of descent from Kether of the 25th Tree to this SL =
103 – (4n–3) = 106 – 4n. The sum:
_{25}
24700 = Σ
2n(106 – 4n)
^{n=1}
The sum of the (700+700) integers in the two stacks
is
49600
=
10 = 1 + 2 +
3 + 4
100
= 1^{3} + 2^{3} +
3^{3} +
4^{3}
This 10fold array of integers 1–100
is a beautiful illustration of how the Pythagorean Decad and integers 1, 2, 3 & 4
define this superstring number. The sum of the rows of the twentyfive ‘1's and
twentyfive ‘3's = 100 = 1^{3} + 2^{3} + 3^{3} +
4^{3}. The number of
integers 5–103 in the
50 remaining rows of a stack
is
65
65
65
t
_{2} T
_{2}
543
26
650 =
65
65
65
=
=
65
65
65
65
t_{3}
T_{3}
31
50
19
where
65 is the number value of ADONAI, the Godname of Malkuth,
and 543, 26, 50 &
31 are the number
values of the Godnames of the first four Sephiroth, illustrating once more the defining role of
the Tetrad. Their sum is
2470
2470
2470
24700
=
2470
2470
2470
2470
2
470
2470
2470,
where
_{4}
2470 =
Σ (t_{
n} + T_{n})
= 1^{2} +
2^{2} + 3^{2} + ... +
19^{2}.
^{n=1}
The number of integers
5–103 in the
remaining (50+50) rows of both stacks =
1300 = 26×50 = 1^{5} + 2^{5} + 3^{5} +
4^{5}
1^{4}
2^{4}
2^{4}
^{ }
=
3^{4}
3^{4}
3^{4}
4^{4} 4
^{4}
4^{4}
4^{4}
.
This illustrates the
Tetrad Principle, through which archetypal patterns of numbers such as that
in Figure 5 are prescribed by the number 4. Their sum is
_{4}
_{ }
49400 =
(1^{3} + 2^{3} + 3^{3} +
4^{3})
Σ (1^{n} + 2^{n} +
3^{n} + 4^{n}
) .
^{
n=1}
The number of integers on the
edge of each stack is 76, demonstrating how the Godname number of Tiphareth also
defines the number 24800. It is remarkable that the sum of the 152 integers
on the boundary of both stacks is 5456 because this is the sum of the
first 31 triangular numbers. This shows how the Godname EL of
Chesed with number value 31 prescribes the pattern of integers generating the number
24800.
Another way of seeing how
ELOHIM, the Godname of Binah, prescribes the number 24800 and thus the
dimension 248 of the superstring symmetry group
E_{8} is as follows: any square number n^{2} is the
sum of the (n–1)th and (n+1)th triangular numbers because
n^{2} = ½(n–1)n + n(n+1).
Applying this property to the
squares of the 25 integers 4, 6, 8... 52:
4^{2}
= 6 +
10.
6^{2}
=
15
+
21.
8^{2} = 28
+ 36.
●
●
●
52^{2} =
1326 + 1378.
The sum:
24800 = 4^{2} + 6^{2} +
8^{2} + ... + 52^{2}
20
is the sum of the first
50 triangular numbers after 3. Therefore, the number of
ELOHIM prescribes the number 248 as the ‘kernel’ of 24800.
Since 24800 = 496×50, the
crucial dimension
496 of any gauge symmetry group that is free of quantum
anomalies is the arithmetic mean of the first 50 triangular numbers after 3. This is the remarkable way in
which ELOHIM prescribes the number of gauge fields that mediate the unified
superstring interaction.
Finally, the number
155
of ADONAI MELEKH, the Godname of
Malkuth, is related by the Pythagorean Tetrad to the number
248
by
155
= 1+2+3+4 × 248.
4^{2}
In other words, a 4×4
square array of the number value of ADONAI MELEKH generates the 2480
spacetime components of the 10dimensional fields of the
248 gauge bosons of E_{8}. This fact demonstrates the profound Pythagorean
principle whereby the Tetrad and its geometrical symbol — the square —
define the basic, physical properties of
nature.
