ARTICLE 64
by
Stephen M. Phillips
Flat 4, "Oakwood House," 117119 West Hill Road. Bournemouth. BH2 5PH.
England.
Website: http://smphillips.mysite.com
Abstract
Previous articles have revealed in sacred geometries
analogous patterns of division of shared parameters. This article will analyse
those divisions that have particular relevance to the account given by Annie
Besant and C.W. Leadbeater of their remote viewing of what they took to be the
basic constituents of atoms. The divisions:
48 = 24 + 24,
336 = 168 + 168
and
168 = 84 + 84
in these shared numbers will be examined for the
following examples of holistic systems/sacred geometries:
1. the inner form of the Tree of Life;
2. the dodecagon;
3. the Sri Yantra;
4. the 64 hexagrams known to students of the I
Ching;
5. the five Platonic solids;
6. the disdyakis triacontahedron;
7. the {3,7} tessellation of the 3torus;
8. the seven diatonic musical scales.
The exact correspondences in their properties are too
extensive to be attributed, plausibly, to chance. Instead, these holistic
systems should be recognised as triangular, polygonal, polyhedral, topological
or musical manifestations of a universal, mathematical pattern that governs such
systems and which is realised in the subatomic world in the
E_{8}×E_{8} heterotic superstring that was
paranormally described over a century ago by Besant and
Leadbeater.

1
Table 1. Gematria number values of the ten Sephiroth in the four Worlds.

SEPHIRAH

GODNAME

ARCHANGEL

ORDER OF ANGELS

MUNDANE CHAKRA

1

Kether
(Crown)
620

EHYEH
(I am)
21

Metatron
(Angel of the
Presence)
314

Chaioth ha
Qadesh
(Holy Living Creatures)
833

Rashith ha
Gilgalim
First
Swirlings.
(Primum
Mobile)
636

2

Chokmah
(Wisdom)
73

YAHWEH,
YAH
(The
Lord)
26, 15

Raziel
(Herald of the
Deity)
248

Auphanim
(Wheels)
187

Masloth
(The Sphere
of
the
Zodiac)
140

3

Binah
(Understanding)
67

ELOHIM
(God in
multiplicity)
50

Tzaphkiel
(Contemplation of
God)
311

Aralim
(Thrones)
282

Shabathai
Rest.
(Saturn)
317


Daath
(Knowledge)
474





4

Chesed
(Mercy)
72

EL
(God)
31

Tzadkiel
(Benevolence of
God)
62

Chasmalim
(Shining
Ones)
428

Tzadekh
Righteousness.
(Jupiter)
194

5

Geburah
(Severity)
216

ELOHA
(The
Almighty)
36

Samael
(Severity of
God)
131

Seraphim
(Fiery
Serpents)
630

Madim
Vehement
Strength.
(Mars)
95

6

Tiphareth
(Beauty)
1081

YAHWEH
ELOHIM
(God the
Creator)
76

Michael
(Like unto
God)
101

Malachim
(Kings)
140

Shemesh
The Solar
Light.
(Sun)
640

7

Netzach
(Victory)
148

YAHWEH
SABAOTH
(Lord of
Hosts)
129

Haniel
(Grace of
God)
97

Tarshishim
or
Elohim
1260

Nogah
Glittering
Splendour.
(Venus)
64

8

Hod
(Glory)
15

ELOHIM
SABAOTH
(God of
Hosts)
153

Raphael
(Divine
Physician)
311

Beni Elohim
(Sons of
God)
112

Kokab
The Stellar
Light.
(Mercury)
48

9

Yesod
(Foundation)
80

SHADDAI EL
CHAI
(Almighty Living God)
49, 363

Gabriel
(Strong Man of
God)
246

Cherubim
(The
Strong)
272

Levanah
The Lunar
Flame.
(Moon)
87

10

Malkuth
(Kingdom)
496

ADONAI
MELEKH
(The Lord and
King)
65, 155

Sandalphon
(Manifest
Messiah)
280

Ashim
(Souls of
Fire)
351

Cholem
Yesodeth
The Breaker of the
Foundations.
The
Elements.
(Earth)
168

(All numbers in this table that the article
refers to are written in boldface).
The Sephiroth exist in the
four Worlds of Atziluth, Beriah, Yetzirah and Assiyah. Corresponding to them
are the Godnames, Archangels, Orders of Angels and Mundane Chakras (their
physical manifestation). This table gives their number values obtained by the
ancient practice of gematria, wherein a number is assigned to each letter of
the alphabet, thereby giving a number value to a word that is the sum of the
numbers of its letters. Numbers from the table are written
in boldface in
the article.

2
1.
Introduction
Previous articles have established that sacred geometries
embody in their mathematical properties a common set of patterns relating a shared set of
structural parameters. Some of these numbers — like 26, 248 and 496 — have become
established in the research literature of theoretical physics in the form of superstring
physics, although whether they apply to the real world remains experimentally uncertain,
despite the beautiful, mathematical ideas that surround them. Other parameters have not
(yet) appeared, giving sceptics scope to dismiss as the product of chance all evidence of
correlations between these geometries and recent discoveries in theoretical physics. This
article will analyse and compare those mathematical correspondences that have particular
relevance to the remoteviewing investigations of subatomic particles by Annie Besant and
C.W. Leadbeater. The number relationships in question are:
336 = 168 + 168,
168 = 84 + 84 = 24 +
144.
The upward continuations of these divisions:
672 = 336 + 336
and
1344 = 672 + 672,
also appear
in sacred geometries. As they have been analysed in detail and given scientific
interpretation at the author’s website [1], they need not be discussed here. The purpose of
this article is more than just to map out the correspondences. It is to convince the reader
that they are too
detailed and
too
Figure 1.
The inner form of the Tree of Life.


Figure
2. Each set of 7 enfolded polygons in the inner Tree of Life shares with its
outer form three corners & two sides of its triangles (coloured
green).

pervasive to
be due to chance. The fact that this demonstration provides highly convincing
support for the objective character of the paranormal observations of Besant &
Leadbeater is incidental, although highly pertinent.
2. The Inner Tree of
Life Implicate in the geometry of the outer form
of the Tree of Life (Etz Hayim, עץ חיים) known to traditional Kabbalah is its inner
form. It consists of two sets of seven enfolded, regular polygons:
triangle,
square, pentagon, hexagon, octagon, decagon, dodecagon.

Figure 3. The 7 separate
polygons with 48 corners is divided into two sets, each with 24
corners. 


Figure 4. The two sets of
enfolded polygons: S_{1} & S_{2}.

Figure 5. The two sets of
enfolded polygons: S_{1} & S_{2} and their mirror
images S_{1}′ &
S_{2}′.

