ARTICLE 53
by
Stephen M. Phillips
Flat 4, Oakwood
House, 117119 West Hill Road. Bournemouth. Dorset BH2 5PH. England.
Website: http://smphillips.mysite.com
Abstract
Five sacred
geometries , — the inner form
of the Tree of Life, the first three Platonic solids, the 2dimensional Sri
Yantra, the disdyakis triacontahedron and the
1tree — are shown to possess
240 structural components or geometrical elements. They correspond to the
the 240 roots of the rank8 Lie group group E_{8} because in each case they divide into 72
components or elements of one kind and 168 of another kind, in analogy to
the 72 roots of E_{6}, the
rank6 exceptional subgroup of E_{8}, and to
the remaining 168 roots of E_{8}.
Furthermore, the 72 components are proved to form three sets of 24 and
the 168 components are shown to form seven sets of 24, so that all 240
components form ten sets of 24. Each sacred geometry therefore has a
tenfold division, indicating a similar division in the holistic systems
that they represent. The bestknown example of this is the Kabbalistic
Tree of Life with ten Sephiroth that comprise the the Supernal Triad and
the seven Sephiroth of Construction. A less wellknown example is the
"ultimate physical atom," or UPA, the basic unit of matter paranormally
described over a century ago by the Theosophists Annie Besant and C.W.
Leadbeater. This has been identified by the author as the
E_{8}×E_{8} heterotic superstring constituent of up and down
quarks. Its ten whorls (three major, seven minor) correspond to
the ten sets of structural components making up each sacred
geometry. The analogy suggests that 24 E_{8} gauge
charges are spread along each whorl as the counterpart of each set of 24
components. The tenfold composition of the E_{8}×E_{8} heterotic superstring predicted by this analogy
with sacred geometries is a consequence of the tenfold nature of God,
which is most explicitly represented in the Tree of Life.

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. The inner Tree of Life The
10fold UPA/superstring as the 10fold division of the (7+7) polygons
The 240 roots of the Lie group
E_{8} can be regarded as consisting of the 72 roots that are also roots of its exceptional subgroup
E_{6} and 168 other roots
(Fig. 1). When their 47 sectors are tetractyses, the seven enfolded polygons of the inner Tree
of Life have 120 yods on their 42 sides, where
120 =
2^{2} + 4^{2} + 6^{2} + 8^{2}.
They comprise
36 (= 6^{2}) black yods at
the corners of polygons and 84 (= 2^{2} + 4^{2} + 8^{2}) red
hexagonal yods. The two separate sets of seven enfolded polygons have
(2×36=72) black yods at their corners and
(2×84=168) red hexagonal yods.
The distinction between corners and hexagonal yods corresponds to E_{6} being a
subgroup of E_{8}. This is the sacred geometrical counterpart of the
symmetrybreakdown of E_{8} into E_{6}
Figure 1. The 72 corners
and 168 hexagonal yods on the sides of the two sets of seven enfolded
polygons symbolize the 72 roots of E_{6} and the
168 extra roots of E_{8}.
that has been considered by superstring
theorists in order to reproduce the physics of the Standard Model. Each yod on
the boundaries of the polygons symbolizes a root of E_{8} to which corresponds a gauge
charge and its associated gauge boson that mediates the unified force between
superstrings.
The group E_{6} contains the
exceptional subgroup F_{4} with dimension 52. It has four simple roots and 48
roots. As the first six enfolded polygons have 24 corners outside their shared root edge, the
counterpart of these 48 roots are the (24+24=48) black yods at the corners of the
two sets of the first (6+6) enfolded polygons that are outside their root edges. The two
separate dodecagons have (12+12=24) corners. We see that the 72 corners divide
naturally into three groups of 24 corners. Do the 168 red hexagonal yods
also split up naturally into seven groups of 24? Indeed, they do! Below are shown the
numbers of hexagonal yods on the sides of one set of seven enfolded polygons:
triangle square
pentagon hexagon octagon decagon dodecagon
4
6
8
10
14
18
24
(the root edge must be associated with the
dodecagon because this then generates its 24 hexagonal yods). The 18 hexagonal yods on the
sides of the decagon can only be grouped with the six hexagonal yods of the square in order
to form a set of 24, the 14 hexagonal yods of the octagon can only be grouped with the ten
hexagonal yods of the hexagon and the eight hexagonal yods of the pentagon can only be
grouped with the four hexagonal yods of the remaining triangle to generate 24 hexagonal yods
in both pairs. Hence, there are two sets of 24 (dodecagons), two sets of 24 (decagon &
square), two sets of 24 (octagon & hexagon) and one set of 24 (triangle & pentagon).
No other combinations are possible. Amazingly, both the 72 corners and the 168
hexagonal yods group naturally into ten sets of 24 yods. The significance of this for
superstrings is as follows: in some of
Figure 2.
3
his research articles on his website, the
author identified the UPA described by Besant & Leadbeater as the subquark state of the
E_{8}×E_{8} heterotic superstring. Its ten whorls "carry" the 240 gauge
charges of E_{8} corresponding to the 240 roots of this group. In the spirit of
heterotic string theory, 24 gauge charges are regarded as spread out along each whorl. The
three major whorls carry 72 gauge charges and the seven minor whorls carry
168 gauge charges. This
72:168 division is precisely what we see in the
distinction between the 72 corners
and the 168 hexagonal yods on the
sides of the two sets of seven enfolded polygons! The corners of a tetractys are analogous
to the three major whorls, which are the microphysical manifestation of the three Sephiroth
of the Supernal Triad, and its hexagonal yods correspond to the seven minor whorls, which
are the microphysical manifestation of the seven Sephiroth of Construction (Fig. 2).
Remarkably, this analogy extends even to individual whorls, for the ten sets of 24
gauge charges of E_{8} carried by the ten whorls correspond to the ten sets of
corners and hexagonal yods.
The
counterpart in the first six enfolded polygons (which constitute a holistic set in themselves)
of each set of 24 gauge charges are the 24 intrinsic corners associated[1] with them.
