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**How the Tetrad & tetractys generate the dimension 248
of the rank-8, exceptional Lie group E _{8}**

We found on page 16 that, when its 595 triangles are tetractyses, the **49**-tree
mapping the cosmic physical plane contains 2480 yods. According to E_{8}×E_{8} heterotic
superstring theory, the unified force between superstrings of ordinary matter is transmitted by the
**248** gauge fields of E_{8}, the rank-8, exceptional Lie group. These gauge fields
exist in 10-dimensional space-time, so that they have 2480 space-time components. Each component can be thought
of being symbolised by a yod in the **49**-tree constructed from tetractyses. *This establishes a remarkable conjunction of group theory, superstring physics, the mystical tradition
of Kabbalah and the teaching of Theosophy that the cosmic physical plane consists of seven planes, each with
seven subplanes*. For the lattermost is mapped by the **49**-tree, namely, the lowest
**49** of the 91 overlapping Trees of Life making up the Cosmic Tree of Life (CTOL). This
shows how EL CHAI, the Godname of Yesod with number value **49** (see here) prescribes the dimension **248** of
E_{8}.

Consider the tetractys arrays t_{n} & T_{n} of the integers 1, 2,
3 & 4 raised to the power n (see diagram). The table lists their values for n = 0, 1, 2, 3 & 4.
Since T_{0} = t_{0} = 10 and T_{n} ≠
t_{n} for n = 1, 2, 3 & 4, there are nine *different* numbers generated by the tetractys
arrays for these first five powers. Writing down the two types of arrays in order of increasing n, their sum
is:

2480 = t_{0} (or T_{0}) +
(t_{1} + t_{2}) + (t_{3} + t_{4} + T_{1} + T_{2} + T_{3}
+ T_{4})

= 10 + 70 + 2400 = **80** +
2400.

Together with zero, the four Pythagorean
integers 1, 2, 3 & 4 have the amazing property of generating both the root composition of E_{8} and
the population of the **49**-tree. t_{0} or T_{0} (= 10) denotes the 10 yods added when the
Tree of Life with 70 yods becomes the 1-tree, (t_{0} + t_{1}) (= 30) is the number of yods in
its Lower Face (the kite-shaped quadrilateral with Tiphareth, Netzach, Hod & Malkuth at its corners)
and t_{2} (= **50**) is the number of yods in the rest of the 1-tree outside the Lower Face. Hence,
(t_{0} + t_{1} + t_{2}) (= **80**) is the yod population of the 1-tree (see here) and (t_{3} + t_{4} +
T_{1} + T_{2} + T_{3} + T_{4}) (= 2400) is the population of the
**48** Trees of the **49**-tree above the 1-tree. The **80** yods denote the
**80** space-time components of the eight gauge fields associated with the eight simple roots of
E_{8} and the 2400 yods denote the 2400 space-time components of its 240 roots. In other words, the
distinction between the lowest Tree of Life and the **48** Trees in the **49**-tree
above it corresponds to the difference between the so-called eight "zero roots" and the 240 "non-zero roots" of
E_{8}, as well to the difference between the first three tetractys arrays (t_{0},
t_{1}, t_{2}) and the last six (t_{3}, t_{4}, T_{1}, T_{2},
T_{3}, T_{4}). As t_{0} is identical to T_{0}, it makes no difference to
the sum (2480) of the nine tetractyses which one of these two tetractyses should be included in them.
However, given (see here) that the yod population of the Tree of
Life = 70 = t_{1} + t_{2}, the more *natural* choice for the difference of 10 yods
between the Tree of Life and the 1-tree is t_{0} because, then, the nine tetractyses divide into
the first five of one type: t_{0}, t_{1}, t_{2}, t_{3} & t_{4}, and
the last four of the other type: T_{1}, T_{2}, T_{3} & T_{4}, whereas
choosing T_{0} would mix the two types in the three tetractyses adding to
**80**.