3
The polygons in one set are the mirror images of their
counterparts in the other set (Fig. 1) and share the socalled “root edge” separating the
two sets. The 14 polygons lie in the plane formed by the two side pillars of the outer Tree
of Life, namely, the righthand Pillar of Mercy and the lefthand Pillar of Judgement (Fig.
2). The top, bottom and centre of each hexagon coincides with corners of triangles belonging
to the outer Tree of Life, and the two internal, vertical sides of their sectors are sides
of these triangles. Hence, 10 of the geometrical elements making up the inner Tree of Life
are shared with its outer form.
It has been pointed out in previous articles that these seven
separate, regular polygons with 48 corners comprise two
sets of polygons with 24 corners: the first set (let us call it “S_{1}”) consists of the triangle, square, pentagon &
dodecagon; the second set (call it “S_{2}”) consists of the hexagon, octagon & decagon (Fig. 3). (remember
that in
situ (Fig. 4), S_{2} has the sector of its hexagon missing that is
occupied by the violet triangle in S_{1} because all the polygons lie in the same plane).
Similarly, Figure 5 indicates that each red polygon in S_{1} has its mirrorimage in S_{1}′ and that each blue polygon in
S_{2} has its mirror image in
S_{2}′. As we shall discover later, two
other sets of polygons with 24 corners exist, but — unlike S_{1}
& S_{2} — they do
not exactly divide in half the
collective properties of the seven polygons — whether the latter are separated or
enfolded.
This 24:24 division of the parameter
48 is characteristic of all holistic systems. It manifests
in the 4‑dimensional projection of the 4_{21} polytope as the 120 vertices of each of the two
600cells whose compound is the 4dimensional projection of this polytope in the Coexter
plane [1]. This is because the 120 hexagonal yods in the four separate polygons of
S_{1} constructed from tetractyses
symbolise the 120 vertices in one 600cell and the 120 hexagonal yods in the three polygons
of S_{2} symbolise the 120 vertices
in the other 600cell. If the inner form of the Tree of Life is thought of as the "DNA"
encoding how its outer form replicates, then S_{1} and S_{2} are the "strands" of this DNA and have their
microscopic realisation in the outer and inner halves of the 10 helical whorls of the
UPA/E_{8}×E_{8} heterotic superstring, the two halves being generated
by the 240 vertices of the two 600cells (or, rather, the 240 E_{8} gauge charges that these vertices represent). It is
worthwhile, therefore, to examine the geometrical and yod compositions of
S_{1} and
S_{2}. Later, it will be shown that
these combinations are unique in the sense that no other grouping of the seven polygons
into two sets with equal numbers of yods and equal numbers of geometrical elements leads
to equal numbers when they become enfolded in one another as the inner form of the Tree
of Life.
Geometrical composition of
S_{1} & S_{2}
Listed in Table 2 are the numbers of corners, sides &
triangles in the seven separate, regular polygons of the inner
Tree of Life when they are divided into their sectors:
Table 2. Geometrical composition of the seven separate
Type A polygons.

Triangle

Square

Pentagon

Hexagon

Octagon

Decagon

Dodecagon

Total

Corners

4

5

6

7

9

11

13

55

Sides

6

8

10

12

16

20

24

96

Triangles

3

4

5

6

8

10

12

48

Total

13

17

21

25

33

41

49

199

When they are
separate, S_{1} and S_{2} have — by definition — the same number of corners
(24), the same number of sides (48) and the same number of triangles (24). When they
are enfolded (Fig. 6), the righthand corner of the triangle coincides with the centre of
the hexagon, its two sloping sides coincide with sides of sectors of the hexagon and the
righthand corner of the pentagon coincides with the centre of the decagon. Associating the
root edge with
S_{1},


Figure 6.
The 7 enfolded polygons have 47 sectors with 41 corners & 88
sides. 
Figure
7. S_{1} (red polygons) & S_{2} (blue
polygons).

the
polygons in both S_{1} and S_{2} have 18 corners and 21 sides.
In terms of just their boundaries, S_{1} and S_{2} are the two halves of the
righthand half of the inner Tree of Life. The same is true for their respective mirror
images S_{1}′ and S_{2}′ in its lefthand
half. But what happens when their sectors are taken into account? Table 3 lists the
geometrical compositions of the root edge and what is outside it in S_{1} and in S_{2}
when their enfolded polygons are divided into their sectors (Fig. 7):
4
Table 3. Geometrical composition of
S_{1} & S_{2} outside the root edge.

Root edge

S_{1}

S_{2}

Total

Corners

2

20

19

41

Sides

1

44

43

88

Triangles

0

24

23

47

Total

3

88

85

176

S_{1} and S_{2} cannot have the same
number of corners of their sectors because the total number (41) is an odd integer, not an
even one, and is, therefore, not exactly divisible by 2. However, if we associate the root
edge with S_{2}, the latter has the same
number of sides (44), as well as the same number of corners & triangles (44) and the
same number of corners, sides & triangles (88), as S_{1}. Whether as combinations of just polygons or as combinations of polygons divided into
their sectors, S_{1} and S_{2} can be regarded as the two halves of the set of seven
enfolded polygons provided the root edge is assigned to S_{2}.
Intrinsic geometrical elements
The righthand corner of the triangle coincides with Chesed
in the Tree of Life, whilst the two internal vertical sides of sectors of the hexagon form
its side pillars. This means, referring to Table 2, that, outside the root edge,
S_{1} has 87 geometrical elements unshared with the Tree of Life
and that S_{2} has 81 unshared
geometrical elements, so that S_{1} & S_{2} have
(87+81=168) geometrical elements outside the root edge that
are intrinsic to them. The
number 87 is the number
of Levanah, the Mundane Chakra of
Yesod, which is the penultimate Sephirah. The number 168 is
the number of Cholem Yesodeth, the
Mundane Chakra of the last Sephirah. Amazingly, it is embodied in the inner Tree of Life in two different ways
because the topmost corner of the hexagon coincides with the lowest corner of the hexagon
enfolded in the next higher Tree, so that (176−1−7=168) elements intrinsic to each
set of seven polygons surround their centres.
Yod populations of
S_{1} & S_{2}
Let us now consider the yod compositions of
S_{1} and S_{2}. Listed in Table 4 are the numbers of yods in the
seven separate, regular polygons whose
sectors are constructed from tetractyses:
Table 4. Yod populations of the seven separate, Type A
polygons.

Triangle

Square

Pentagon

Hexagon

Octagon

Decagon

Dodecagon

Total

Corners

4

5

6

7

9

11

13

55

Hexagonal yods

15

20

25

30

40

50

60

240

Total

19

25

31

37

49

61

73

295

Table 5 lists the yod populations of the root edge and what
is outside it in S_{1} and in
S_{2}:
Table 5. Yod compositions of
S_{1} & S_{2}
outside the root edge.

Root edge

S_{1}

S_{2}

Total

Corners

2

20

19

41

Hexagonal yods

2

112

109

223

Total

4

132

128

264

If we associate the root edge
and its four yods with S_{2}, the latter has the same number
of yods (132) as S_{1}. In other words,
the yod population of the seven enfolded polygons then becomes evenly divided between
S_{1} and S_{2}, just as we saw earlier that their number of geometrical
elements is. The split of the seven separate polygons into two sets with the same numbers of
corners, sides & triangles surrounding their centres and the same numbers of yods
surrounding their centres persists when they are enfolded in the form of
S_{2} having the same number of yods as
S_{1} has outside the root
edge.
Proof of uniqueness of
S_{1} & S_{2} as the two halves of the seven
polygons
In terms of the numbers of corners of the seven regular
polygons making up the inner Tree of Life:
3, 4, 5, 6, 8, 10, 12
four combinations of numbers add up to
24:
1.
3 + 4 + 5 + 12
2.
6 + 8 + 10
3.
3 + 5 + 6 + 10
4.
4 + 8 + 12
5
This allows
two possible sets of polygons with 24 corners: either (1) & (2) or (3) & (4). The
sets (1) & (2) are, respectively, S_{1} and S_{2}. Let us call the
sets (3) & (4), respectively, S_{3} & S_{4}. The geometrical
compositions of the root edge and what is outside it in S_{3} and in
S_{4} are shown in Table 6:
Table 6. Geometrical compositions of S_{3} &
S_{4} outside the root edge.