The 60 polygons enfolded in ten overlapping Trees of Life, each mapping a whorl, have 240
associated corners that correspond to the 240 roots of E_{8} and their
associated gauge charges (Fig. 3). The (240+240) intrinsic corners associated with both
sets of 60 polygons enfolded in
ten overlapping Trees of Life denote the (240+240=480) roots of
E_{8}×E_{8}, i.e., the mirror symmetry of the inner Tree of Life is
responsible for the direct product nature of
the
Figure 3. The (240+240) intrinsic corners of the first
(6+6) polygons enfolded in 10 overlapping Trees of Life denote the (240+240) roots of
E_{8}×E_{8}.
Each Tree represents one of the 10 whorls of the subquark state of the
E_{8}×E_{8} heterotic superstring.
anomalyfree gauge symmetry
group of the unified force between E_{8}×E_{8} heterotic
superstrings.
[1]
The two endpoints of the root edge are shared by both sets of
seven polygons. One can be associated with one set and one with the other set. The topmost
corner of each hexagon coincides with the lowest corner of a hexagon enfolded in the next
higher Tree. Both sets of the first six polygons have 50 corners. 25 of them are associated
with each set, but only 24 belong exclusively to each one in the sense of being unshared
with polygons enfolded in the next higher Tree. In other words, only 24 corners are
intrinsic to a set.
4
As


3





5 
7 

11^{2} − 1 = 120
=

9

11

13



15 
17 
19 
21, 
i.e.,
as 120 is the sum of the first ten odd integers after 1, the number 240 is the sum of the
ten even integers that start with 6 and are four units apart:
6
10
14
240 = 18 22
26
30
34 38 42 .
In terms of the numbers 1, 2, 3 & 4
symbolized by the four rows of dots in the tetractys,the factorization of 240 as 10×24 is
expressed as (1+2+3+4)×1×2×3×4.
As 5^{2} − 1 = 24 = 3 + 5 + 7 + 9,
the number 240 is the sum of the tetractys arrays of the first four odd integers after
1:
3
5
7
9
3
3
5
5
7
7
9 9
240
= 3 3
3
+ 5
5 5 +
7 7 7 +
9 9 9
3 3 3
3 5 5 5
5 7 7 7
7 9 9
9 9 .
The sum of the 12 odd integers at the
corners of these tetractys is 72 and the sum of the remaining 28 odd integers is
168. This is the arithmetic counterpart of the 72 corners and 168
hexagonal yods that line the two sets of the seven enfolded polygons. A beautiful harmony
always exists between number and geometry when the latter is sacred. At the end of this
article, it is proved that the number 16800 — the number of 1storder spirillae in the
UPA/superstring — is the sum of 240 odd integers.
The 10fold UPA/superstring as the 10fold
division of the two Type A dodecagons
As
168 extra yods are needed to
construct the Type A dodecagon from tetractyses, i.e., it embodies the very number that
determines the structure of the UPA/heterotic superstring, it should come as no surprise
that the geometry of the dodecagon also embodies the 10fold pattern of this particle, as
now proved.
The pair of
Type A dodecagons (Fig. 4) has 24 corners, 24 red sides & 24 blue sides of its
sectors, i.e., three sets of 24 geometrical elements. Each sector is divided into three
triangles with three green sides and one internal corner (apart from the shared centre of
the dodecagon). Hence, the 24 sectors are, additionally, composed of 24 internal corners,
three sets of 24 green sides and three sets of 24 triangles, i.e., seven sets of 24
geometrical elements. The
two Type A dodecagons, therefore, naturally comprise ten sets of 24 geometrical
elements.
These sets
are the counterpart of the ten whorls of the UPA/heterotic superstring, each of which
carries 24 gauge charges of the superstring gauge symmetry group E_{8}. The three
sets of 24 corners & sides that create the 24 sectors of the two
dodecagons correspond to the three major whorls of the UPA, each carrying 24 E_{8}
gauge charges. The seven extra sets of
24 geometrical elements needed to turn the
dodecagon into a
Type
A
dodecagon correspond to the seven minor whorls, each
carrying 24 E_{8} gauge charges. We saw previously that the 120 yods lining the
42
sides of the seven enfolded polygons divide
into ten sets of 12, further divided into three sets (36 corners) and seven sets
(168 hexagonal yods). Each yod corresponds to one of the geometrical elements making up
the Type A dodecagon.
Figure 4. The two Type A dodecagons.
The 10fold UPA/superstring as the 10fold
division in the yods on edges of tetractyses in the pair of Type A
decagons The sixth regular
polygon in the inner form of the Tree of Life is the decagon. Constructed from tetractyses, it
is the polygonal representation of ten overlapping Trees of Life because its 61 yods correspond
to the 61 SLs up to Chesed of the tenth Tree. Each sector is the counterpart of a Tree, the six
yods per sector being the counterparts of the six SLs per Tree. The centre of the Type A
decagon is surrounded by 120 yods that line the 50 sides of its 30 tetractyses. There are 12 boundary yods
per
5
Figure 5. The 240 yods on sides of
tetractyses in the two Type A decagons correspond to the 240 corners associated with the first
six polygons enfolded in ten Trees of Life.
sector. The two Type A decagons have
(120+120=240) yods surrounding their centres that line the 100 sides of their 60 tetractyses
(Fig. 5). They comprise (12+12=24) boundary yods per pair of sectors. Therefore, the 240
boundary yods consist of ten sets of 24 yods. Three pairs of sectors (coloured white, grey
& dark grey) have (3×24=72)
boundary yods. The remaining seven pairs of sectors (coloured the seven colours of a rainbow)
have (7×24=168) boundary yods.
Starting from a point (the centre of a decagon), 240 yods are needed to delineate the
tetractyses that make up two Type A decagons, 24 per pair of sectors. This
72:168 pattern is analogous to what was discussed
earlier. Instead of ten geometrical elements, each repeated (12+12) times, we now have
(12+12) yods that are repeated ten times.
