ARTICLE 54
by
Stephen M. Phillips
Flat 4, Oakwood
House, 117119 West Hill Road. Bournemouth. Dorset BH2 5PH. England.
Website: http://smphillips.mysite.com
Abstract
The Tree of Life known to Kabbalah is but the outer form
of the true cosmic blueprint. Its geometry generates an inner form that
consists of two sets of seven enfolded, regular polygons, one set being the
mirror image of the other. This article analyzes the properties of the
combined outer and inner forms of the Tree of Life when the triangles in the
former are either simple triangles, Type A or Type B and the polygons in the
latter are either Type A or Type B. The geometrical and yod compositions of
these combinations are numbers found to characterize other sacred
geometries, confirming their equivalence to the Tree of Life demonstrated in
previous articles. For example, 384 geometrical elements are outside the
root edge when the outer Tree of Life has simple triangles and the inner
Tree has Type A polygons. They consist of 192 corners & triangles and
192 sides. This pattern exists in other sacred geometries, e.g., each half
of the 3dimensional Sri Yantra with its 43 triangles turned into
tetractyses has 192 yods. When triangles in the outer Tree are tetractyses,
the combined Trees contain 496 hexagonal yods. This number is the dimension
of the two Lie groups that define the unified superstring interaction. It
demonstrates the Tree of Life basis of superstring theory. Furthermore, the
496 hexagonal yods consist of: 1. the 248 hexagonal yods either in the outer
Tree, at the centres of the tetractyses in the first (6+6) enfolded polygons
or in the two dodecagons, and 2. the 248 hexagonal yods surrounding the
centres of tetractyses in the first (6+6) enfolded polygons. This
corresponds to the 248:248 division of roots in
E_{8}×E_{8}′, one of these superstring symmetry
groups. 384 hexagonal yods line the sides of tetractyses as 192 pairs. This
192:192 division is characteristic of sacred geometries. With Type A
triangles in the outer Tree and Type A polygons in the inner Tree, 480
geometrical elements are outside the root edge. They consist of 240 corners
& triangles and 240 sides. This is analogous to the 480 nonzero roots
of E_{8}×E_{8}' (one of the two Lie groups),
which comprise 240 nonzero roots of E_{8} and 240 nonzero
roots of E_{8}'. The number of hexagonal yods in the combined Trees
is that in the seven separate Type B polygons. The gematria number 620 of
Kether is the number of yods outside the root edge that are hexagonal yods
in both combined Trees. The tetractyses in the separate outer & inner
Trees with Type B triangles and polygons have 248 corners, where 248 is the
dimension of E_{8}. There are 620 yods in the outer Tree other than
corners and centres of its Type B triangles. 496 yods line its tetractyses
that are not Sephiroth shared with its inner form. When their triangles and
polygons are Type B, there are 240 corners outside the root edge in the
combined outer & inner Trees of Life. ADONAI MELEKH, the Godname of
Malkuth, prescribes the number of unshared yods outside the root edge that
line tetractyses but are not located at the positions of the ten Sephiroth.
With Type A triangles in the outer Tree and Type B polygons in the inner
Tree, there are 192 corners outside the root edge. With Type B triangles in
the outer Tree and Type A polygons in the inner Tree, there are 384 corners
& triangles and 384 sides outside the root edge. Different combinations
of these transformed Trees embody numbers characterising other sacred
geometries. It is powerful evidence that the outer & inner Trees of Life
constitute sacred geometry when combined as well as when separate.