Root edge

S_{3}

S_{4}

Total

Corners

2

18

21

41

Sides

1

42

45

88

Triangles

0

23

24

47

Total 
3

83

90

176

We see that
the geometrical compositions of S_{3} and S_{4} (either for each
type of geometrical element or in total) cannot be made equal by associating the three
geometrical elements in the root edge with either S_{3} and S_{4}.
Listed in Table 7 are their yod populations:
Table 7. Yod
populations of S_{3} & S_{4} outside the root
edge.

Root edge

S_{3}

S_{4}

Total

Corners

2

18

21

41

Hexagonal yods

2

107

114

223

Total

4

125

135

264

We see that their various
yod populations cannot be made equal by associating the four yods in the root edge with
either S_{3} or S_{4}. Only the 24:24 division of the seven separate
polygons into S_{1} and S_{2} creates equal total numbers of geometrical
elements and equal yod populations when the two combinations with 24 polygonal corners
become enfolded. This remarkable fact indicates that only S_{1} and S_{2}
can be regarded as true halves of the set of seven polygons and vindicates the use of only
these two combinations in various analyses in previous articles of the inner Tree of Life
basis of holistic parameters, such as the superstring structural parameters 1680, 336 &
168. As this has shown, they are in fact the only two sets of polygons that have
equal populations of geometrical elements and yods.

Figure 8. The root edge
with 4 yods separates the two sets of 7 enfolded polygons containing 260
yods. 
Further
evidence of the unique status of S_{1} and S_{2} visàvis the
inner Tree of Life is provided by the (S_{1}+S_{1}′) and the
(S_{2}+S_{2}′) making up the (7+7) enfolded polygons. According to Table 5,
(S_{1}+S_{1}′) have (2×132=264) yods outside the root edge and
(S_{2}+S_{2}′) have (2×128=256=4^{4}) yods outside the root edge,
i.e., 260 yods when the latter is included. Compare this with the fact that the seven
enfolded polygons have 264 yods, of which 260 yods lie outside the root edge (Fig. 8).
(S_{1}+S_{1}′) without the root edge have the yod population of the seven
enfolded polygons, i.e., one half of the inner Tree of Life, and
(S_{2}+S_{2}′) with the root edge have the yod population of the other set
of seven enfolded polygons outside their root edge. Table 8 lists the numbers of yods lining
the tetractyses in S_{1} and S_{2}:
Table 8.
Numbers of boundary yods in S_{1} & S_{2}.

S_{2}′

S_{1}′

Root edge

S_{1}

S_{2}

Yod
population

128

132

4

132

128

Number of hexagonal
yods at centres of tetractyses

23

24

−

24

23

Number of boundary
yods

105

108

4

108

105

6


Figure 9.
216 yods intrinsic to a set of 7
enfolded polygons line their 47 tetractyses. 
Figure 10.
216 yods outside the root edge
line the 48 tetractyses in S_{1} &
S_{1}′.

We see that
S_{1} and S_{1}′ have 216 boundary yods outside the root edge.
The seven enfolded polygons have 264 yods, of which 47 are hexagonal yods at centres of
tetractyses. This leaves 217 boundary yods, of which one is the top of the hexagon and
coincides with the lowest corner of the hexagon enfolded in the inner form of the next
higher Tree of Life. This means that 216 yods lining the sides of tetractyses
are intrinsic to the inner form of each overlapping Tree (Fig. 9). This is the number of
boundary yods in S_{1} & S_{1}′ (Fig. 10).
According to Table 8, S_{2} &
S_{2}′ have 214 boundary yods when the root edge is included. This is the number of
yods in the outer Tree of Life with Type A triangles (Fig. 11). S_{2} has 109 boundary yods when the
root edge is included. As it
contains the hexagon, the topmost corner of which coincides with the lowest corner of the
hexagon enfolded in the next higher Tree of Life, 108 such yods are intrinsic to S_{2}.
This is also the number of boundary yods in S_{1} outside the
root edge.
The Tree of Life contains
214 yods when its 16 triangles are Type A.

214 yods line the 46
tetractyses in S_{2} & S_{2}′.

Figure 11.
(S_{2}+S_{2}′) are lined by as many
yods as the Tree of Life contains yods when its triangles are Type
A.

Not only are
the yod populations of S_{1} and S_{2} equal, but so, too, are the numbers
of yods on the boundaries of their tetractyses when the root edge is associated with
S_{2}. The 216 boundary yods intrinsic to the seven enfolded polygons are
divided equally by S_{1} and S_{2}, whilst
(S_{1}+S_{1}′) and
(S_{2}+S_{2}′) embody the same holistic parameter. It is
further evidence that S_{1} and S_{2} uniquely embody properties that are
quantified by the same numbers that quantify properties of the complete form of the
inner Tree of Life. The implication of this is that these two halves of the inner Tree
should be regarded as holistic structures in themselves. It should, therefore, come as no
surprise that S_{1} and S_{2} can be regarded as representing the two
600cells that are the two halves of the Petrie projection onto the Coxeter plane of the
4_{21} polytope which in turn generate the outer and inner halves of the UPA, the
subquark state of the E_{8}×E_{8} heterotic superstring (see the discussion
under Fig. 4 here). They can
further be associated with the 120 vertices & edges of the 24cell making up each
600cell and with its 120 faces & octahedral cells, that is, with its 0 &
1polytopes and with its 2 & 3polytopes, for the polytope composition of the 24cell
conforms to the archetypal pattern of the seven regular polygons in the inner Tree of Life
(see beginning of #4 here). This 120:120
division exists in S_{1} and S_{2} not only when their polygons are regarded
as enfolded but also when they are separate. The number of hexagonal yods in an ngon is 5n,
so that each combination of separate polygons has 5×24 = 120 yods. Assigning a different
colour to each of the five hexagonal yods per sector of an ngon, every colour appears 24
times in S_{1} and 24 times in S_{2}. For the 600cell, each hexagonal yod
denotes a vertex of a 24cell, the five
7
colours
signifying the five such cells whose compound is the 600cell (Fig. 12).

Figure 12. The five sets of
24 hexagonal yods in S_{1} denote the five sets of 24 vertices in the
compound of five 24cells that is the 600cell. Similarly, for S_{2}.
(S_{1}+S_{2}) represents the 240 vertices of the compound of
two 600cells that is the Coxeter plane projection of the 4_{21}
polytope whose 240 vertices map the 240 roots of E_{8}. Similarly,
(S_{1}′+S_{2}′) represents the 240 roots of E_{8}′.
Therefore, the complete inner form of the Tree of Life represents
E_{8}×E_{8}′. 
The number of
corners, sides & triangles surrounding the centre of a Type B ngon is 10n. The 10
geometrical elements per sector comprise five sides and five corners & triangles. The
240 geometrical elements in either S_{1} or S_{2} that surround centres of
polygons comprise 120 sides and 120 corners & triangles (Fig. 13). So they display the
120:120 division characteristic of holistic systems in both their sides and their corners
& triangles.

Figure 13. Containing Type B polygons, S_{1} and S_{2} each has
240 geometrical elements comprising five sets of 24 sides and five sets of 24
corners & triangles. This is the geometrical analogue of the (240+240)
roots of E_{8}×E_{8}′. The five
types of sides, repeated 24 times, correspond to the five 24cells, each with
24 vertices, making up a 600cell. The five types of vertices or triangles,
repeated 24 times, correspond to the five 24cells making up a second 600cell.
S_{1} is geometrically analogous to the compound of two 600cells whose
240 vertices are the projections of the 240 vertices of the 4_{21}
polytope representing the 240 roots of E_{8}. Similarly, S_{2}
is analogous to E_{8}′. (S_{1}+S_{2}) is analogous to
E_{8}×E_{8}′. The division of the seven enfolded polygons into
S_{1} & S_{2} generates, therefore, a pattern of
geometrical elements that is analogous to the root composition of
E_{8}×E_{8}′. This is the symmetry group of one of the two
types of heterotic superstrings that has been shown to be free of quantum
anomalies.