The first six enfolded polygons up to the
decagon have 26 corners. One of the
corners is an endpoint of the root edge shared not only amongst the six polygons but also with
the second set of six polygons — the mirror image of the first set. When sets of polygons
enfolded in overlapping Trees of Life are considered, the topmost corner of the hexagon
coincides with the lowest corner of the hexagon enfolded in the next higher Tree. This means
that 25 corners of
a set of six enfolded polygons are intrinsic to them. One corner (an endpoint of the root edge)
can be associated with the set and one (the other endpoint) can be associated with the second
set. Hence, 24 corners are associated with and intrinsic to the set of six polygons enfolded in
each Tree. The 60 polygons of the first six types enfolded in ten overlapping Trees of Life
have (10×24=240) corners, that is, 24 corners repeated ten times. This is the inner Tree
of Life basis of the 24 E_{8} gauge charges spread along each whorl of a
UPA/subquark.
The Decad defines both the decagon with ten
sides and the tenth polygon — the dodecagon.
The pair of either polygon embodies ten sets of 24 geometrical elements or yods. Their
subatomic counterparts are the 24 E_{8} gauge charges carried by each of the ten whorls
of a UPA, which is the subquark state of an E_{8}×E_{8} heterotic superstring.
Here is yet more evidence provided by sacred geometry of the tenfold nature of the basic unit
of matter paranormally described by Annie Besant & C.W. Leadbeater.
The 10fold UPA/superstring as the 10fold
division in the 240 hexagonal yods in the seven separate polygons of the inner Tree of
Life
The seven separate polygons of the
inner Tree of Life have 48 sectors, where 48 is the number value of
Kokab, the Mundane Chakra of Hod (see Table 1). As tetractyses, they contain 240
hexagonal yods that naturally group into ten sets of 24, as now explained.
6
Figure 6. The
24:24 division of the sectors of the seven separate polygons corresponds to the major/minor
distinction in the ten whorls of the UPA/subquark state of the E_{8}×E_{8}
heterotic superstring.
As a parameter of holistic systems, the number
48 displays a 24:24 division that always characterizes them. For the seven separate
polygons, the triangle, square, pentagon & dodecagon have 24 sectors and the hexagon,
octagon & decagon have 24 sectors (Fig. 6). The only other possible combinations of
polygons with 24 sectors are firstly, the triangle, pentagon, hexagon & decagon, and
secondly, the square, octagon & dodecagon (the two possibilities arise because the
triangle and pentagon have the same number of sectors as the octagon). Just as the seven
Sephiroth of Construction comprise two triads: ChesedGeburahTiphareth and
NetzachHodYesod, and Malkuth, so research by the author has confirmed that this 3:3:1
pattern is shown by the seven polygons of the inner Tree of Life, i.e., they group into
trianglesquarepentagon, hexagonoctagondecagon & dodecagon. Only the former
combinations of polygons respect this grouping because the latter combinations are each
taken from both sets of three polygons. Although, arithmetically speaking, either
pair of combinations is possible because each combination has 24 corners, the former will be
favoured here because it preserves the analogy between the outer and inner Trees of Life.
There are five hexagonal yods per sector. Therefore, the 240 hexagonal yods comprise 24 sets
of five in the triangle, square, pentagon & dodecagon and 24 sets of five in the
hexagon, octagon & decagon. Alternatively, they may be regarded as five sets of 24 and
another five sets of 24, i.e., as ten sets of 24. This 5:5 differentiation reflects
the division of the Tree of Life into the socalled "Upper Face" spanned by the highest five
Sephiroth and the "Lower Face" spanned by the lowest five Sephiroth. It arises here from the
primary division of the 48 sectors into two sets of 24.
Figure 6 displays the ten sets of 24 hexagonal
yods, the yods in each set having the same colour. Each colour corresponds to one of the ten
Sephirah. The white, light grey & dark grey yods correspond to the Supernal Triad,
which, being part of the Upper Face, means that these yods belong to the triangle,
square, pentagon & dodecagon. The question is: which of the five hexagonal yods in each
sector should be coloured white, light grey & dark grey? For overlapping Trees of Life,
Kether, Chokmah & Binah of a given Tree become Tiphareth, Netzach & Hod of the next
higher Tree, whilst Tiphareth is Malkuth of the next higher Tree, the Sephirah that is
symbolized by the hexagonal yod at the centre of a tetractys. These Sephiroth of
Construction belong to different triads. Hence, the white, light grey and dark grey
hexagonal yods must be, respectively, the central hexagonal yod and the two hexagonal yods
either on a side of a polygon or on an internal side of a tetractys. The latter possibility
has been chosen here.
The ten sets of 24 hexagonal yods symbolize
the ten whorls of the UPA/subquark state of the E_{8}×E_{8} heterotic
superstring. The white hexagonal yods at the centres of the 24 sectors in the triangle,
square, pentagon & dodecagon denote the 24 E_{8} gauge charges carried by the
first major whorl, the 24 light grey hexagonal yods denote the 24 E_{8} gauge
charges carried by the second major whorl, the 24 dark grey hexagonal yods denote the 24
E_{8} gauge charges carried by the third major whorl, etc. The tenfold pattern of
the distribution of hexagonal yods arises from the simple facts that there are five such
yods per sector and that the 48 sectors of the seven polygons divide into two sets of
24.
7
2. The Platonic
solids Constructed from tetractyses,
the Platonic solids have three types of hexagonal yods: 1. hexagonal yods at centres of
tetractyses; 2. hexagonal yods on edges of solids, and 3. hexagonal yods on sides of
tetractyses inside the boundaries of faces. Their numbers are calculated and tabulated
below:
Table 2. Hexagonal yod populations in the faces
of the five Platonic solids.
(the numbers of yods are further
divided into their two halves in order to indicate the numbers of hexagonal yods in each
half of the polyhedron). The first three Platonic solids have 240 hexagonal yods comprising
120 hexagonal yods either at centres of tetractyses or on edges and 120 hexagonal yods on
interior sides (Fig. 7). This 120:120 division in the 240 hexagonal yods in the 18 faces of
the first three Platonic solids reoccurs in both the icosahedron and the dodecahedron because each halfpolyhedron has 120 hexagonal yods. As we saw in Section 1, each of the two separate sets of
seven enfolded polygons has 120
Figure 7
yods on their boundaries. Both sets have 240 yods on their boundaries that
comprise 72 corners
and 168 hexagonal
yods. We saw there that these 240 boundary yods symbolize the 240 roots of the superstring
gauge symmetry group E_{8}, the 72 corners denoting the
72 roots of its
exceptional subgroup E_{6}, leaving 168 roots represented by the
168 hexagonal yods.