1
Table 1. Gematria number values of the ten Sephiroth in the four Worlds.
(Numbers in this table referred to in the article will be written in
boldface).
2
1. Shared geometrical elements & yods
The outer Tree of Life
consists of 16 triangles arranged in space in such a way that they have 10 corners and 22
sides. The inner Tree of Life comprises two sets of seven regular polygons: triangle,
square, pentagon, hexagon, octagon, decagon & dodecagon. One set is the mirror image of
the other, so that they lie in a single plane — that containing the Pillar of Mercy and the
Pillar of Judgement. This means that the two vertical sides of sectors of the two hexagons
coincide with the sides of triangles in the outer Tree that form the two outer pillars. It
also means that the top and bottom corners of these two hexagons coincide with corners of
triangles located at the positions of Chokmah, Netzach, Binah & Hod, whilst the outer
corner of each triangle belonging to a set of polygons coincides with the corner of
triangles that is located at Chesed and at Geburah, Hence, six green corners and four green
sides, i.e., 10 corners & sides, of the outer Tree are shared with its inner form (Fig.
1). No triangles in the outer Tree are sectors of the 14 polygons, which number 94. This
leaves it with 16 unshared triangles with four unshared corners and 18 unshared sides, i.e.,
38 geometrical elements. The sectors have 80 corners (six shared, 74 unshared) and
175 sides (four shared, 171 unshared). Superposed, the outer & inner Trees of Life
consist of 84 corners (six shared, 78 unshared), 193 sides (four shared, 189 unshared) &
110 unshared triangles. The 387 geometrical elements comprise 10 shared elements and 377
unshared ones.
Figure 1. The outer & inner
Trees of Life have 10 shared geometrical elements and 16 shared yods.
Transformed into tetractyses, the 16 triangles
of the outer Tree contain 70 yods (10 corners, 60 hexagonal yods) and the 94 sectors of the
14 enfolded polygons contain 524 yods (80 corners, 444 hexagonal
yods). There are eight hexagonal yods on the four vertical sides of the sectors of the two
hexagons that are shared with the outer Tree of Life because they lie on its side pillars.
Although the two hexagonal yods on the ChesedGeburah Path are also the centres of the
triangles belonging to the two sets of enfolded polygons, they still count as hexagonal yods
in determining the hexagonal yod population of the combined Trees of Life. Although they are
shared as yods along with eight hexagonal yods in the hexagons, these two yods are not
shared as hexagonal
yods. This is important to realize when we
consider later the hexagonal yod population of the combined Trees. As well as these ten
yods shared with polygons, three corners of tetractyses on each side pillar coincide with
either the top and bottom of the hexagons or with the outer corner of the triangle
belonging to each set of polygons. Hence, the outer & inner Trees share 16 yods. This
leaves 54 green unshared yods in the former and 508 unshared yods in the latter (254 blue
yods in the lefthand set of polygons in Figure 1 and 254 red yods in the righthand
set). Superposed, the outer & inner Trees have 578 yods (16 shared, 562
unshared).
We shall now analyze the properties of the
combined Trees of Life when the outer one (henceforth to be called “OTL”) is composed of
either simple triangles, Type A triangles or Type B triangles and when the
3
inner Tree of Life (to be
called “ITL”) consists of either Type A polygons or Type B polygons.
2. Simple triangles in OTL & Type A
polygons in ITL
Geometrical composition
As stated earlier, OTL has 16 triangles with
10 corners & 22 sides, i.e., 48 geometrical elements,
where 48 is the number value of Kokab, the Mundane Chakra of Hod.
ITL has 94 triangular sectors with 80 corners & 175 sides,
i.e., it has 349 geometrical elements. Combined, they comprise 387 geometrical elements made
up of 84 corners, where
84 = 1^{2} +
3^{2} + 5^{2} +
7^{2},
193 sides & 110
triangles. Outside the root edge are 384 geometrical elements: 192 corners & triangles
and 192 sides. This 192:192 division manifests in ITL as the 192 corners & triangles in the 14 separate polygons and as
their 192 sides (alternatively, as
Figure 2. The 192:192
pattern in various sacred geometries.
the 192 geometrical
elements in each set of seven polygons). Figure 2 shows how this division appears in other
sacred geometries.
Figure 3. The combined outer & inner Trees of Life have 496
hexagonal yods. This is the dimension of the superstring symmetry groups
E_{8}×E_{8} and SO(32).
4
Yod
composition
Combined, OTL & ITL contain 578 yods (see
Fig. 1). Sixteen of them are shared and 562 are unshared. OTL has 60 hexagonal yods, eight
of which are hexagonal yods in the pair of hexagons. ITL has 444 hexagonal yods. Remembering
our earlier comment that the two hexagonal yods on the ChesedGeburah Path still count as
hexagonal yods even though they are corners of tetractyses in the two triangles belonging to
ITL, the number of hexagonal yods in the combined OTL & ITL = 60 + 444 − 8
= 496 (Fig. 3). This is both the number value of Malkuth and the dimension of
E_{8}×E_{8} & SO(32) — the two symmetry groups with dimension
496 that allow interactions between superstrings that are free of
quantum anomalies. What more explicit proof could there be that the Tree of Life encodes
superstring theory — or, rather, Mtheory, the socalled 'theory of everything' that
contains the five versions of superstrings as approximations?
Figure 4.
Outer Tree of Life
Turned into tetractyses, the 16 triangles
contain 60 hexagonal yods consisting of the 16 brown hexagonal yods at their centres and the
44 hexagonal yods on their 22 sides (Fig. 4). As four of these sides are sides of the
sectors of the two hexagons belonging to the inner Tree of Life, its outer form has
(22−4=18) sides with (18×2= 36) green hexagonal yods that are unshared with it as hexagonal yods and
eight white hexagonal yods that are shared. Hence, the outer Tree has
(16+36=52) unshared hexagonal yods. 52 is
the 26
th even integer, showing how YAHWEH with number
value 26 determines the population of intrinsic hexagonal yods in the
outer Tree. The two hexagonal yods on the ChesedGeburah Path are the centres of the two
triangles belonging to the (7+7) enfolded polygons. This means that there
are 50 hexagonal yods in the outer Tree that are hexagonal yods from the
perspective of both Trees, showing how ELOHIM, the Godname of Binah with number
value 50, prescribes this property.
Inner Tree of Life
Turned into tetractyses, the (47+47) sectors
of the (7+7) enfolded polygons have 524 yods. They consist
of 80 corners and 444 hexagonal yods,
where 80 is the number of Yesod. The latter comprise (47+47=94) hexagonal
yods at the centres of tetractyses and (444−94=350) hexagonal yods that line their sides.
Each dodecagon has 48 hexagonal yods lining sides of tetractyses. The 46 yods outside the
root edge consist of 22 pink hexagonal yods on its sides and 24 red hexagonal yods on
internal sides of its 12 tetractyses. Hence, 46 = 22 + 24 and 350 = 46 + 2 + 46 + 256, where
"2" denotes the two purple hexagonal yods on the root edge, there being 256 hexagonal yods
outside it that line tetractyses in the first (6+6) enfolded polygons. Each set of the first
six enfolded polygons has 35 sectors and the dodecagon has 12 sectors. Seventy of the 94
hexagonal yods at centres of tetractyses belong to the first (6+6) enfolded polygons; they
are coloured purple. The 24 remaining hexagonal yods belong to the two dodecagons; they are
coloured blue. Therefore,
444 = 350 + 94
= 46 + 2 + 46 + 256 + 70 +
24
= 22 + 24 + 2 + 22 + 24 +
256 + 70 + 24
= (70+2) + 24 + 22 + 24 + 22
+ 24 + 256.
Combined outer & inner Trees of
Life When they are combined, the eight hexagonal yods on the side pillars of
the outer Tree coincide with some of the hexagonal yods in the two hexagons. It was found
above that there are 52 hexagonal yods in the outer Tree that are unshared as such with the
inner Tree. The number of hexagonal yods in the combined Trees = 52 + 444
= 496. This is the dimension of both E_{8}×E_{8} & SO(32), the two
symmetry groups that govern the anomalyfree interactions between heterotic superstrings.
This property constitutes amazing evidence for the Tree of Life basis of superstring theory
because, although it might be coincidental, only the most extreme debunker would be willing to
consider this a plausible possibility, especially given the additional properties of this
number embodied by the combined Trees of Life to be discussed next.
Collecting together the various
results discussed above:
496 = 52 + 444 = (16+36) + (70+2) + 24 + 22 + 24 + 22 + 24 + 256.
The 256 hexagonal yods lining
tetractyses in the first (6+6) enfolded polygons comprise eight black hexagonal yods on the
four sides of the pair of triangles outside the root edge
and 248 khaki hexagonal yods (let us write this set
as 248′).
Therefore,
496 = (16+36) + (70+2) + 24 + 22 + 24 + 22 + 24 + 8 + 248′
= 248 + 248′,
where
248 = (16+36) + (70+2) + 24 + 22 + 24 + 22 + 8 = 8 + (70+2) +
(36+24+24) + (16+24+22+22)
= 8
+ 72
+ 84 + 84 = 8
+ 72
+ 168.
We find that the hexagonal yod
population 496 naturally splits into 248 & 248′. This is analogous to how 496 is the sum of the dimension 248 of E_{8} and the
dimension 248′ of E_{8}′. That the geometry of the combined Trees allows such
a natural splitting of this number into two equal numbers is evidence that it
is not by chance that they have 496 hexagonal yods. This conclusion is confirmed by the fact that the
number 248 further divides naturally into the number 8 denoting the eight
simple roots of E_{8}, the number 72 denoting the 72 roots of E_{6}, an exceptional subgroup of
E_{8}, and
the number 168 denoting the 168 remaining roots of E_{8}. 168 further divides into two equal numbers 84 — a bifurcation that is
displayed in all sacred geometries/holistic systems, as is
the 72:168 division. What
we are encountering here is not the highly improbable result of miraculous chance but the
manifestation of the group mathematics of heterotic E_{8}×E_{8}^{′}
superstring theory in the amazing sacred geometry of the outer & inner Trees of
Life.
Let us next inspect whence the
contributions to the 248 roots of E_{8} originate:

8 simple roots → 8 hexagonal yods on sides
of the two triangles that are part of the 14 polygons making up the inner Tree
of Life;

72 roots of E_{6} → two
hexagonal yods on the root edge + 70 hexagonal yods at the centres of the 70
tetractyses in the first (6+6) enfolded polygons;

168
roots → 52 hexagonal yods in the
outer Tree of Life unshared with the inner Tree + 116 hexagonal yods in the
two dodecagons outside the root edge.
Notice that the 12 polygons
embodying the number 72 constitute a holistic subset of the 14 polygons making up the inner
Tree of Life, being characterized by the same parameters as those embodied in the complete
set. Notice also that the hexagonal yods symbolizing the 168 roots come from the outer Tree (16+36) and from the dodecagon (24+22+24+22+24), which, too, is a holistic
object. All the hexagonal yods symbolizing the second group E_{8}′ come from the hexagonal yods
in the first (6+6) enfolded polygons that surround the hexagonal yods at the centres of the
70 tetractys sectors of these polygons. In terms of the formal correspondence between yods
and Sephiroth (see here), such yods correspond to the six Sephiroth of
Construction above Malkuth in the outer Tree. Hence, the encoding of
E_{8}×E_{8}′ in the combined Trees of Life suggests the following
association:
E_{8}
↔
Malkuth;
E_{8}
′ ↔ 6 Sephiroth of Construction
above
Malkuth.
It would indicate that the
symmetry group E_{8} applies to the forces
operating in the familiar, physical universe of ordinary matter, namely, Malkuth, and that
E_{8}′ applies to forces operating in a more
subtle (although still physical) universe. This is what superstring theorists have called
the parallel universe of invisible "shadow
matter" that is separated from the ordinary universe by a narrow gap extending along the
seventh compactified dimension predicted by supergravity theories. This type of matter is
sometimes confused by popular science writers with "mirror matter" — hypothetical particles
that are the mirror images of the known subatomic particles. Although shadow matter is
similar to this parityreversed form of matter in not strongly interacting with ordinary
matter, it is otherwise completely different, quite apart from being a rigorous,
mathematical prediction of E_{8}×E_{8}′ heterotic
superstring theory and not just a theoretical speculation.
OTL has 16 hexagonal yods at the centres of
its 16 tetractyses and ITL has 94 hexagonal yods at the centres of its 94 tetractyses. The
number of hexagonal yods lining the 193 sides of the (16+94=110) tetractyses in the combined
Trees = 496 – 110 = 386. Outside the root edge of the (7+7) enfolded polygons
are 384 hexagonal yods lining the 192 sides of the 110 tetractyses as 192 pairs. This number
is a parameter of holistic systems, and its 192:192 division is characteristic of them,
e.g.,

the 64 hexagrams used
in the ancient Chinese I Ching system of divination have 384 lines & broken
lines;

each half of the 3dimensional Sri Yantra whose 43 triangles are
tetractyses contains 192 yods;

192 yods are associated with, and intrinsic to, each set of the
first six enfolded polygons;

the 384 yods up to the top of the 7tree divide into two groups of
192 yods;