Whether we
are speaking of hexagonal yods in Type A polygons or geometrical elements in Type B
polygons, the number 240 is present. In the case of the latter, because the 10 geometrical
elements per sector divide into two sets of five, S_{1} and S_{2}, the two
halves of the seven separate polygons, whose centres are surrounded by 480 geometrical
elements, each comprise 240 such elements. As the parameter 240 always refers in sacred
geometries to the 240 roots of E_{8}, we see that the two halves
signify the two identical E_{8} Lie groups that appear in the direct product
E_{8}×E_{8}, one of the two symmetry groups that were discovered in 1984 to
describe the interactions of superstrings that are free of quantum anomalies. We saw earlier that, even
though they are, separately, only half of the whole set of polygons making up the
inner Tree of Life, S_{1} and S_{2} possess properties quantified by numbers
that refer to this whole set. Here, once again, is evidence that S_{1} and
S_{2} can be regarded as wholes in themselves. They remind one of the division of
the Tao into the polar opposites of Yin and Yang — complementary components of the Whole,
yet complete in themselves.
Superstring structural parameters 84, 168 & 336 embodied in
S_{1} & S_{2}
According to
Table 3, S_{1} has 88 geometrical elements outside the root edge, i.e., 84 elements
surround the centres of its four polygons. S_{2} has 85 geometrical elements outside
the root edge. Of these, one is the centre of the octagon, the centres of the hexagon and
decagon being counted as belonging to S_{1} because the righthand corner of the
triangle coincides with the centre of the former and the righthand corner of the pentagon
coincides with the centre of the latter. Therefore, 84 geometrical elements surround the
centres of its three polygons. Surrounding the centres of S_{1} & S_{2}
in Figure 7 are (84+84=168) geometrical elements outside the root edge.
Similarly, (84+84=168) geometrical elements outside the root edge surround the
centres of their mirror images S_{1}' & S_{2}′.
(S_{1}+S_{2}) and (S_{1}′+S_{2}') consist of
(168+168=336)
8
geometrical
elements outside the root edge that surround the (5+5) centres that are not also corners of
polygons. They comprise four sets of 84 geometrical elements, so that
168 geometrical elements surround the centres of not only S_{1} and
its mirror image S_{1}′ but also S_{2} and its mirror image S_{2}′
(Figs. 14 & 15).


Figure 14. (84+84=168) geometrical elements surround the centres of the
eight polygons in S_{1}&
S_{1}′. 
Figure 15. (84+84=168) geometrical elements surround the centres of the six
polygons in S_{2} &
S_{2}′. 
According to Table 3, the seven
enfolded polygons have 176 geometrical elements, including the root edge. Therefore,
(176−7=169) geometrical elements surround their centres. One of them is the topmost corner
of the hexagon in S_{2}, which coincides with the lowest corner of the hexagon enfolded
in the next higher Tree. The number of geometrical elements surrounding the centres of the 7n
polygons enfolded in the ntree = 168n + 1. 168 such elements are intrinsic
to each set of polygons. S_{2} has 85 geometrical elements outside the root edge,
so that 83 elements intrinsic to it surround its centres. Including the root edge,
S_{1} has 91 geometrical elements, so that (91–4–2=85) elements are not centres.
The 168 intrinsic elements that are not centres comprise 85 elements in
S_{1} and 83 in S_{2}. The 84:84 division of the holistic parameter
168 exists for the generic set of seven polygons enfolded in a single Tree. It does
not exist in the inner forms of successive Trees of Life. The
S_{1}:S_{2} division does not lead to an 840:840 division in 10 Trees of
Life. But this should not be a surprise, because we know that holist systems embody the number
336, which divides into two halves, and this is not possible in this context because the number
168 includes the root edge, so that it cannot have a mirrorimage set of
168 elements. The correct embodiment in the inner Tree of Life that permits
generalisation to multiple Trees is the pattern of 168 geometrical elements outside
the root edge in each set that are unshared with the outer Tree of Life. S_{1} has
88 geometrical elements outside the root edge, of which one (corner of triangle) is shared with
the Tree of Life. The 168 elements intrinsic to the inner form of each Tree
comprise 87 elements in S_{1} and 81 elements in S_{2}.
87 is the number value of Levanah, the Mundane Chakra of Yesod, which is the
Sephirah preceding Malkuth, whose Mundane Chakra has the number value 168! Here is a
remarkable example of how the gematria number values of the Sephiroth, their Godnames, etc.
quantify properties of whatever is a holistic system (in this case, that part of the inner Tree
of Life that is intrinsic to it).
Both the
84:84 division of the holistic parameter 168 and the
168:168 division of the holistic parameter 336 have been discussed in
many earlier articles as characteristic of sacred geometries. Later sections of this article
will review their embodiment in the Sri Yantra, disdyakis triacontahedron, the four Platonic
solids, the table of 64 hexagrams used in I Ching, the seven diatonic, musical
scales and the 3torus. The 8:6 division of the 14 regular polygons making up the inner Tree
of Life splits in half the 336 intrinsic geometrical elements surrounding their centres. As
discussed in Section 10, we find a similar division in the 14 types of notes that make up
the seven diatonic scales [2] and whose tone ratios are:
1
256/243 9/8 32/27 81/64
4/3 1024/729 │ 729/512
3/2 128/81
27/16 16/9
243/128 2
The vertical
line divides the first seven notes from their inversions.[1] Eight
notes with their tone ratios written in black) constitute the Pythagorean scale and six
notes written in red are nonPythagorean notes. It occurs also in the outermost group of 14
triangles of the Sri Yantra. Six of these triangles touch the circular ring of eight lotus
petals that surrounds the 43 triangles and eight triangles do not touch it (see Fig.
34).
The inner
Tree of Life has 336 intrinsic geometrical elements outside the root edge surrounding the
centres of its 14 polygons. It is divided into two halves (S & S′), with S′ the mirror
image of S. Each half has 168 such geometrical elements and is further split into two
halves: S_{1} & S_{2}. Each "quarter" of the inner Tree of Life has 84
geometrical elements surrounding centres of polygons. Compare this with the following
structural properties of the UPA: each helical whorl with 1680 turns twists five times
around the axis of spin. It makes five halfrevolutions, each of 168 helical turns,
in the larger spiral of its outer half and five halfrevolutions, each of 168 turns,
in the narrower spiral of its inner half. Each turn is a circularly polarised oscillation
consisting of two perpendicular plane waves vibrating 90° out of phase, so that a
halfrevolution
[1] The inversion of a musical note with tone ratio m is a note with tone
ratio 2/m.
9
comprises 336
plane wave oscillations grouped into 168 pairs. A factor of 2 manifests in a
halfrevolution of a whorl, firstly, as its two quarterhalves and, secondly, as the two
perpendicular oscillations generating each turn. Two schemes of correspondence between the
inner Tree of Life and a halfrevolution of a whorl are possible:


Case
A

Case
B

168 oscillations

→ 
168 geometrical
elements in S

168 geometrical
elements in S_{1} &
S_{1}′

168 oscillations with 90° phase difference

→ 
168 geometrical
elements in S′

168 geometrical
elements in S_{2} &
S_{2}′

84 oscillations in first
¼revolution

→ 
84 geometrical elements in
S_{1}

84 geometrical
elements in S_{1}

84 oscillations in first
¼revolution with 90° phase difference

→ 
84 geometrical elements in
S_{1}′

84 geometrical
elements in S_{2}

84 oscillations in second
¼revolution

→ 
84 geometrical elements in
S_{2} 
84 geometrical
elements in S_{1}′

84 oscillations in second
¼revolution with 90° phase difference

→ 
84 geometrical
elements in S_{2}′

84 geometrical
elements in S_{2}′

Intuitively speaking, case A seems the correct correspondence, with S_{1} and
S_{2} denoting the two quarterrevolutions and the two halves of the inner Tree
corresponding to the two plane wave oscillations that are 90° out of phase. This is because
each half of the inner form of the Tree of Life constitutes a whole in itself and is a more
fitting representation of each set of 168 orthogonal plane waves that makes up the
halfrevolution of a whorl. This is consistent with the Tree of Life nature of a whorl because
the latter makes 10 halfrevolutions around the spin axis of the UPA, each being analogous to
one of the 10 Sephiroth, so that a whorl is represented by 10 Trees of Life, each corresponding
to a Sephirah, and its 10 halfrevolutions are also represented by 10 overlapping Trees. Figure
16 summarises the analogy between the geometries of the inner Tree of Life and the UPA. Figure 17 shows
the inner form of 10
overlapping Trees of

Figure 16. The (7+7) enfolded
polygons of the inner Tree of Life represent a halfrevolution of the 10 whorls
of the UPA/subquark state of the E_{8}×E_{8} heterotic
superstring.

Life divided
into the sets of polygons S_{1}, S_{2}, S_{1}′ & S_{2}′.
Each of the 3360 intrinsic geometrical elements in the (79+70=140) polygons that make
up this inner form corresponds to a plane wave component of the 1680 circularly polarised
oscillations present in a halfrevolution of the 10 whorls of the UPA. That there are four
components of the inner Tree of Life and that each half of it results from pairing of these
components only in one way:
S_{1} → S_{2} and
S_{1}′ → S_{2}′
10
11
not:
S_{1} → S_{2}′ or
S_{2} → S_{1}′
reminds us of
the four types of nitrogen bases in the DNA molecule: A, T, G & C, and how base A always
pairs with base T through two hydrogen bonds and base C with base G through three hydrogen
bonds when they form a rung of base pairs in the helical DNA ladder. It is as though the
complementary pairs of bases (A, T) and (C, G) are analogous, respectively, to
(S_{1}, S_{2}) and (S_{1}′, S_{2}′), so that the two helical
halves of the DNA molecule are analogous to the two sets of sinusoidal waves in each helical
whorl of the E_{8}×E_{8} heterotic superstring that generate its 1680
circularly polarised waves because the waves in one set are 90° out of phase with those in
the other set. A turn in a whorl would then correspond to a rung of base pairs, whilst the
complementary nitrogen bases at each end of a rung would correspond to the two sinusoidal
oscillations of the same amplitude but 90° out of phase that constitute a turn. The pairing
of complementary bases in the molecule of life is an expression of the Taoist principle that
the yang aspect of nature is always balanced by its yin aspect. The supposition of two
orthogonal, sinusoidal oscillations in every turn of a whorl is another manifestation of
this cosmic balance. In the inner form of the Tree of Life, it is expressed by its two
halves, one associated with the righthand Pillar of Mercy, which embodies the male
principle (Siva, or Yang), and the other associated with the lefthand Pillar of Judgement,
which embodies the female principle (Shakti, or Yin). These principles, of course, are
entirely metaphysical.
S_{2} & S_{2}′ embody the
dimension 248 of E_{8}
According to
Table 4, S_{1} has 132 yods outside the root edge, where 132 = 4×(1!+2!+3!+4!).
S_{2} has 128 yods outside the root edge. This is the sum of the Godnames of
the first four Sephiroth:
21 + 26 +
50 + 31 = 128.
S_{2}
& S_{2}′ have 256 yods outside the root edge, where 256 = 4^{4}. These
properties demonstrate the role of the Tetrad in determining the yod population of the inner
Tree of Life. Of the 128 yods in S_{2}, six yods lie on the two vertical sides
of tetractyses inside the hexagon that are shared with triangles in the outer Tree of Life
because they form its Pillar of Mercy and Pillar of Judgement. (128−6=122) yods outside the
root edge are intrinsic to S_{2}.
Including the root edge,

Figure 18.
The 295 yods in the seven polygons denote the 295 SLs up to Chesed of the
49th Tree. The S_{1}:S_{2}division corresponds to the
middle of the 49tree and is marked by the 148th SL. This is
Netzach of the 25th Tree. It is the SL most characteristic of this Sephirah,
whose gematria number value is 148. 
the number of yods intrinsic to S_{2}+S_{2}′ in the inner form
of the Tree of Life = 4 + 2×122 = 248. This is the
dimension of E_{8}, the Lie group associated with the E_{8}×E_{8} heterotic superstring. Of the 132 yods in
S_{1},
one yod (righthand corner of the triangle) is shared with the outer Tree because it is
located at the Sephirah Chesed; this leaves 131 intrinsic
yods. 131 is the number value of Samael, the Archangel of
Geburah, whose number 216 was found earlier to
be the number of yods lining the 48 tetractyses in the two
S_{1}, 48 being the number
of Kokab, the Mundane Chakra of Hod.
The
significance of the S_{1}:S_{2} division for CTOL
According to
Table 4, the seven separate Type A polygons have 295 yods. 288 yods surround their centres,
where
12
288 =
1!×2!×3!×4!.
There are 295
SLs up to Chesed of the 49th Tree (Fig. 18). In order of emanation in the Tree of Life,
Chesed is the first of the seven Sephiroth of Construction. The 49th Tree is the first
of the 49 Trees mapping the 49 subplanes of the seven planes of consciousness. As
each Tree expresses the essence of one of these Sephiroth of Construction, the 49th Tree
expresses Chesed. So
the 295th SL expresses par
excellence the quality of this Sephirah. Each yod in a
polygon denotes one of the 295 SLs up to the starting point for the emanation of the
49tree representing the cosmic physical plane. According to Table 3, the four separate
polygons in S_{1} contain 148 yods and S_{2} has 147 yods. The
S_{1}:S_{2} division therefore splits in half the yod population of the seven
separate polygons of the inner Tree of Life. The number 148 is the gematria number value
of Netzach. The 148th SL is Netzach of the 25th Tree, which is the middle of the
lowest 49 Trees. The 25th Tree maps the fourth subplane of the fourth
plane of consciousness, both the subplane and the plane corresponding to Netzach because it is
the fourth Sephirah of Construction. So the number 148 — the halfway point in the
emanation of the 295 SLs —locates the very SL in the 49tree that most expresses this
Sephirah, just as the number 295 locates the SL most expressive of the Sephirah Chesed! This
remarkable property cannot be a mere coincidence. Instead, because of the correspondence
between yods and SLs, the 24:24 division of the corners of the seven polygons, which splits the
168 intrinsic geometrical elements surrounding their centres into two sets of 84
elements, also divides in half the SLs that their yods symbolise and which belong to the
49 Trees mapping the cosmic physical plane. The following correspondences between yods
and SLs exist:

4
centres of S_{1} → 4 SLs from bottom of 1tree to Netzach of 1tree;

144
yods surrounding centres of S_{1} → 144 SLs up to Netzach of 25th
Tree;

144
yods surrounding centres of S_{2} → 144 SLs above Netzach of 25th Tree to
Netzach of 49th Tree;