The table indicates that their counterparts in the first three Platonic solids are the
72 hexagonal yods in
one half of the tetrahedron and the octahedron and the 168 hexagonal yods in their other half
and in the cube. The 72:168 division in the holistic parameter
240 manifests in the first three Platonic solids. Moreover, just as the
72 corners were
found to be composed of three sets of 24 corners of a unique combination of polygons, so
the 72 hexagonal
yods consist of 24 hexagonal yods in one half of the tetrahedron, 24 hexagonal yods that
either line edges of the octahedron or are centres of tetractyses and 24 hexagonal yods
on sides of tetractyses inside the faces of the octahedron. Finally, just as the
168 hexagonal yods
on the sides of the two sets of polygons were found earlier to be distributed amongst the
polygons in a way that naturally generates seven sets of 24, so, too, Table 2 shows that
the 168 hexagonal
yods in the other half of the tetrahedron & octahedron and in the cube group
naturally into seven sets of 24:
8
1. 24 (6+6+12)
hexagonal yods that are in that half of the tetrahedron;
2. 24 (12+12) hexagonal yods
that are either centres of tetractyses or line edges of the other half of the octahedron;
3. 24 hexagonal yods that are sides of tetractyses in that
half;
4. 24 (12+12) hexagonal yods at
centres of tetractyses in the cube;
5. 24 (12+12) hexagonal yods
that line its edges;
6. 24 hexagonal yods lining sides of tetractyses in one half of
the cube, and
7. 24 hexagonal that line sides in the other half of the
cube.
(the red numbers are those that appear as such
in the table). The mathematical analogy between the two systems of sacred geometries exists
because one encodes in the yods shaping the twodimensional, regular polygons the same
pattern as that what exists in the faces of the threedimensional, regular polyhedra. As
with the two sets of seven enfolded polygons, the 240 hexagonal yods in the 18 faces of the
tetrahedron, octahedron & cube can be grouped into ten sets of 24. Here, therefore, are two
sacred geometries that encode in analogous ways the 10fold structure of the UPA as ten
whorls, each carrying 24 gauge charges of E_{8}. The importance of this discovery for the future development
of Mtheory cannot be emphasized too strongly. This as yet
undiscovered 10fold composition of the superstring is the manifestation in the
subatomic world of the 10fold nature of God, as represented by both the Tree of Life and
Pythagoras' tetractys. Predicting this 10foldness will be an authentic sign of the
validity of any future candidate for the much sought Mtheory that unifies the five
superstring theories with General Relativity.
The 240 hexagonal yods repeated in the
fourth Platonic solid symbolize the 240 roots of the second, similar Lie group E_{8}′
that is part of the gauge symmetry group E_{8}×E_{8}′ governing the unified
force between one of the two types of heterotic superstrings. The first four Platonic solids
embody not only the dimension 248 of
E_{8} as the 248 corners &
sides of the sectors of its faces (see p. 12, Article 3) but also the numbers of roots of E_{8} and
E_{8}×E_{8}′. It has been proved on page 5 at Superstrings as sacred
geometry/Platonic solids that 1680 geometrical elements surround their
axes when their faces are divided into their sectors and the face centres are joined to the
centres of the solids. Here is irrefutable evidence that the UPA paranormally described by
Besant & Leadbeater is an E_{8}×E_{8}′ heterotic superstring (actually,
an E8'singlet state), for both the basic structural parameter of the UPA and the root
composition of this symmetry group are embodied in the first four Platonic solids
symbolising the physical Elements Earth, Air, Fire & Water — a remarkable property that
is too implausible to attribute to chance.
It was pointed out in Section 1 that the
48 corners outside the root edge of
the two separate sets of the first (6+6) enfolded polygons denote the
48 roots of F_{4}, the
rank4 exceptional subgroup of E_{6} whose 72 roots are denoted by the 72 corners of the two separate sets of the (7+7)
enfolded polygons. What is their counterpart in the first three Platonic solids? Table 2
above indicates that the 72
hexagonal yods in the six faces making up half the tetrahedron and half the octahedron
are:
Hexagonal
yods
Hexagonal
yods
Hexagonal yods
Total
at
centres
on
edges
on
interior
sides
Tetrahedron:
6
6
12
24
Octahedron:
12
12
24
48
The
48 hexagonal yods in the four faces
in a half of the octahedron denote the 48 roots of F_{4}.
In Section 4, we shall discuss the
properties of the disdyakis triacontahedron, the 120face Cartan solid which was shown in
Article 23 to be the outer form of the polyhedral version of the
Tree of Life. We shall find that it embodies both these properties in the same way that the
first four Platonic solids do, namely, 1680 geometrical elements surround an axis of the
disdyakis triacontahedron, whilst 240 hexagonal yods are in the 60 faces making up each half
of it when they are turned into tetractyses. Like these Platonic solids, the disdyakis
triacontahedron also embodies the number 248 (see Article 44). The reappearance of the paranormallyobtained
structural parameter of the superstring in a polyhedron that Article 23 identified as the
outer form of the "Polyhedral Tree of Life" cannot be a coincidence. Instead, it confirms
the archetypal nature of this number whose significance awaits discovery by theoretical
physicists.
3. The 2dimensional Sri Yantra
The geometrical makeup of the 2dimensional
Sri Yantra (Fig. 8) is tabulated below:
Table 3. Geometrical composition of the
2dimensional Sri Yantra.