192 yods line the 56 hyperbolic triangles in the {3,7} tessellation
of the 3torus and its version turned inside out when they are regarded as
deformed tetractyses.
(See Fig. 2). As the combined Trees contain 578
yods, there are (578–386=192) corners of tetractyses and hexagonal yods at their
centres.
Figure 5. The combined outer & inner Trees of Life constitute a
holistic system because the number of unshared yods lining their boundaries is the number of
yods in the seven enfolded polygons.
Of the 578 yods in the
combined OTL & ITL, 110 hexagonal yods are at centres of tetractyses, leaving 468 yods that
line their 193 sides. They are made up of 54 yods in OTL ((54–16=38) unshared with ITL) and 414
yods in ITL. The counterpart in the latter of the 54 yods on the boundaries of the triangles is
the set of 236 yods that line the sides of the 14 enfolded polygons. They include six yods at
the positions of the Sephiroth on the two side pillars of OTL that also belong to the set of 54
yods. Therefore, the number of yods lining triangles in OTL and polygons in ITL = 54 – 6 + 236
= 284. The number of such boundary
5
yods outside the root edge = 284 – 4
= 280.
This is the number value of Sandalphon, the Archangel of
Malkuth. They include the 16 yods shared between OTL and ITL. There are
(280–16=264) unshared, boundary yods outside the root edge. This is the number
of yods in the seven enfolded polygons (Fig. 5). They include the four remaining Sephiroth
on the central pillar: Kether, Tiphareth, Yesod & Malkuth. There are 260
(=26×10) unshared boundary yods other
than Sephiroth in the combined OTL & ITL. This is the number of yods in the seven
enfolded polygons outside their root edge. It demonstrates how YAHWEH, the Godname of
Chokmah with number value 26, prescribes
the combined Trees as well as ITL.
3. Type A triangles in OTL & Type A
polygons in ITL
Let us, next, consider the case where both the
triangles of OTL and the polygons of ITL are Type A.
Geometrical
composition
Dividing each of the 16 triangles in OTL into
their three sectors adds 16 corners and (3×16=48) sides, so that there are now 48
triangles with (10+16=26) corners (six shared, 20 unshared) and (22+48=70) sides
(four shared, 66 unshared), i.e., 144 geometrical elements (10 shared, 134 unshared). The 94
sectors of the ITL have 80 corners (six shared, 74 unshared) and 175 sides (four shared,
171 unshared), i.e., 349 geometrical elements (10 shared, 339 unshared). The combined Trees of
Life have 100 corners (six
Figure 6.
The 7 separate polygons have 192 hexagonal yods on sides of 48 tetractyses and
48 hexagonal yods at their centres.
shared, 94 unshared), 241
sides (four shared, 237 unshared) and 142 triangles (all unshared), i.e., 483 geometrical
elements (10 shared, 473 unshared).
Analogy with E_{8}×E_{8}
Outside the root edge in the combined OTL
& ITL are 480 geometrical elements consisting of (98+142=240) corners & triangles and
240 sides. Compare this property with the fact that the 480 roots of the heterotic superstring
symmetry group E_{8}×E_{8} consist of 240 roots of the first group
E_{8} and 240 roots of the second group E_{8}. Replacing the 16 triangles in
OTL by Type A triangles increases the 192 corners & triangles outside the root edge to 240
(16 extra corners & 32 extra triangles) and the 192 sides outside the root edge to 240
sides (48 extra sides). The counterpart of this 192→240 increase in the seven separate polygons is the fact
that their 48 tetractys sectors have 192 hexagonal yods on their sides and 240 hexagonal
yods in total (Fig. 6).
Figure 7. The 7 separate Type B
polygons have 624 hexagonal yods.
6
Figure 8. ADONAI prescribes the 7 enfolded Type B polygons.
7
Yod composition
Inside each of the 16 Type A triangles in OTL
are one corner and nine hexagonal yods. We have seen that the combined Trees have 578 yods
when the triangles in OTL are tetractyses. Now, nine yods are added per triangle, i.e., the
yod population of the combined Trees = 578 + 16×9 = 722. The number of hexagonal
yods in OTL = 22×2 + 16×9 = 188. ITL has 444 hexagonal yods, of which eight are shared with
OTL. The number of hexagonal yods in the combined Trees of Life = 188 + 444 − 8 = 624. This
is the number of hexagonal yods in the seven separate Type B polygons (Fig. 7).
The 16 triangles in OTL have
(10+16=26) corners of 48 tetractyses; the 94 tetractyses in ITL have 80
corners, of which six corners are shared as such and two corners (centres of the two triangles)
coincide with hexagonal yods on the ChesedGeburah Path. Combined, OTL & ITL have
(26+80−6−2=98) corners. Therefore, they have (624+98=722) yods, i.e., 718 yods
other than the four yods in the root edge. Of these, 68 are corners of the (7+7) enfolded
polygons. This leaves (718−68=650=65×10) yods outside the root edge that are not corners
of polygons. Alternatively, the combined Trees have 650 yods other than corners of polygons and
Sephiroth. This shows how ADONAI, the Godname of Malkuth with number value 65,
prescribes the form of the combined Trees of Life.
Figure 9. The (7+7) enfolded polygons have 444 hexagonal
yods.
ADONAI also prescribes the
650 yods associated with each set of seven enfolded Type B polygons that are not their
corners, for both sets of enfolded polygons have 1370 yods, of which 70 are corners of
polygons, leaving 1300 yods that are not corners, i.e., 650 yods associated with each set
(Fig. 8). In each case, this number is the extra number of yods needed to construct
the Trees, given their Sephirothic points and/or polygonal corners.
The 48 tetractyses in OTL with
Type A triangles have 48 hexagonal yods at
their centres. The 94 tetractyses in ITL with Type A polygons have 94 hexagonal yods at
their centres. The number of hexagonal yods on the sides of the (48+94=142) tetractyses in the
combined Trees of Life = 624 − 142 = 482. Just as there are 480 geometrical elements in
them outside the root edge, so there are 480 hexagonal yods on the sides of the
tetractyses outside their root edge. This is an amazing property. Eight of the 482
hexagonal yods are shared as such by both Trees, leaving (482−8=474) unshared hexagonal yods
that line sides of tetractyses. 474 is the number value
of Daath ("knowledge"), the nonSephirah.
As calculated above, the combined Trees of
Life have 722 yods. In the context of the two sets of the seven separate Type B polygons
separated by the root edge, this is the sum of the two yods in the latter associated with
each set and the 720 yods surrounding their seven centres.
The number value of Kether
The Type A triangle has 19 yods. Inside it are
10 yods. Inside the 16 Type A triangles of OTL are (16×10 = 160) yods that comprise 16
corners of tetractyses and (160−16=144) hexagonal yods. 44 hexagonal yods are on their 22
sides. The number of hexagonal yods in OTL = 144 + 44 = 188. They
comprise 140 hexagonal yods on the 70 sides of the 48 tetractyses
and 48 hexagonal yods at the centres of the
latter. 140 is the number value of Masloth, the Mundane Chakra of
Chokmah, and 48 is the number of Kokab, the Mundane Chakra of
Hod.
ITL has 444 hexagonal yods
when the 94 sectors of its (7+7) enfolded polygons are turned into tetractyses (Fig. 9).
When OTL is superposed onto ITL, the centres of the two enfolded triangles, which
8
Figure 10. The outer Tree, the combined Trees
& the decagon embody the gematria number 620 of Kether.
are corners of tetractyses, coincide with the
two hexagonal yods on the Path connecting Chesed and Geburah, thereby removing them from the
count of the number of hexagonal yods in the
combined OTL & ITL, although they remain, of course, hexagonal yods as far as the former
is concerned. Four white hexagonal yods on each side pillar coincide with hexagonal yods on
the vertical, internal sides of the sectors of the two hexagons. The number of yods in the
combined OTL & ITL that are hexagonal yods in both = 188 − 2 + 444 − 4 − 4
= 622. There are 620 such yods in the combined Trees outside the root edge of the (7+7)
enfolded polygons (Fig. 10). This is the gematria number value of Kether, the first Sephirah
of the Tree of Life.
4. Type B triangles in OTL & Type B polygons in
ITL
Outer Tree of Life
We found in Section 3 that the 16 Type A
triangles in OTL consist of 26 corners and 70 sides
of 48 triangles. Converting the Type A triangles into Type B triangles,
i.e., dividing the 48 new triangles into their sectors,
adds 48 corners and (3×48=144) sides, so that there
are now (3×48=144) triangles with
(26+48=74) corners and (70+144=214) sides, i.e., 432 geometrical elements.
They include (74+214=288) corners & sides, where 288 = 1^{1} +
2^{2} + 3^{3} +
4^{4}.
Notice that the construction of the outer Tree of Life from Type B triangles generates
(74−10=64=4^{3}) new corners and (214−22=192) new sides, i.e., (64+192=256=4^{4}) new corners & sides.
Here we see how the Pythagorean Tetrad expresses the new properties of the sacred
geometry of the Tree of Life when it undergoes higherorder transformations.
Inner Tree of Life
The geometrical composition of the (7+7)
enfolded Type B polygons is tabulated below:

Corners 
Sides 
Triangles 
Total 
7 enfolded polygons 
88 
229 
141 
458 
(7+7) enfolded polygons 
174 
457 
282 
913 
ELOHIM, the Godname of Binah with number
value 50, prescribes the 229 sides of the 141 triangles in the seven enfolded Type
B polygons because 229 is the 50th prime number. The (7+7) enfolded polygons have 910
(=91×10)
geometrical elements outside their root edge, where
91 = 1^{2} +
2^{2} + 3^{2} +
4^{2} + 5^{2} +
6^{2}
is one of the defining parameters of holistic
systems, being, for example, the number of Trees of Life in the Cosmic Tree of Life. They
include 282 triangles, where 282 is the number value
of Aralim, the Order of Angels assigned to Binah.
Geometrical composition of the combined
Trees
Separately, OTL & ITL
have (74+174=248) corners, (214+457=671) sides &
(144+282=426) triangles,
where 248
is both the number value
of Raziel, the Archangel of Binah, and the dimension of
E_{8}, the rank‑8, exceptional Lie group appearing in
E_{8}× E_{8} superstring theory. Amazingly, the
separate outer and inner forms of the Tree of Life embody the dimension of the symmetry
group generating the unified superstring force! When they are superimposed, the six
Sephiroth on the two side pillars coincide with the
9
top and bottom corners of the two hexagons and
their centres, whilst the two Paths on each side pillar coincide with internal sides of
their sectors. To avoid double counting, these 10 geometrical elements must be subtracted.
When combined, OTL & ITL, therefore, have 426 triangles with (248−6=242) corners
and (671−2−2=667) sides, i.e., 1335 geometrical elements. There are 240 corners outside the
root edge that consist of the (174−6–2=166) corners of
the 282
triangles in ITL that are unshared with OTL and the
74 corners of the 144 triangles in the latter. The number 240 is a parameter of sacred
geometries. The six corners on the side pillars of OTL that disappear when it is combined
with ITL correspond to the six simple roots of E_{6} and the endpoints of
the root edge correspond to the two remaining simple roots of E_{8}.
The combined Trees of Life have (242+426=668)
corners & triangles. There are (668−2–6=660) corners & triangles outside the root
edge that are not shared between the two Trees. 660 = 66×10, where 66 is
the 65th integer after 1. This shows how ADONAI, the Godname of Malkuth with
number value 65, prescribes the form of the combined Trees.
The number of corners, sides & triangles
in the combined Trees = 242 + 667 + 426 = 1335. The number of corners & sides = 242 +
667 = 909. The number of corners & sides outside the root edge that do not include
shared, Sephirothic points = 909 − 3 − 6 = 900 = 90×10. The number 90 is a holistic
parameter, being the sum of the 10 integers making up the Lambda Tetractys (1). As the table
indicates that the (7+7) enfolded polygons have 913 corners, sides & triangles, of which
two corners and one side make up the root edge and 10 corners & sides are shared with
OTL, ITL has 900 intrinsic corners, sides & triangles outside the root edge. As there
are 1332 geometrical elements outside the root edge, (1332–900=432) geometrical elements
belong to OTL (422 unshared). It has (422+6=428) geometrical elements other
than the four sides of triangles making up the two side
pillars. 428 is the number value of Chasmalim, the Order of Angels
assigned to Chesed.
The number of sides & triangles in the
(7+7) enfolded polygons = 457 + 282 = 739. There are 369 sides & triangles on either side of the
root edge. This is the sum of the number values of the Godnames of the first seven
Sephiroth:
21 + 26 + 50 + 31 + 36 + 76 + 129 = 369.
Yod composition of the combined
Trees
When the 282 triangles of ITL with Type B polygons are tetractyses, they contain
1370 yods. Inside each of the 16 primary triangles in OTL are 37 yods when it is a Type B
triangle. Each of their 22 sides has two hexagonal yods. The number of yods in OTL when its
16 triangles are Type B = 10 + 22×2 + 16×37 = 646. Starting with the 10 corners of these
triangles, its conversion into Type B triangles generates (646−10=636) yods, where 636 is the number value of Rashith
ha Gilgalim, the Mundane Chakra of Kether. Separately, OTL & ITL have
(646+1370=2016) yods. The seven yods on each side pillar are shared with ITL because they lie
on sides of tetractyses in each hexagon. So are the two hexagonal yods on the ChesedGeburah
Path, because they are the centres of the two enfolded triangles. Hence, these have to be
subtracted when the Trees are superposed in order to avoid double counting. The number of yods
in the combined Trees = 2016 − 7 − 7 − 2 = 2000. 1370 yods belong to ITL
and 630 yods belong to OTL, where 630 is the number value of Seraphim, the Order of Angels assigned to Geburah. The 14 polygons of ITL have 68
corners outside their shared root edge, six of which are shared with ITL. Hence,
(68−6=62) polygonal corners outside the root edge are intrinsic to ITL
(31 for each set of seven polygons),
where 31
is the number value of EL, the Godname of Chesed
and 62 is the number value of Tzadkiel, the Archangel of this Sephirah. The outermost corners of the two
pentagons coincide with the centres of the decagons. Hence, 60 intrinsic corners outside
the root edge are pure corners of polygons. The number of yods in the combined Trees of
Life other than such corners = 2000 − 60 = 1940 = 194×10, where 194 is the number value of Tzadekh, the Mundane Chakra of Chesed. Here is a remarkable instance of three
numbers appearing naturally in the same context that refer to thesame Sephirah!
Of the 646 yods in OTL with 16 Type B
triangles, 10 yods are at their Sephirothic corners and 16 yods are at their centres. There
are (646−10−16=620) yods in OTL other than corners and centres of its triangles (Fig.
10). 620 is the number value of Kether, the first Sephirah of the Tree of
Life. The polygonal counterpart of this property is the set
of 620 hexagonal yods making up the decagon with 2ndorder tetractyses as
its sectors.
The number of yods lining the 144 tetractyses
in the 16 Type B triangles of OTL = 646 − 144 = 502. Six of these located at the positions
of Sephiroth are shared with the (7+7) enfolded polygons. Hence, there are
(502−6=496) yods on sides of tetractyses in OTL that are not Sephirothic points
shared with ITL. This is how the outer
Tree of Life embodies the number value 496 of Malkuth and the dimension of E_{8}×E_{8} &
SO(32), the two symmetry groups describing the forces between heterotic
superstrings.
The number of yods lining
the 282 tetractyses of ITL = 1370
− 282= 1088. There are (502+1088=1590) yods that
line
10
tetractyses in the separate Trees of Life.
Remembering that the two hexagonal yods on the ChesedGeburah Path become the centres of the
two triangles belonging to the 14 enfolded polygons, the number of yods on sides of the
(144+282=426) tetractyses in the combined Trees = 1590 − 7 − 7 − 2 = 1574. The
number of such yods that are not shared = 1574 − 16 = 1558. They include the four yods of
the root edge and the four Sephiroth on the central Pillar of Equilibrium. Hence, 1550
(=155×10) yods outside the root edge that
line tetractyses are neither shared nor Sephiroth of OTL. This shows how ADONAI MELEKH, the
Godname of Malkuth with number value 155, prescribes how many yods are needed to shape all the sectors of triangles making up the outer and inner forms of
the Tree of Life.
Inside each Type B triangle are 33
hexagonal yods. The number of hexagonal yods in OTL = 22×2 + 16×33 = 572. The 282
tetractyses in the (7+7) enfolded polygons have 174 corners and 1370 yods. Therefore, they
have (1370−174=1196) hexagonal yods. Eight of these are shared with OTL as
hexagonal yods. The number of hexagonal yods in the combined Trees of Life = 572 + 1196 − 8 = 1760. The
seven enfolded Type A polygons have 176 hexagonal yods. This means that the 70 Type A
polygons

