Article 52

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 2-d 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 filled-in 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:

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 means that there are binomial coefficent3

=½(N-1) (N-2) right angles and binomial coefficient4

= (N-1) angles which are either 120º, 135º, 150º or mixtures thereof [N.B. binomial coefficient1

]. Tabulated below is the number of angles enclosed by all pairs of simple roots defining all possible Lie algebras:

numbers of angles between pairs of simple roots

The Godname number 21 of Kether is the number of pairs of orthogonal roots of the superstring symmetry group E₈ and the number of angles between the seven simple roots of E₇. The Godname number 15 of Chokmah is the number of pairs of orthogonal roots of E₇ and the number of angles between the six simple roots of E₆, a subgroup of E₈ favoured by many string theorists as the probable product of symmetry breaking of E₈.

Turning to the non-exceptional groups:
1. for n = 6, number of angles between the simple roots of SU₇, SO₁₂, SO₁₃ & Sp₁₂ = 15;
2. for n = 7, number of right angles between the simple roots of SU₈, SO₁₅, SO₁₄ & Sp₁₄ = 15;
3. for n = 8, number of right angles between the simple roots of SU₉, SO₁₇, SO₁₆ & Sp₁₆ = 21.

Since the superstring symmetry group is E₈, any other groups picked out by the Godnames of Kether or Chokmah must be subgroups of E₈. These are: SU₉, SO₁₆ (n = 8), SU₈ (n = 7) & SU₇, SO₁₃, SO₁₂ (n = 6).

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:

dimensions of Lie algebras

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:

n = 16 determines the anomaly-free group SO₃₂ 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. B₃, (SO₇) and C₃ (Sp₆) have dimension 21, A₃ (SU₄) and D₃ (SO₆) have dimension 15, B₄ (SO₉) and C₄ (Sp₈) have dimension 36 and D₉ (SO₁₈) 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 B₁₀ (SO₂₁) and C₁₀ (Sp₂₀) have the dimension:

                                                                                                                              21

                                                                                                                           21    21

                                                                                                     d_G =       21 21 21
                                                                                                                    21   21   21 21

and SU₁₉ 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 E₈ are SO₇ and Sp₆ (with dimension 21), SU₄ and SO₆ (with dimension 15), and SO₉ and Sp₈ (with dimension 36). Since

E₈⊃SO₁₆⊃SO₉⊃SO₇⊃SU₄, E₈⊃E₆⊃Sp₆⊃SU₂×Sp₆, and SU₄≈SO₆,

the Godname numbers 21, 15 and 36 define dimensions of subgroups of E₈.

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:

Dimensions of Lie groups defined by Godname numbers

Godnames numbers of four Sephiroth each define four, non-exceptional Lie algebras:

SO₁₄ (91) SU₈ (63)

n = 7:

= 21
Sp₁₄ (105) SO ₁₃ (78)

SO₁₂ (66) SU₇ (48)

n = 6:

= 15

Sp₁₂ (78) SO ₁₃ (78) (number in brackets is dimension of Lie algebra)

SO₁₈(153) SU₁₀ (99)

n = 9:

= 36

Sp₁₈ (171) SO₁₉ (171)

SO₃₆ (630) SU₁₉ (360)

n = 18:

= 153
Sp₃₆ (666) SO₃₇ (666)

21 also defines E₇ and 15 defines E₆. The Lie algebras corresponding to 153 have rank 18, which exceeds the rank 8 of E₈ and the rank 16 of E₈×E₈ and SO₃₂. 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 E₈, all of which are its subgroups.

For what Lie algebras of rank N is the number of right angles between pairs of simple roots binomial coefficient8

= N_GOD, and what are their dimensions? Below are tabulated values of n

determining N_GOD and the corresponding dimension d_G of the Lie algebra:

dimensions of Lie algebras defined by Godname numbers

The Godname numbers of four Sephiroth (the same as above) define four non-exceptional Lie algebras:

SO₁₆(120) SU₉( 80)

n = 8:

= 21

Sp₁₆(136) SO₁₇(136)

SO₁₄(91) SU₈(63)

n = 7:

= 15

Sp₁₄(105) SO₁₅(105) (number in brackets is dimension of Lie algebra)