Number of vertices 
Number of sides 
Number of triangles 
Total 
Central 'triangle' 
1 
3 
0 
4 
1st group of triangles 
4 + 8 = 12 
8×3 = 24 
8 
44 
2nd group of triangles 
6 + 10 = 16 
10×3 = 30 
10 
56 
3rd group of triangles 
2 + 10 = 12 
10×3 = 30 
10 
52 
4th group of triangles 
28 
14×3 = 42 
14 
84 
Total 
69 
129

42 
240 
9
Mathematically speaking, the central,
downwardpointing, white triangle does not count as a true, triangular area (hence the
two inverted commas enclosing the word ‘triangle’ in the first row of the table) because the
bindu at the centre of the 2dimensional Sri Yantra is a point that lies in its
plane, so that this triangle does not enclose an area of space but is really only three
lines and three points, the upper two of which coincide with corners of violet triangles
belonging to the first group. The 42 triangles surrounding the centre have 126 sides, i.e.,
they comprise 168 lines & triangles. Including the lowest corner of the central
‘triangle,’ 69 points surround the centre of the 2dimensional Sri Yantra. Three straight
lines create the shape of the innermost 'triangle.' Hence, the 240 points, lines &
triangles that surround its centre comprise (69+3=72) points & lines and
168 lines & triangles. This is the counterpart of the similar
72:168 division manifested by the two separate sets of seven enfolded polygons
making up the inner Tree of Life and by the first three Platonic solids. We shall see in the
next section that it also appears in the disdyakis triacontahedron — the outer form of the
polyhedral Tree of Life. It is the counterpart of the 72 roots of E_{6} and
the remaining 168 roots of E_{8}. Notice that the 168 lines &
triangles comprise the 42 lines in the fourth (red) group of triangles and 42 triangles,
i.e., 84 lines & triangles, as well as 84 lines in the first three groups.
Alternatively, the four groups of triangles in each half of the Sri Yantra comprise 63 sides
and 21 triangles, i.e., 84 sides & triangles. This 84:84 division is
characteristic of sacred geometries, e.g., in the case of the two sets of seven enfolded
polygons discussed in Section 1, there are 84 hexagonal yods on the sides of the polygons in
each set.
The first (violet) & third (green) groups
of triangles have (12+12=24) vertices. The central ‘triangle’ comprises one unshared vertex
and three straight lines. They group into two sets: one that is made up of this vertex and
the straight line opposite it and one that comprises the two lines that meet at this point.
The second (blue) & fourth (red) groups of triangles have (16+28=44) vertices, i.e., 22
vertices in each half of the Sri Yantra, making two sets of 22 vertices. Combining them with
the set of point & line and with the set of two lines creates two sets of 24 vertices
& sides. Hence, the 72 points & lines discussed in the previous paragraph
divide into three sets of 24:
1. 1st & 3rd groups have 24
vertices;
2. Central ‘triangle,’ 2nd & 4th groups
have 23 vertices & one side;
3. Central ‘triangle,’ 2nd & 4th groups
have 22 vertices & two sides.
Figure 8. The Sri Yantra.
The
third group of triangles has ten sets of three sides and the fourth group has 14 sets of
three sides. Together, they form three sets of (10+14=24) sides and one set of (10+14=24)
triangles. The first and second groups in the Sri Yantra have (8+10=18) triangles, i.e.,
nine in each half. The second group has 15 sides in each half. There are
therefore (9+15=24) sides & triangles in each half of the first and second
groups, as well as a set of 24 sides in the former group. The 168 sides &
triangles in the four groups of triangles divide into seven sets of 24:
1. 3rd & 4th groups have set of 24
sides;
2. 1st & 2nd groups have 24 sides &
triangles in one half of Sri Yantra;
3. 1st & 2nd groups have 24 sides &
triangles in other half of Sri Yantra;
4. 3rd & 4th
groups have set of 24 sides;
5. 3rd & 4th groups have set of 24
sides;
6. 3rd & 4th
groups have set of 24 sides;
10
7. 3rd & 4th groups have set of 24
triangles;
In conclusion, the 240 geometrical elements
surrounding the central bindu in the 2dimensional Sri Yantra consist of ten sets of 24
elements that divide into three sets of vertices & lines and into seven sets of sides
& triangles. They are the counterpart of the three major whorls and seven minor whorls
of the UPA/subquark.
4. The disdyakis
triacontahedron As the outer form of the
Polyhedral Tree of Life (see Article 23), the disdyakis triacontahedron is a Catalan solid
[1]
with 62 vertices, 180 edges & 120 triangular faces. The vertices comprise 30 A
vertices, 12 B vertices & 20 C vertices (Fig. 9). The 180 edges consist of 60 AB, 60 BC
& 60 AC edges. When its axis is a straight line passing through two diametrically
opposite A vertices, its central 12gon is formed by four A, four B & four C vertices.
Four B vertices are above this plane and four B vertices are below it. Hence, there are
three sets of four B vertices. Figure 8 shows that the 12 B vertices are those of an
icosahedron
Figure 9. The rhombic
triacontahedron & the disdyakis triacontahedron.
with 30 green edges, whilst the 20 C vertices are those of a
dodecahedron with 30 red edges. The 32 B & C vertices forming a rhombic triacontahedron
with 30 Golden Rhombic faces. Three AC edges (raised, red halfedges of the dodecahedron)
are associated with each of the 20 icosahedral triangular faces with three green sides.
Hence, the 60 AC edges are composed of three sets of 20 AC edges. Adding them to the three
sets of B vertices generates three sets of 24 geometrical elements (four B vertices & 20
AC edges):
1. 4
B vertices in upper half & AC edge associated with 20 icosahedral faces;
2. 4 B vertices in central 12gon & AC
edge associated with 20 icosahedral faces;
3. 4 B vertices in lower half & AC edge
associated with 20 icosahedral faces.
Associated with each of the 12 faces of the
underlying red dodecahedron are five green halfedges, which become raised into five AB
edges, and five blue edges, which are BC edges (see the rhombic triacontahedron). Hence,
there are 12 sets of five AB edges and 12 sets of five BC edges. This may be seen as five
sets of 12 AB edges and five sets of 12 BC edges, i.e., as five sets of 24 edges (12 AB
& 12 BC).
The 20 C vertices consist of 10 C vertices in
one half of the polyhedron and 10 C vertices in its other half. The 28 A vertices
surrounding the axis passing through two opposite A vertices comprise 14 A vertices in each
half. Hence, (10+14=24) A & C vertices surround the axis in each half, i.e., there are
two sets of 24 A & C vertices.
There are therefore seven more sets of 24
geometrical elements: two sets of 24 A & C vertices and five sets of 12 AB & 12 BC
edges.