Figure 11.
The 7 enfolded polygons have 176 geometrical elements and 176 hexagonal yods on
the sides of their tetractys sectors. 
enfolded in
10 overlapping Trees of Life on each side of the central Pillar of Equilibrium
have 1760 hexagonal yods. The combined Trees of Life with Type B triangles have as many
hexagonal yods as are in the inner form of 10 Trees of Life with Type A polygons! This is
remarkable, for it demonstrates how a single Tree of Life encodes its potential 10fold
development through a different order of transformation of its geometry. There are eight
hexagonal yods shared by OTL & ITL and two hexagonal yods in the root
edge.
Figure 12. Decagonal representation of the gematria
number value of ADONAI MELEKH.
Hence, (1760−8−2=1750=175×10) hexagonal
yods outside the root edge in the combined Trees are unshared. What emerges here is a
differentiation that is analogous to the 175/176 distinction, reported by C.W. Leadbeater in
his book Occult Chemistry (2), in the augmentation of major and minor whorls of the
UPA, now identified by the author as the subquark state of the
E_{8}×E_{8} heterotic superstring.
The number 175 determines the number of yods unshared by the inner and outer forms of 10
Trees and the difference 1 between 175 and 176 determines both shared hexagonal yods and
hexagonal yods in the root edge. This quality of being shared by both the outer
& inner forms of the Tree of Life is characteristic of this difference when it manifests
in other sacred geometrical contexts. For example:
11

the seven enfolded Type A polygons in ITL have 176 hexagonal yods on
the 88 sides of their 47 sectors (Fig. 11). One of the hexagonal yods on their
shared root edge can be associated with one set of seven enfolded polygons and
the other hexagonal yod on the root edge can be associated with the other set.
This means that 175 hexagonal yods can be associated with each set. The
complete set of 176 hexagonal yods includes one hexagonal yod that is
associated not with this set but with its mirrorimage set;

the seven enfolded polygons of ITL are composed of 176 corners,
sides & triangles (41 corners, 88 sides & 47 triangles). The top
corner of the hexagon coincides with the lowest corner of the hexagon
belonging to the set of polygons that is the inner form of the next higher
Tree. 175 geometrical elements are intrinsic to the set of seven enfolded
polygons (in fact, to the set in successive Trees) and one element is shared
with the set associated with the next higher Tree;