SO₂₀(190) SU₁₁(120)

n = 10:

= 36

Sp₂₀(210) SO₂₁ (210)

SO₃₈(703) SU₂₀(399)

n = 19:

= 153

Sp₃₈(741) SO₃₉(741)

Comments
1. 21 also defines E₈ and 15 defines E₇, as found earlier. SO₁₇ is excluded because SO₁₆ is a maximal subgroup of E₈. Since E₈ contains the others as subgroups, the group selected by 21 with the largest dimension is E₈.
2. 15 selects the rank-7 groups SU₈, SO₁₅, Sp₁₄, SO₁₄ and E₇.
3. 36 selects only rank-10 groups, which are forbidden, having higher rank than E₈.

Similarly, the rank-19 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 E₈ 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 E₆, E₇ and E₈. 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 E₇. The value 10 of the letter Y is the number of right angles between the root vectors of E₆ and the value 5 of letter H is the extra number of right angles between the simple root vectors of E₇.
2. The number 21 of EHYEH (Hebrew: AHIH) is the number of right angles between the eight simple root vectors of E₈. 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 E₈, YH specifies E₇ and I (or Y) specifies E₆.

According to the Dynkin diagram of E₈, 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

vectors of E₈ 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 E₈ 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 E₈ 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 space-time requires the existence of so-called '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 E₈×E₈, where E₈ is the so-called '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 so-called ‘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 E₈ is 21, as

is the total number of angles between the lines representing the simple roots of E₇ (a group whose symmetries are part of the larger set of symmetries of E₈). The number of right angles between the lines representing the simple roots of E₇ is 15 , as is the total number of angles between the simple roots of E₆ (a group that belongs to E₈ 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 E₈ and the dimension 496 of E₈×E₈ 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 = 1² + 2² + 3² + ... + 15².

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 (=4⁴) integers comprising a diagonal row of 16 (=4²) zeros and

                                                                                                                  24
                                                                                                               24 24
                                                                        4⁴ – 4² = 240 =         24 24 24                                                                             (24 = 1×2×3×4)
                                                                                                         24 24 24 24

non-zero integers 1, 2, 3... 15, where

2⁰ 2 ¹

15 =

2³ 2 ²

is the number of YAH. Remarkably, the square representation of the dimension of E₈ also encodes as non-zero integers its number of non-zero roots. The Pythagorean Tetrad prescribes this arithmetic representation of the superstring parameter 248.

The identity:

4960 = 2×2480 = 4(1² + 2² + 3² + ... + 15²).

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

cross pattee representation of 168

integers form its boundary. Once again, the number 168 of the Mundane Chakra of

4960 =

Figure 2. Integers 1–15 in a cross pattée array sum to 4960.

4960 =

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 E₈×E₈ and SO(32) with dimension 496. This cross pattée representation of 4960 — the number of space-time components of the 496 gauge particles predicted by superstring theory — encodes the number 480 of non-zero roots of this group as the number of non-zero integers summing to 4960. Alternatively, a 31×31 square 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 E₈ in the following way: using the identity

6201 = 1² + 2² + 3² +… + 26²,

then

24800 = 4×6200 = 4(2² + 3² +… + 26²).

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

24800 =

Cross pattee representation of 24800

Figure 4. Integers 2–26 in a cross pattée array sum to 24800.

                                                                                 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 four-fold, 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 E₈, SO(32) and E₈×E₈. 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 = 4² + 6² + 8² + ... + 52².
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

representation of 24800 & 49600

Figure 5

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 space-time, we see that the number 496 naturally splits up into two 248s, reproducing what mathematicians call the "direct product" of two similar E₈ groups with dimension 248. These Godnames prescribe in an arithmetic way the E₈×E₈ heterotic superstring.

The number of yods in the n-tree (the lowest n Trees of Life) with all their triangles turned into tetractyses is given by

Y(n) = 50n + 30.

The 49-tree 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 space-time components of the 248 10-dimensional gauge fields of E₈. 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
                                              1ⁿ  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 right-hand 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 vis-à-vis 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 26-tree

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 vis-à-vis the 26-tree that is the counterpart of 26-dimensional space-time. 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