Surrounding the AA axis of the disdyakis
triacontahedron are (60+180=240) vertices & edges. They consist of 72 B vertices
& AC edges grouped into three sets of 24 and 168 vertices & edges grouped
into seven sets of 24 (two sets of 24 vertices and five sets of edges). In other words, the
240 vertices & edges form ten sets of 24 that divide into groups of 72 and
168. This is the same pattern as found earlier for the inner Tree of Life, the two
Type A dodecagons, the two Type decagons, the first three Platonic solids and the
2dimensional Sri Yantra. It is intrinsic to all sacred geometries because they
embody the root
[1]
Catalan solids are the duals of the 13 different Archimedean
solids created by replacing their vertices with faces and vice versa.
11
composition of the superstring gauge symmetry
group E_{8} and its subgroups, in particular E_{6}. This fundamental 10fold
division manifests in the subquark state of the E_{8}×E_{8} heterotic
superstring as the ten whorls of the UPA described by Besant & Leadbeater. The symmetry
breakdown from E_{8} to E_{6} appears in the difference that Leadbeater
reported between the major and minor whorls, which carry, respectively, the
72 E_{8} gauge charges of E_{6} and the
remaining 168 gauge charges of
E_{8}. The major whorls are made thicker by the
inclusion of an extra (n+1)thorder spirilla in every 25 nth spirillae, so that, instead of
the latter being composed of 175 (n+1)thorder spirillae (as in the minor whorls), in a
major whorl they contain 176 (n+1)thorder spirillae, where n = 16).
10fold geometrical composition of the disdyakis triacontahedron
Article 27 showed that, when both the internal triangles and the
faces of the disdyakis triacontahedron are Type A triangles, 2400 corners, sides &
triangles surround any axis passing through two diametrically opposite vertices (see Table
1, p. 4). It will now be proved that the 10fold division of vertices & edges found
above for this polyhedron extends to its total geometrical composition. In other
words, the 2400 geometrical elements divide naturally into ten sets of 240 elements
that divide further into three sets of 240 and seven sets of
240. The meaning of this 10fold division is that, associated with the 240 roots of
E_{8} are 240 gauge fields, each with ten components, so that 240 components of
these fields are defined by each of the 10 dimensions of superstring spacetime. Each
geometrical element making up the disdyakis triacontahedron that surrounds its axis
therefore bears a formal correspondence to a component of the 240 gauge fields of
E_{8}.
Figure 10 shows one of the 120 faces with
centre O. a, b & c are the internal sides of its three sectors that terminate,
respectively, on A, B & C vertices. It also depicts an internal triangle with polyhedral
vertices at two corners and the centre of the polyhedron at its third corner. The three
sides meeting at O′ are of two types: one (X) that terminates on a vertex and one (Y) that
ends at the centre. The three sectors of an internal triangle are of two types: one (p) that
has a polyhedral edge as one side and two (q) that do not have an edge as a side. The edges
are of three types: AB, BC & AC. As the disdyakis triacontahedron has 180 edges, there
are 60 edges of each type, each one generating in an internal triangle two X sides & one
Y side, one triangle of class p & two triangles of class q, as well as a centre O′.
Table 4 shows the numbers of geometrical elements in the faces and interior of the
polyhedron that surround its axis:
Table 4. Geometrical composition of
the disdyakis triacontahedron.
In the faces are three sets of 120 sides and
three sets of 120 triangles, i.e., three sets of 240 sides & triangles, or 720
(=72×10) geometrical elements. There are also 120
corners that are centres of faces and 60 vertices, as well
as 60 sides joining the latter to the centre of the
polyhedron. These form a set of 240 corners & sides. There are three sets of 120 Y sides
& 120 q triangles (indicated in orange, yellow & green), i.e., three sets of 240
geometrical elements, and there are three sets of 60 edges, 60 internal corners, 60 internal
sides & 60 internal triangles (indicated in blue, indigo & violet), i.e., three sets
of 240 elements. Hence, there are seven more sets of 240 geometrical elements.
We see
that the 2400 geometrical elements comprise ten sets of 240 that are made up of three sets
in the faces and seven sets that comprise two triplets of sets and a single set. This
3:3:3:1 pattern reproduces the pattern of Sephiroth of the Tree of Life:
12
Figure
11. The 720:240:720 pattern of yods in the two sets of seven polygons and the 1tree
corresponds to the pattern of distribution of the 1680 geometrical elements that surround an
axis of the disdyakis triacontahedron.
13
KetherChokmahBinah
ChesedGeburahTiphareth
NetzachHodYesod Malkuth
There are 720 elements in the faces that are
unshared with the remaining 1680 elements. This pattern is the counterpart of the 72
roots of E_{6} and the 168 other roots of E_{8}. The 1680 elements
comprise 60 vertices & 180 edges, i.e., 240 elements in the faces, leaving 1440 elements
in the interior, 720 elements being in each half of the disdyakis triacontahedron. The
division:
1680 = 720 + 240 + 720
Is embodied in the 240 yods other than SLs in
the 1tree and the 720 yods that surround the centres of the two sets of seven Type B
polygons of the inner Tree of Life (Fig. 11). Every yod or geometrical element corresponds
to a circular turn of each helical whorl of the UPA/subquark superstring.
5. The
1tree
The 10fold UPA/superstring as the 10fold
division in the 1tree
When
its 19 triangles are turned into Type A triangles, i.e., when each triangle is divided into
its three sectors which are then changed into tetractyses, the 1tree contains 251 yods
(Fig. 12). Their 11 corners are “Sephirothic levels” (SLs), as Daath is Yesod of the next
higher Tree. Hence, the transformation generates (251−11=240)
yods that belong to (19×3=57) tetractyses. The 19 triangles have 25 sides. Inside them are
(19×3=57) sides of tetractyses.
(251−57=194) yods are on the (25+57=82) sides of these tetractyses,
where 194 is the number value of Tzadekh, the
Mundane Chakra of Chesed (see Table 1). The
Figure
12. The 251 yods in the 1tree generated when its 19 triangles are Type A. (Circles denote
yods that are directly behind other yods).
194 yods
comprise (2×82=164) hexagonal yods and (11+19=30) corners. The 1tree contains (251−30=221)
hexagonal yods.
Figure 13. The 1tree embodies the
248 roots of E_{8}.