3
centres of S_{2} →3 SLs above Netzach of 49th Tree to Chesed of
49th Tree.
Here is remarkable confirmation of this conclusion: the numbers of corners,
sides, triangles & tetrahedra making up the ntree are shown below:
Number of
corners = 6n + 5;
Number of sides = 16n + 9;
Number of triangles = 12n + 7;
Number of tetrahedra = n +1.
Total number of
points, lines, triangles & tetrahedra in the ntree = 35n + 22.
Every
48 overlapping Trees of Life contain (48×35=1680) geometrical elements and
every 24 Trees contain 840 elements. Dividing any section of 48 Trees into two sets
of 24 Trees generates two sets of 840 geometrical elements. Apart from the Pythagorean
factor of 10, here is the 84:84 division


Figure 19. The 42 blue “yods”
in the Cosmic Tetractys symbolise the 42 subplanes above the cosmic physical
plane; the 42 red yods denote the 42 subplanes above the physical plane, whose
7 subplanes are symbolised by the 7 grey yods in the central
tetractys. 
Figure 20. 84 yods in the
2ndorder tetractys surround the central black yod denoting Malkuth, or the
physical universe. 
13
of
geometrical elements found earlier for the division of the seven polygons with
48 corners into S_{1} and S_{2}, each with 24 corners. In
fact, in the encoding of CTOL in the 12 separate polygons other than one square and one
triangle, their corners mark out Trees of Life, so that S_{1} with 24 corners
is associated with 24 Trees and S_{2} with 24 corners is associated with the
next higher 24 Trees. The same 840:840 division exists in the inner form of
48 Trees. This is because the number of corners of the 7n polygons enfolded in
the ntree = 35n + 1, so that the 336 polygons enfolded in 48 Trees have
(48×35=1680) corners and the 168 polygons enfolded in 24 Trees have 840
such corners. Their counterparts in the UPA are the 840 turns in either an outer or an inner
half of a single whorl or in a quarterrevolution of all 10 whorls. This mathematical
analogy between the subquark superstring and the cosmic physical plane is a powerful and
beautiful illustration of the hermetic principle: "As above, so below." The 70 enfolded
polygons making up the inner form of 10 Trees of Life consist of 10 S_{1}, with 840
intrinsic geometrical elements surrounding the centres of 40 polygons, and 10 S_{2},
with 840 intrinsic geometrical elements surrounding the centres of 30 polygons.

336 = 

Figure 21. The Tetrad
expresses the holistic parameter 84. 
Figure 22. The Tetrad
expresses the 4fold pattern within the holistic parameter 336.

Returning to
CTOL, the number of SLs on the central Pillar of Equilibrium of the ntree = 2n + 3. For n
overlapping Trees, the number = 2n + 2. Seventeen SLs span the central pillar of the 7tree
mapping the physical plane and 184 SLs span the 91 Trees of CTOL. 167 SLs span the 84 Trees
above the 7tree, that is, the apex of the 7tree is the 168th SL on the central
pillar, counting from the top of CTOL. The apex of the 49‑tree mapping the cosmic
physical plane is the 101st SL on the central pillar, that is, the 84th SL, counting
from the top of CTOL. Fittingly, the number value 168 of Cholem Yesodeth, the
Mundane Chakra of Malkuth, locates the Malkuth level of CTOL. The 84:84 division created by
the geometrical composition of S_{1} and S_{2} in the inner Tree of Life
appears in CTOL as the 84 SLs on the central pillar down to the top of the 49tree
and the next 84 SLs down to the top of the 7tree. Each set of 84 SLs belongs to 42 Trees
(Fig. 19).
As the next
higher order tetractys above the wellknown tetractys, or 1storder tetractys, the 2ndorder
tetractys (Fig. 20) is a representation of holistic systems. Its 10 tetractyses contain 85
yods, where
85 =
4^{0} + 4^{1} + 4^{2} + 4^{3}.
84 yods
surround its central yod, where
84 =
1^{2} + 3^{2} + 5^{2} + 7^{2}.
(Fig. 21).
The pattern of four groups of 84 structural units will always be found in a holistic system.
It is represented by a cross pattée whose four arms are 2ndorder tetractyses (Fig. 22) with
84 yods outside their shared apex. The diagram aptly illustrates how the Tetrad expresses
parameters characterising holistic systems and their fourfold division.
Compare this
with the result obtained earlier that the centres of the four polygons in
S_{1} are surround by 84 geometrical elements, as are the centres of the three
polygons in S_{2}. The two halves of the set of seven enfolded polygons are
themselves holistic because they require 84 bits of information in the form of points, lines
& triangles to create them, starting in each case from a mathematical point. This is
what the 2ndorder tetractys signifies: 84 bits of information (whatever their meaning)
symbolised by the 84 yods surrounding its centre are required to generate a holistic system,
the outer, material form of which is symbolised by this central yod, signifying Malkuth. In
the case of CTOL, they are the 84 Trees/subplanes whose final manifestation at the level of
Malkuth is the 7tree that maps the physical plane, or physical universe. In the case of the
UPA, they are the 84 turns in an outer or inner halfrevolution of each whorl. The factor of
4 results from each turn being a circularly polarised oscillation with two plane wave
components and two halfrevolutions forming a complete, 360° revolution of a
single whorl: 4 = 2×2.
14
3. The 24:24 division in the 1st (6+6)
polygons
Article 4 analysed how the
gematria number values of the Godnames of the 10 Sephiroth prescribe the properties not only of
the (7+7) enfolded polygons making up the inner Tree of Life but the first six polygons and
their mirror image counterparts on the other side of the root edge. The first six
separate polygons have 36 sectors with 42 corners, so that the first (6+6)
separate polygons have 72 sectors with 84 corners. Whether they are separate or
enfolded, let us call them “H.” Their embodiment of the holistic parameter 84 expressed by the
2ndorder tetractys in Fig. 20 indicates that this subset of the 14 polygons constitutes a
holistic system in itself. The 42:42 division of its corners is analogous to the same division
of Trees in CTOL above the 7tree shown in Fig. 19 that is created by the six cosmic
superphysical planes with 42 subplanes and by the six superphysical planes with 42 subplanes.
Their holistic character is confirmed by their enfoldment to produce a geometrical structure
with 50 corners, where 50 is the number value of ELOHIM, the Godname of Binah.
YAHWEH, the Godname of Chokmah on the other side of the Tree of Life with number value
26, prescribes the first six enfolded polygons because they have 26 corners. Each
set has 24 corners outside the root edge, so that both sets have (24+24=48) external
corners (Fig. 23). The 24:24 division that splits in half the seven separate polygons with
48 corners reappears by dividing H in half. As the topmost corner of each hexagon
coincides with the lowest corner of its counterpart enfolded in the next higher Tree of Life,
the H enfolded in each Tree has 48 unshared corners (24 associated with each set of
six). Unlike that displayed in Fig. 23, this 24:24 division gets repeated in every Tree, which
means that, as the correct way of dividing the 48 corners, the two endpoints of the root
edge must be included, one with each set of 24 corners.
The 168:168 division in the 1st (6+6)
polygons
When its (35+35=70) sectors are
turned into tetractyses, H has 386 yods [3]. The topmost corners of the two hexagons coincide
with the lowest corners of the two hexagons enfolded in the next higher Tree, leaving 384 yods
intrinsic to the H in each Tree. This is another indication of its holistic nature, for
previous articles have revealed how this number appears as a global parameter of wellknown
sacred geometries, as well as objects for which considerable evidence has been accumulated in
the author’s articles and website to indicate that they possess a similar status. Surrounding
the hexagonal yods at the centres of the 70 tetractyses are (384–70=314) yods intrinsic to H.
Outside the four yods in the root edge are 310 (=31×10) intrinsic yods lining these
tetractyses. This shows how EL, the Godname of Chesed with number value 31, prescribes
H. 155 yods outside the root edge lie on the boundaries of the 35 tetractyses in
each set of six polygons. This shows how ADONAI MELEKH, the complete Godname of Malkuth with
number value 155, prescribes the form of H.