Embodied, therefore, in the Tree of Life map
of the cosmic physical plane is the number of space-time components of the **248** gauge fields of
E_{8}. Here is how superstring physics with unbroken E_{8} symmetry is related to the
mathematical description of the levels of physical and superphysical reality. If the sceptic wants to argue that
the **49**-tree embodies the number **248** by coincidence, then he or she has to
accept that it must also be just a matter of chance that the nine possible tetractys arrays of the powers of 1,
2, 3 & 4 add up to 2480 and that the values 10, 20 & **50** of the first three tetractyses
are, respectively, the 10 yod difference between the Tree of Life and the 1-tree, the 20 yods in the tetrahedron
of the 1-tree and the **50** remaining yods of the 1-tree (the number of yods in n Trees of Life =
**50**n + 20 and the number of yods in the n-tree = **50**n +30, where 20 =
t_{1}, 30 = t_{0} + t_{1} and **50** = t_{2}). That *all*
these mathematical counterparts could exist by coincidence is highly implausible. A more reasonable view is
that we are encountering here yet another remarkable example of the harmony of Pythagorean geometry and number
in the context of sacred geometry and its manifestation in group theory in the superstring gauge symmetry group
E_{8}. The significance of the **49**-tree is that it maps the cosmic physical plane, so that it corresponds on a cosmic level to the last
Sephirah of Construction, namely, Malkuth. Hence, it should not come as a surprise that this section of CTOL
should embody the number of gauge fields that transmit the unified force between superstrings of ordinary
matter making up the physical universe. We say "ordinary" because, according to E_{8}×E_{8}
heterotic superstring theory, this type of matter is a singlet state of the *second* E_{8} group,
which is the invariance group for the interactions between shadow matter superstrings mediated by another set of
**248** spin-1 gauge fields with 2480 space-time components. The total number of such components of
the **496** gauge fields is 4960. This number is not, of course, represented in the yod
population of CTOL, which has 4570 yods, because its context is the *physical* universe, which means that
it should be expected to be embodied in only that section of CTOL that bears a correspondence to Malkuth, the
last Sephirah — namely, the **49**-tree. However, it is readily shown* that the number of yods in n
overlapping Trees of Life that are either on the Pillar of Equilibrium or directly in line with it = 10n + 6, so
that **49** overlapping Trees have **496** such yods, where **496** is
the number value of Malkuth:

The **49**-tree therefore
embodies the dimension **248** of E_{8} and **49** overlapping Trees embody
the dimension **496** of E_{8}×E_{8}.** The appearance of these two
group-theoretical numbers characterising one of the superstring theories in the Tree of Life mapping of the
cosmic physical plane is very remarkable and cannot, plausibly, be due to coincidence. Instead, it further
demonstrates the Pythagorean geometrical basis of superstring theory itself. It also illustrates the
hermetic axiom "As above, so below" because, as the representation of the
cosmic physical plane, the **49**-tree is the counterpart of the 7-tree mapping the physical plane,
that is, the 10-dimensional space-time of superstrings.

**E _{8} embodied in the 5
Platonic solids**Discussed here is the geometrical composition of the five Platonic solids with
their faces divided into their sectors and with the sides of the latter regarded as also sides of internal,
Type A triangles with the centre of the polyhedron as one of their corners. It is found
that, excluding their vertices, 2480 points, lines & basic triangles surround the axes of the five
regular polyhedra passing through their centre and any two vertices diametrically opposite it. Given the

* Proof: The 1-tree (shown here) has 10 yods either on or aligned with the central Pillar of
Equilibrium inside the lower kite shape with Tiphareth, Netzach, Hod & Malkuth at its corners. The upper
kite shape in the 1-tree with Kether, Chokmah, Binah & Tiphareth at its corners also has 10 such yods above
its lowest point. The n-tree has (n+1) kites joined end-to-end, each with 10 yods either on this
pillar or aligned with it. The number of yods either on or aligned with the central pillar of the n-tree ≡ N(n)
= 10n +10. Comparing #39 & #40, the upper kite in the Tree of Life has 4 fewer yods of this
type than the upper kite in the 1-tree. The number of yods either on or aligned with the central pillar of
n overlapping Trees of Life ≡ Ň(n) = 10n + 10 − 4 = 10n + 6. Hence, **49** overlapping
Trees have (Ň(**49**) = **496**) yods that are either on the Pillar of
Equilibrium or aligned with it in the line of sight.

** **49** overlapping Trees also
embody in their geometry the superstring structural parameter 1680. The (12n+4) triangles in n overlapping Trees
have (6n+4) corners and (16n+6) sides, i.e., (34n+14) geometrical elements. Therefore, **49**
overlapping Trees are composed of (34×**49** + 14 = 1680) geometrical
elements. *The Tree of Life map of the cosmic physical plane embodies
both* *the structural parameter 1680* *and the dynamical parameters 2480 & 496 of
E _{8}×E_{8} heterotic superstrings, as manifested in the UPA. The sceptic should ask
himself: what is the likelihood of such a triple conjunction of numbers relevant to superstring theory
occurring by chance? Why should the first number, derived by C.W. Leadbeater by remote viewing UPAs and
counting the 1680 turns in each of its whorls, turn up in the same context alongside two numbers connected
with superstring theory if the UPA was NOT a real superstring but merely the product of his imagination?!
Here is convincing evidence that the UPA is, indeed, a superstring, as interpreted by the author in his book
ESP of Quarks & Superstrings — in particular, the
E_{8}×E_{8} heterotic superstring.*

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