The 1tree is the lowest Tree of a set of overlapping Trees of Life. Below its apex (Kether)
and outside the 1tree on each side are four red hexagonal yods (Fig. 13). One is at the centre
of a tetractys, two are on a side of this tetractys and another is on the vertical side of
another tetractys. These eight hexagonal yods belonging to the next higher Tree of Life consist
of two that are at centres of two tetractyses and six that are on four sides of four
tetractyses. Below the apex of the 1tree are therefore eight red, hexagonal yods that belong
to the second Tree and 240 black yods other than Sephiroth that belong to the lowest
Tree,
14
that is, 248 yods other than
Sephiroth, where 248 is the number value of Raziel, the Archangel of Chokmah.
The eight red hexagonal yods outside the 1tree
symbolize the eight simple roots of E_{8}, the rank8, exceptional Lie
group with dimension 248; the 240 black yods denote its 240 roots. As the latter include 221
hexagonal yods in the 1tree, there are (8+221=229) hexagonal yods in the
248 yods that
are below the top of the 1‑tree. 229 is the 50th prime number, showing how
ELOHIM, the Godname of Binah with number value 50, prescribes the number of
hexagonal yods below the top of the lowest Tree of Life. As the eight hexagonal yods outside
it belong to four tetractyses, the 248 yods belong to (57+4=61)
tetractyses, where 61 is the 31st odd integer and
31 is the
number value of EL, the Godname of Chesed. They comprise 19 corners, (57+2=59) hexagonal
yods at centres of 61 tetractyses & (164+6=170) hexagonal yods on their (82+4=86)
sides.
Figure 14. The Tree of Life & its
trunk.
The trunk of the
Tree of Life (Fig. 14) consists of the mathematical sequence of the point (Kether), straight
line (ChokmahBinah Path), triangle (corners at Chesed, Geburah & Tiphareth) &
tetrahedron (vertices at Netzach, Hod, Yesod & Malkuth). It expresses the emergence of
basic, 3dimensional form from the dimensionless point. The ten points in space needed to
achieve this represent the integers 1, 2, 3 & 4, which are also symbolized by the four
rows of the Pythagorean tetractys. The trunk is composed of 26
geometrical elements (ten points, ten lines, five triangles & one tetrahedron),
where 26 is the number value of YAHWEH, the Godname of
Chokmah (see Table 1). When
these triangles are Type A triangles, the trunk has
15 corners of 15 triangular sectors with 25 sides,
i.e., 55 geometrical elements, where 55 is the tenth triangular
number:
55 = 1 + 2 + 3 +
4 + 5 + 6 + 7 + 8 + 9 + 10
and the tenth Fibonacci number. This
demonstrates how the Decad (10) arithmetically determines the geometrical composition of the
trunk of the Tree of Life that represents this number. Table 5 lists the numbers of
yods in the trunk other than those coinciding with Sephiroth:
Table 5. The nonSephirothic yod population of the trunk of the Tree of Life.

Corners 
Hexagonal yods
at centre 
Hexagonal yods
on sides 
Total 
Point 
0 
0 
0 
0 
Line 
0 
0 
2 
2 
Triangle 
1 
3 
12 
4 + 12 
Tetrahedron 
4 
12 
12 + 24 = 36 
4 + (12+12) + 24 
Total 
5 
15 
50 
(2+4+4=10) + 12 + 24 + 24 = 70 
ADONAI,
the Godname of Malkuth with number value 65, prescribes the trunk because its
15 tetractyses have 65 hexagonal yods. ELOHIM, the Godname of Binah with
number value 50, shapes the trunk because 50 hexagonal yods line its 25 sides.
The branches of the 1tree are all those parts of it other than its trunk. They have
(240−70=170) yods. As the 240 yods comprise 19 corners of 57 tetractyses & 221 hexagonal
yods, the branches contain (19−5=14) corners & (221−65=156) hexagonal yods in
(57−15=42) tetractyses, i.e., (14+156=170) yods. The distinction between the trunk
& branches of the 1tree therefore generates the division: 240 = 70 + 170, where 70 = 5
corners + 65 hexagonal yods and 170 = 14
corners + 156 hexagonal yods. The 240 yods include (19+57=76) corners & centres
of tetractyses, where 76 is the number value of YAHWEH ELOHIM, the Godname of
Tiphareth.
According to Table 5, the numbers of yods in
the trunk are:
70 = 10 + 12 + 24 + 24.
Suppose that we add the two hexagonal yods on
the Path joining Daath and Tiphareth. The projection of this line onto the plane of the
(7+7) enfolded polygons making up the inner Tree of Life is the side that they all
share — what we have called the "root edge." This Path is its counterpart in the outer Tree
of Life. It is what may be called the "root" of the Tree of Life. Its trunk and root
constitute a set of 72 yods, where
72 = (10+2) + 12 + 24 + 24 = (12+12) + 24 + 24 = 24 + 24 + 24.
The 72 yods consist of three sets of
24 yods (Fig. 15). This leaves (170−2=168) yods in the branches made up of 14
corners and 154 hexagonal yods in 14 Type A triangles with two hexagonal yods on each of
their 14 sides (there are actually 15 sides but one is the line between Daath &
Tiphareth,
15
Figure 15. The 1tree embodies the 240 roots of the superstring gauge symmetry E_{8}.
Circles represent yods that are behind other yods.
the two hexagonal yods on which have been
added to the 70 yods in the trunk). There are nine hexagonal yods inside each Type A
triangle, so that the 168 yods comprise 14
corners, 14 sets of nine hexagonal yods and 14 pairs of hexagonal yods, i.e., 14 sets of 12
yods, or seven sets of 24
yods. Each set of 24 yods comprises two centres of
triangles of the 1tree, four hexagonal yods on two Paths & 18 hexagonal yods inside two
triangles. The latter consist of six hexagonal yods at centres of six tetractyses & 12
hexagonal yods on six sides of them, so that the 24 yods comprise two corners of
tetractyses, six hexagonal yods at their centres and (4+12=16) hexagonal yods on their
sides. This 2:6:16 pattern is identical to what is found for the 24 transverse
dimensions of bosonic strings, for they comprise two largescale dimensions, six dimensions
of the compactified space of superstrings and 16 higher dimensions. The following
correspondences exist:

2 corners of tetractyses ↔ 2 largescale, transverse
dimensions;

6 hexagonal yods at centres of tetractyses ↔ 6
superstring, compactified dimensions;

16 hexagonal yods on sides of tetractyses ↔ 16
bosonic, compactified dimensions.