Figure
24. 24 intrinsic corners & 168 yods are associated with the first
6 polygons enfolded in each Tree.

The 384 yods
intrinsic to H comprise 48 corners of its 12 polygons and 336 other yods,
where
336 = 4×84 =
2^{2} + 6^{2} + 10^{2} + 14^{2}.
Associated
with each half of H are 24 corners and 168 other yods (Fig. 24). The
division:
48 = 24 + 24
15
in the
polygonal corners intrinsic to H automatically generates the division:
336 =
168 + 168
in its yod
population. Each set of 168 yods consist of 78 yods in the triangle, square
& octagon and 90 yods in the square, hexagon & decagon (Fig. 25).

Figure 25.
The yod populations of subsets of polygons are the gematria number values of
Cholem & Yesodeth appearing in the Kabbalistic name of the
Mundane Chakra of Malkuth 
These numbers
are the gematria number values of, respectively, Cholem and Yesodeth,
which are the two words making up the Kabbalistic name of the Mundane Chakra of
Malkuth.
The 84:84 division in the 1st (6+6) enfolded polygons
Just as the
24:24 division of the corners of the seven separate polygons generates an 84:84 division of
the yods in S_{1} & S_{2}, so, too, the 24:24 division of the corners of
H corresponds to a 84:84 division in the yods lining the sides of its polygons, as now
shown. The first six separate polygons have 36 sides. Enfolded, they have 31 sides,
showing how ELOHA, the Godname of Geburah with number value 36 and EL, the Godname of
Chesed opposite it with number value 31, prescribe this holistic set of polygons. Two
hexagonal yods lie on each side, totalling 62, where 62 is the number value of Tzadkiel, the
Archangel of Chesed.

Figure
26. (84+84=168) yods outside the root edge line the sides of the 1st
(6+6) enfolded polygons. They include (24+24) corners.

Notice that, as H contains 384 intrinsic yods, 192 yods are associated with each half of it, so
that the first six enfolded polygons contain (192+2=194) yods that are intrinsic to
them, where 194 is the number value of Tzadekh, the Mundane Chakra of Chesed.
Both sets of six separate polygons have 72 corners, where 72 is the number value
of Chesed. Here is a remarkable conjunction of four gematria numbers referring to the
same Sephirah! The 12 polygons in H have 61 sides lined by (61×2=122) hexagonal yods and 50 corners,
i.e., 172 yods. Hence, 168 yods outside the root edge lie on the sides of these polygons
(Fig. 26). Each set of six polygons has 84 such yods (coloured red or blue in the diagram).
(172–2=170)

Figure 27.
Examples of the number 70 as a holistic parameter.

16
boundary yods
are intrinsic to H, and 87 such yods line the sides of each set of six enfolded
polygons. 87 is the number value of Levanah, the Mundane Chakra of
Yesod.
The 84 yods
lining the sides of each set of six enfolded polygons divide into 42 yods on the sides of
the triangle, pentagon & decagon and 42 yods lining the square, hexagon & octagon.
These are the only possible combinations of polygons showing this division. The sequence of
divisions:
336→168→84→42
Is always
displayed by holistic systems. The way other sacred geometries exhibit it will be discussed
in the remaining sections. Their higherorder transformations reveal that the sequence
proceeds in the opposite direction as well. The most remarkable example of this is the
Gosset polytope, whose 672 edges divide into the 336 edges in each half and which has
(2×672=1344) hexagonal yods lining its edges when its faces are turned into tetractyses. Its
Tree of Life basis is analysed in Article 62 (see ref. 1).
4. The dodecagon As the
tenth type of regular polygon and the last of the seven polygons making up the
inner Tree of Life, the dodecagon is unique in that it is the single, polygonal
representation of holistic systems. One of the parameters that define such systems is the
number 70. It is embodied in the outer Tree as its 70 yods when its 16 triangles
are tetractyses and as the 70 sides of the sectors of these triangles when they are Type
A (Fig. 27). It is embodied in the inner Tree of Life as the 70 corners of its polygons,
in the first (6+6) enfolded polygons as their 70 sectors and in the dodecagon as the 70
yods associated with each of the two joined dodecagons that surround its centre. The
number 48 of Kokab, the Mundane Chakra of Hod, is embodied in these sacred
geometries because 1. the outer Tree of Life is composed of 48 geometrical
elements (10 corners of 16 triangles with 22 sides), 2. its 16 triangles have 48
sectors, 3. the seven separate polygons have 48 corners, 4. the first (6+6)
enfolded polygons have 48 corners that are intrinsic to them, and 5. 48
geometrical
elements

Figure
28. 24 red hexagonal yods & 24 blue hexagonal yods surround the black
centres of the 12 tetractys sectors of the dodecagon.

surround the centre of a Type B
dodecagon. It is also embodied in the Type A dodecagon, as it contains 73 yods, of
which one is its centre, 12 are corners and 60 are hexagonal yods, 48 of which
surround those at the centres of its 12 tetractyses (Fig. 28). The dodecagon conforms to
the division:
48 = 24 + 24
that is characteristic of holistic systems because 24 hexagon yods line
its sides, inside of which are 24 hexagonal yods surrounding centres of tetractyses. This
24:24 division was shown in Section 2 to divide equally the 264 yods of the seven
enfolded polygons into S_{1}, whose four polygons have 24 corners when they are
separate and 132 yods when they are enfolded, and S_{2}, whose three separate
polygons have 24 corners and 132 yods when enfolded. This same division appears in the
two joined Type B dodecagons, as now explained. Each dodecagon has 181 yods, so that the
two joined dodecagons have 358 yods. Of these, 22 are their corners and 72 are
hexagonal yods at the centres of their tetractyses. This means that
(358–22–72=264) yods line the sides of these tetractyses, 132 boundary yods being
associated with each dodecagon (Fig. 29). The two joined dodecagons display the 24:24
division characteristic of

Figure 29. The (132+132=264) yods other than corners lining sides of tetractyses in
two joined Type B dodecagons correspond to the (132+132=264) yods
in S_{1} &
S_{2.} 
17
holistic
systems because their 24 sectors have 24 corners and their tetractyses have 24 more corners.
There are 131 yods outside the root edge in each dodecagon. This is the number value
of Samael, the Archangel of Geburah, whose Godname ELOHA has the number value
36, which is the number of yods lining the sides of the dodecagon. 130 yods outside
the root edge surround the centre of each dodecagon. This is the 65 th even integer and the 129th
integer after 1, showing how ADONAI, the Godname of Malkuth with number value 65, and
YAHWEH SABAOTH, the Godname of Netzach with the number value
129,


Figure 30. 168 yods are needed to transform the 12 sectors of a
dodecagon into Type A triangles.