the first (6+6) enfolded polygons constitute a holistic subset of
the (7+7) polygons. The first six polygons enfolded in each of seven
overlapping Trees have 176 corners, of which 175 are intrinsic to these 42
polygons and one (the topmost corner of the hexagon enfolded in the seventh
Tree) is shared with the hexagon enfolded in the eighth Tree because it
coincides with its lowest corner. The fact that the seven Trees bear a formal
correspondence to the seven Sephiroth of Construction when each Sephirah of
the Tree of Life is replaced by a Tree is further indication that the number
176 is a parameter of holistic systems. It manifests in the subatomic world in
each major whorl of the UPA as the extra spirilla of a given order in every
175 spirillae of the next lower order.
5. ADONAI MELEKH prescribes the combined Trees
of Life
We found in Section 3 that
the combined Trees of Life with Type A triangles & polygons have 650 yods other than
Sephiroth and corners outside the root edge. As this is the number of yods in 65
tetractyses, ADONAI with number value 65 prescribes the combined Trees. We saw that
it also prescribes ITL with Type B polygons because 650 yods other than polygonal corners
are associated with each set of seven enfolded polygons. We found in Section 4 that ADONAI
prescribes the combined Trees with Type B triangles & polygons because they have 660
(=66×10) corners & triangles outside the root edge
that are unshared between them, where 66 is the 65th integer after 1. We also
found that ADONAI MELEKH prescribes the combined Trees with Type B triangles & polygons
because 155×10 yods outside the root edge that
line their 426 tetractyses are neither Sephiroth nor shared.
As 155 is the sum of the first 10 integers after 10:
11
12 13
155 =
14 15 16
17 18 19 20 .
1550 is 10 times the sum of the first 10
integers after 10. It demonstrates how the Decad arithmetically determines this yod
population of the combined Trees of Life. Figure 11 shows how the 10sided, Type B decagon
represents the number 155 as the gematria
number value of ADONAI MELEKH. The letter values of the Godname denote the numbers of
different types of yods.
ADONAI MELEKH prescribes OTL with Type A
triangles
48 of the 214 yods in OTL are hexagonal yods at the centres of
the 48 tetractyses making up the 16 Type A triangles with 10 corners and 22
sides. Therefore, (214−48=166) yods line the (22+48=70) sides of
its 48 tetractyses. (10+16=26) of these are corners,
leaving 140 hexagonal yods, where 26 is the number value of
YAHWEH, the Godname of Chokmah, and 140 is the number value
of Masloth, the Mundane Chakra of this Sephirah. The seven yods lining each side
pillar of OTL and the two hexagonal yods on the Path connecting Chesed and Geburah are
shared with the polygons of ITL, so that (166−16=150=15×10) yods lining the 70 sides
of the 48 tetractyses in OTL are intrinsic to
it. 15 is the number value of YAH, the older version of the Godname
assigned to Chokmah.
12
Ten of the 166 yods lining the 70 sides of
the 48 tetractyses are Sephiroth, leaving 156 yods, where 156 is
the 155th integer after 1. This shows how ADONAI MELEKH prescribes
the shape of OTL with Type A triangles. Starting with the 10 Sephiroth at the
corners of these 16 triangles and with Daath at the centre of the triangle with Chokmah,
Binah & Tiphareth located at its corners, (166−10−1=155) yods are needed to line the
70 sides of the 48 tetractyses in OTL (Fig. 13). Starting with a single,
mathematical

Figure
13. 155 extra yods line the sides of tetractyses in OTL with Type A
triangles. 
point (Kether), 165 yods are needed to delineate these tetractyses,
where
165 = 1^{2} +
3^{2} + 5^{2} +
7^{2} + 9^{2}.
The 16 Type A triangles in OTL
have 48 sectors and the (7+7) Type B enfolded polygons consist
of 282 triangles. Combined, they comprise 330 triangles, where
33
33
33
330
=
33
33 33
33
33 33 33
and 33 = 1! + 2! + 3! + 4!. This number with
wellknown, Masonic significance is, therefore, a measure of the geometry of the combined
OTL & ITL. Amazingly, the number 33 also determines the yod
population of ITL, for this is 1370, which is the number of yods in 137 tetractyses, where
137 (the mysterious number in atomic physics whose reciprocal is the approximate value of
the finestructure constant) is the 33rd prime number!
As
55 = 1^{2} +
2^{2} + 3^{2} +
4^{2} + 5^{2}
and
385 = 7×55 = 1^{2} +
2^{2} + 3^{2} +
4^{2} + 5^{2} +
6^{2} + 7^{2} +
8^{2} + 9^{2} +
10^{2},
330 = 385 − 55 = 6^{2} +
7^{2} + 8^{2} +
9^{2} + 10^{2}.
The number of yods in n overlapping Trees of
Life is given by
Y(n) = 50n + 20
(3). Therefore, the number of yods in
every 31 Trees = Y(n+31) − Y(n)
= 31×50 = 1550. As we found earlier, this is the number of unshared yods
other than Sephiroth in the combined OTL & ITL that are outside the root edge and line
tetractyses. We conclude that EL, the Godname of Chesed with number
value 31, prescribes the yod population of the combined Trees.
6. Type A triangles in OTL & Type B
polygons in ITL When
separate, OTL with Type A triangles has 26
corners, 70 sides & 48 triangles and ITL
with Type B polygons has 174 corners, 457 sides & 282 triangles. Six corners and four sides become shared when
they combine, so that they then have (26+174–6=194)
corners, (70+457–4=523) sides & (48+282=330)
triangles. 194 is the number value of Tzadekh, the Mundane Chakra of Chesed. Ten of the
1047
13
geometrical elements are shared, leaving 1037
unshared elements that consist of (194–6=188) corners, (523–4=519)
sides & 330 triangles. The combined Trees have 192 corners outside the root edge. This
is a parameter of ITL because 192 geometrical elements surround the centres of the seven
separate polygons. Indeed, it is a defining parameter of all sacred
geometries/holistic systems. For example, the 64 hexagrams in the I Ching
system of divination have 192 lines and 192 broken lines, whilst each half of the
3dimensional Sri Yantra constructed from tetractyses contains 192 yods (4).
Figure 14. The
(7+7) enfolded polygons have 524 yods.
There are (194+330=524) corners &
triangles. This number is par
excellence a defining parameter of ITL because
it is the number of yods in the (7+7) enfolded Type A polygons (Fig. 14). There are
(194+523=717) corners & sides. Ten of them are shared between OTL and ITL,
three of them make up the root edge and four are the corners located at the four Sephiroth
on the central Pillar of Equilibrium. There are (717–10–3–4=700) corners & sides outside
the root edge that are neither Sephiroth nor shared. This is the sum of the four separate
tetractys arrays of integers making up the faces of the Tetrahedral Lambda (5), which is the
arithmetic counterpart of sacred geometries.
OTL with Type A triangles has 214 yods
(26 corners & 188 hexagonal yods). ITL with Type B polygons has 1370
yods (174 corners & 1196 hexagonal yods). Combined, they have (214+1370–16=1568) yods
that comprise (26+174–6–2=192) corners and (188+1196–8=1376) hexagonal yods. They include
six corners and eight hexagonal yods that are shared as, respectively, corners &
hexagonal yods. Hence, they have (1568–6–8=1554) unshared yods that comprise (192–6=186)
unshared corners and (1376–8=1368) unshared hexagonal yods. Outside the root edge with two
endpoints and two hexagonal yods are 1550 (=155×10) unshared yods comprising 184
unshared corners and 1366 unshared hexagonal yods. Once again, the number
value 155 of ADONAI MELEKH appears! The number 1366 is the number of yods
outside the root edge in ITL with Type B polygons.
7. Type B triangles in OTL & Type A
polygons in ITL
OTL with Type B triangles has 74 corners, 214
sides & 144 triangles. ITL with Type A polygons has 80 corners, 175 sides &
94 triangles. Combined, they have (74+80–6=148) corners,
where 148 is the number value of Netzach, (214+175–4=385) sides,
where
1^{2}
2^{2}
3^{
2}
385
=
4^{2}
5^{2}
6^{2}
7^{2}
8^{2}
9^{2}
10^{2} ,
and (144+94=238) triangles. Their 771
geometrical elements comprise 10 shared elements and 761 unshared elements. Outside the root
edge are (148+238–2=384) corners & triangles and (385–1=384) sides. This number is
another parameter of holistic systems, being, for example, the number of lines & broken
lines in the 64 hexagrams. There are (385+238=623) sides & triangles, i.e., 622
sides & triangles outside the root edge. We found in Section 3 that the combined OTL
& ITL with Type A triangles & polygons have 622 hexagonal yods.
OTL with Type B triangles has 646 yods (74
corners & 572 hexagonal yods) and ITL with Type A polygons has 524 yods (80 corners
& 444 hexagonal yods). Combined, they have (646+524–16=1154) yods that comprise
(74+80–6–2=146) corners & (572+444–8=1008) hexagonal yods. They include
six
14
Figure 15. The
number 138 as a holistic parameter.
corners and eight hexagonal yods that are
shared as such. Hence, they have (1154–6–8=1140) unshared yods that comprise
(146–6=140)
unshared corners and (1008–8=1000) unshared hexagonal yods,
where 140 is the number value of Masloth, the Mundane Chakra of
Chokmah. Outside the root edge are 1136 unshared yods that consist of 138 unshared
corners and 998 unshared hexagonal yods. The number 138 is another parameter of holistic
systems, being, for example:

the sum of the 10 integers 1, 2, 4, 8, 3, 9, 27, 4, 16 &
64 located on the raised edges of
the Tetrahedral Lambda (Fig. 15);

the number of yods in the two Type A dodecagons belonging to ITL
that are outside their shared root edge.

The number of corners of the triangles formed by the centres and
edges of the two polyhedra that constitute the polyhedral version of the Tree
of Life (6).
Conclusion
Various transformations of the outer and inner
forms of the Kabbalistic Tree of Life produce combinations that display geometrical and yod
compositions characterized by the same numbers. These numbers have been shown in
previous articles to be parameters of sacred geometries. This refutes the possibility that
they arise merely by chance. Instead, it confirms that both these forms of the Tree of Life
constitute sacred geometry. Two combinations are particularly interesting — Type A
triangles/Type A polygons and Type B triangles/polygons. This is to be expected because only
these combinations are consistent, mathematically speaking (clearly, it is inconsistent,
from a strict mathematical point of view, for example, to treat the triangles in the outer
Tree of Life as Type A but to regard the polygons — including the triangles — in the
inner Tree as Type B). With Type A triangles & polygons, there are 240 corners &
triangles and 240 sides outside the root edge. This 240:240 pattern manifests in the (7+7)
separate Type A polygons as their (240+240) hexagonal yods. It appears in the first four
Platonic solids as the 240 hexagonal yods in the faces of the tetrahedron, octahedron &
cube and as the 240 hexagonal yods in the faces of the icosahedron. Their counterpart in
superstring theory are the 240 roots of the two similar Lie groups
E_{8} in E_{8}×E_{8}. The combined Trees of Life encode E_{8}×E_{8}
heterotic superstrings. That is the natural
conclusion to be drawn for this version of the cosmic blueprint, for the only other
explanation of the pattern is that it arises purely by coincidence and — given that it
appears in these (and other) sacred geometries — this possibility has to be rejected as
highly implausible. The hexagonal yod population of the combined Trees with Type B
triangles & polygons embodies the same holistic parameter 176 as that determining the
geometrical composition of the seven enfolded polygons. It manifests in the thickening,
as reported by C.W. Leadbeater, of the major whorls of the UPA relative to its minor
whorls, namely, every 25 spirillae of a given order are composed of 176 spirillae of the
next higher order, not 175, as occurs in the minor whorls. ADONAI MELEKH, the Godname of
Malkuth with number value 155, prescribes
the shape of the combined Trees because their tetractyses have 1550
unshared yods that are not Sephiroth lining their sides outside the root edge.
References
1. Phillips, Stephen M. Article 11: “Plato’s
Lambda — its meaning, generalisation and connection to the Tree of Life,” http://www.smphillips.mysite.com/pdfs/Article11.pdf.
2. Besant, Annie
& Leadbeater, C.W. “Occult Chemistry,” 3rd ed., Theosophical Publishing House, Adyar,
Chennai, India (1951), p. 23.
3. Proof: when they are turned into
tetractyses, the (12n+4) triangles of n overlapping Trees of Life have (6n+4) corners and
(16n+6) sides, two hexagonal yods lying on each side. The number of yods in n overlapping
Trees = 6n + 4 + 2(16n+6) + 12n + 4 = 50n + 20.
4. Phillips, Stephen M. Article 35: “The Tree
of Life nature of the Sri Yantra and some of its scientific meanings,” http://www.smphillips.mysite.com/pdfs/Article35.pdf.
5. Ref. 1, pp. 13, 14.
6. Phillips, Stephen M. Article 23: “The
polyhedral Tree of Life,” http://www.smphillips.mysite.com/pdfs/Article23.pdf.
15