This is not coincidental, because the 240 yods
symbolize the 240 roots of E_{8}, whose associated gauge charges are spread along the
ten whorls of the E_{8}×E_{8} heterotic superstring, ten to a whorl, and each
gauge charge is the result of an oscillation along each of the 24 transverse dimensions of
space. Nevertheless, this analogy between the hidden yod composition of the 1tree and the
pattern of dimensions of space predicted by the theories of bosonic strings and superstrings is
remarkable.
Notice that the
72 yods, which symbolize the 72 roots of E_{6}, the rank6,
exceptional subgroup of E_{8}, have
16
Figure 16.
The (7+7) triangles making up the branches of the 1tree.
been compounded from a set of 70 yods making up the trunk of the 1tree and the two hexagonal
yods on the DaathTiphareth Path that constitutes its root. Figure 1 indicates that the
72 roots also consist of a set of 70 roots and two roots represented by the
8tuples (½, ½, ½, ½, ½, ½, ½, ½) and (½, ½, ½, ½, ½, ½, ½, ½). So the
grouptheoretical composition of the roots of E_{6} is paralleled by the yod
composition of the trunk and the root of the Tree of Life.
The 240 yods needed to construct the 1tree
from Type A triangles divide into ten sets of 24 yods, which divide further into three
sets of 24 and seven sets of 24. This 3:7 division is the Tree of Life counterpart of the
three major whorls and seven minor whorls of the UPA/subquark state of the
E_{8}×E_{8} heterotic superstring (see Fig. 2). The three major whorls are
the manifestation of the root and trunk of the 1tree and the seven minor whorls are the
manifestation of its branches. The factorization: 168 = 7×24 is made possible by the
facts that: 1. the branches of the 1tree contain 14 triangles, and 2. each side of the
1tree is the mirror image of the other. Five triangles (coloured dark or light shades of
red, orange, yellow, green & blue) are on each side of the central Pillar of
Equilibrium, which is straddled by four triangles coloured light & dark shades of indigo
and violet (Fig. 16). The dark shades of these two colours can be associated with one side
of the 1tree and the light shades can be associated with the other side. Therefore, the
factor of seven in the factorization arises simply from the fact that seven triangles can be
associated with the lefthand side of the 1tree and seven triangles can be associated with
its righthand side. There are two hexagonal yods on a side shared by two triangles. One
hexagonal yod is associated with one triangle and the second one should be regarded as
associated with the other triangle. This allows 12 yods to be associated with each triangle.
Each set of 24 yods consists of 12 yods associated with a triangle in each side of the
1tree, so that the 168 yods divide into two sets of 84 yods associated with each
half of it. This 84:84 division was encountered in earlier analyses of sacred geometries in
this article and has been discussed in previous articles in many different contexts. It
manifests in the subquark as the outer and inner halves of each helical whorl of the UPA,
each having 840 circular turns that make 2½ revolutions around its axis of spin, that is,
168 turns make a halfrevolution and 84 turns make a quarterrevolution. Figure 1
indicates that the 168 roots of E_{8} that are not roots of E_{6}
consist of three sets of 28 (i.e., 84 roots), a set of 56 & another set of 28 (i.e., 84
roots). So these 168 roots divide naturally into two sets of 84 roots. Their
counterparts in the branches of the 1tree are the 84 yods that belong to the seven
triangles associated with its lefthand or righthand half.
Four corners of tetractyses, four hexagonal
yods at their centres & eight hexagonal yods on their sides (i.e., 16 yods) are aligned
with the central Pillar of Equilibrium straddled by the four triangles. Hence,
(168−16=152) yods belong to the branches in both halves of the 1tree. 76 yods
in the branches belong to each half, where 76 is the number value of YAHWEH ELOHIM,
the Godname of Tiphareth. The eight hexagonal yods symbolizing the eight simple roots of
E_{8} consist of four yods outside the 1tree on either side of the Pillar
of Equilibrium (see Fig. 13). Outside the trunk are (76+4=80) yods on either
side, where 80 is the number value of Yesod. This is also the number of yods in the
trunk when the (10+1) Sephirothic corners of the 19 triangles are included.
The 1tree consists of 30 corners of 57
triangles with 82 sides, i.e., 112 corners & sides,
where 112 is the number value
of Beni Elohim, the Order of Angels assigned to Hod, and 87 corners & triangles,
where 87 is the number value of Levanah, the Mundane Chakra of
Yesod. The 1tree consists of 169 points, lines & triangles. In other words, below its
apex are 168 geometrical elements. The number value
of Cholem Yesodeth, the Mundane Chakra
of Malkuth, is the number of geometrical elements needed to construct the 1tree,
starting from the mathematical point represented by its apex. Here is yet more, truly
amazing evidence that the gematria number values of the Hebrew names of the Sephiroth,
their Godnames, etc have a geometrical basis.
In conclusion, the natural, tenfold splitting
into ten sets of 24 geometrical elements of the number 240 embodied in five sacred
geometries has its realization in the microscopic world as the ten whorls of the
E_{8}×E_{8} heterotic superstring, each carrying 24 gauge charges of
E_{8}, as described over a century ago by Annie Besant & C.W. Leadbeater in the
particle that they claimed is the basic unit of matter.
Endnotes Proof that 16800 is the sum of 240 odd integers
Using the identity:
17
168 = 13^{2} – 1,
16800 = 168×10^{2} =
(13^{2}– 1)10^{2} = 130^{2} – 10^{2} =
(2×65)^{2} – (2×5)^{2} = 4(65^{2} −
5^{2}).
As
65^{2} = 1 + 3 + 5 +... +
129
and
5^{2} = 1 + 3 + 5 + 7 + 9,
therefore,
16800 = 4(11 + 13 + 15 +... +
129),
i.e., 16800 is the sum of [4×(65–5) =
4×60 = 240] odd integers, the largest of which is 129. This is the number of
YAHWEH SABAOTH, the Godname of Netzach, which is the fourth Sephirah of
Construction, counting from Malkuth (see Table 1).
18