Figure 31. The two joined Type B dodecagons have (168+168=336)
yods other than their 22 corners.

prescribes
the Type B dodecagon. These are examples of how the Godnames mathematically prescribe
the properties of holistic systems.
The 168:168 division
A Type B
dodecagon with 181 yods requires 168 more yods to transform its 12 sectors into Type
A triangles (Fig. 30). Joined together, two such dodecagons contain 358 yods — 336 yods in
addition to its 22 corners (Fig. 31). 168 such yods are associated with each
dodecagon. Their 24 sectors have 24 corners and their 72 tetractyses have another 24
corners. This is how the 24:24 division generates the 168:168 division in the
outermost polygons of the inner Tree of Life. There are (358–48=310=31×10)
hexagonal yods, showing how EL with number value 31 prescribes the pair of
dodecagons. 155 hexagonal yods are associated with each dodecagon, where 155
is the number value of ADONAI MELEKH, the complete Godname of Malkuth. 154 such yods are
outside each dodecagon, where 154 is the 153rd integer after 1. The number 153
is the number value of ELOHIM SABAOTH, the Godname of Hod.
The 84:84 division
There are 15 yods per
sector of the Type B dodecagon (14 other than a corner of the polygons). Seven blue yods line
internal sides of the three tetractyses in each sector and seven red yods are either hexagonal
yods at their centres or yods lining the sector (Fig. 32). The 168 yods that are not
corners of sectors comprise (12×7=84) red yods and 84 blue yods. When two
dodecagons become joined, one of their 168 yods becomes associated with the other
dodecagon.


Figure
32. The two sets of 84 yods in each Type B dodecagon.

Figure 33. The 84 geometrical elements in each quadrant of a Type C dodecagon
correspond to the 84 intrinsic geometrical elements surrounding the centres of
S1, S2, S1′ & S2′ — the quarters of the (7+7) enfolded
polygons of the inner Tree of Life. The quadrant corresponding to
S_{1}′ is diametrically opposite that for S_{1}; similarly, for
S_{2}′ and S_{2}.

To compensate for this, the centre of the dodecagon must be included to maintain the set of
168 yods in each dodecagon. Each set is a pair of 84 yods. They correspond to the two
sets of geometrical elements that surround the centres of the polygons
18
in
S_{1} & S_{2} and in S_{1}′ & S_{2}′ (see Figs. 14
& 15). Which set corresponds to S_{1} and which corresponds to S_{2} is
an issue that need not be resolved here. It is sufficient to have demonstrated that the pair
of dodecagons constitute a holistic system, embodying the parameter 336 as four sets of 84
yods. However, the 84 red yods include one on the root edge, so that they “need” the root
edge to make up the set, just as, according to Table 5, S_{2} needs it to complete
the 132 yods in half of the seven enfolded polygons. This provides a reason for arguing that
S_{1} in the righthand dodecagon corresponds to the 84 blue yods and S_{2}
corresponds to the 84 red yods and that S_{1}′ corresponds to the 84 light blue yods
in the lefthand dodecagon and S_{2}′ corresponds to the 84 light red
yods.
The
168:168 & 84:84 divisions manifest in the geometrical composition of a
single dodecagon, namely, the Type C dodecagon. The number of corners, sides &
triangles in a Type C ngon = 28 +1. Hence, (12×28=336) geometrical elements surround the
centre of a Type C dodecagon (Fig. 33). 168 elements exist in each half of the
polygon and 84 elements exist in each quarter. Here is further, striking evidence for the
dodecagon being the polygonal version of the inner Tree of Life, for the geometrical
composition of each half is the same as that outside the root edge and surrounding the
centres of the seven enfolded polygons. Each quarter corresponds to S_{1},
S_{2}, S_{1}′ & S_{2}′.
5. The Sri
Yantra
The Sri Yantra (Fig. 34) is
revered by Hindus as the most powerful and sacred of the yantras (images used for meditation).
Five downward pointing triangles symbolizing the feminine, creative energy, or Shakti,
intersect four upward pointing triangles symbolizing the masculine, creative energy, or Siva.
This generates 42 triangles arranged in four layers of eight, 10, 10 and 14 triangles. They
surround a downwardpointing triangle whose corners denote the triple


Figure 34. The Sri
Yantra, 
Figure 35. 336 yods line the sides of the 42 triangles of the 3d Sri Yantra
when they are tetractyses. They form four sets of 84 yods. For clarity,
hexagonal yods at the centres of tetractyses are omitted.

Godhead of
Siva, Brahma and Vishnu. At its centre is a point, or bindu, representing the Absolute. It is
the source of all levels of existence. As such, it ‘exists’ outside them, although this is
hardly the right word to express such an exalted level.
Let us
consider the 3dimensional Sri Yantra, in which the parallel sheets of triangles are stacked on
top of one another so that none of their outward points touch triangles in a different layer.
Let us regard the 42 triangles as tetractyses. There are nine yods intrinsic to each tetractys
because it shares two corners with adjacent tetractyses in the same layer. Eight of these line
its sides. Therefore, (42×8=336) yods line the (42×3=126) sides of the 42 triangles. 168
boundary yods lie in the 21 tetractyses in the upper half of the Sri Yantra and
168 yods line the 21 tetractyses in its lower half (Fig. 35).[1]
Although it
does not amount to exact mirror symmetry in the horizontal plane, there is an isomorphism
in
[1] As
long as its central triangle is excluded, the 3d Sri Yantra can be divided in an upper half
and a lower half with the same number of triangles, although — unlike the inner Tree of Life
— they are not exact mirror images of each other.
19

Figure 36.
(168+168=336) yods line the sides of the 42 triangles in the
3d Sri Yantra. Those in one set of 168 yods are the inverted
images of their counterparts in the other set. (The hexagonal yod at the
centre of each tetractys is omitted for the sake of clarity).


Figure 37. The 336
boundary yods in the Sri Yantra comprise 4 sets of 84 yods. The 4 pairs per
triangle comprise a pair of corners and 3 pairs of hexagonal yods. The 16
circles denote corners of triangles directly above corners of triangles in
the next layer below them. (The hexagonal yod at the centre of each
tetractys is omitted for the sake of clarity).

20
the
distribution of yods in the 42 triangles to the extent that for every yod in one half of the
Sri Yantra there exists a counterpart in its other half that is its inverted image,
reflected not through a horizontal or vertical mirror but through its centre. There are
168 pairs of such yods (Fig. 36). Noting that two black corners are associated with
each tetractys, which has three pairs of hexagonal yods on its sides (two red, two green
& two blue), four pairs of yods per tetractys lie on its boundary. Each set of 21
tetractyses contains (4×21=84) pairs of yods. The 336 boundary yods consist of
168 pairs in two ways.
Comparing
this property with the 168 intrinsic geometrical elements surrounding centres of
polygons in S_{1} & S_{2}, the following correspondences naturally
emerge:
3d Sri Yantra


Inner Tree of Life

336 yods line (21+21)
tetractyses; 
→

336 geometrical elements surround
centres of (S_{1}+S_{2}) and
(S_{1}′+S_{2}′); 
168 yods line 21 tetractyses in each half; 
→

168 geometrical elements surround polygonal centres in each
half; 
168 yods in each half comprise 84 pairs, or two sets of 84
yods; 
→

84 geometrical elements surround
centres in S_{1} (S_{1}′); 84 geometrical
elements surround centres in S_{2}
(S_{2}′); 
The
distinction between pairs of hexagonal yods or corners is what corresponds to the difference
between S_{1} and S_{2} as the two halves of the set of seven enfolded
polygons of the inner Tree of Life. It cannot arise from the 21 triangles themselves
in each half of the Sri Yantra because, being an odd integer, the number 21 cannot be
divided into two integer halves. The possibility of dividing the four pairs of yods per
tetractys into two sets of two pairs is less natural because there does not seem to be any
reason why one pair should join any other pair to generate a group of 84 yods. The fact that
the eight yods are already forming four pairs (one pair of corners, three pairs of
hexagonal yods) argues against this alternative. Fig. 37 shows the 84 pairs of yods (now
coloured red & blue) in the upper half of the Sri Yantra and the 84 pairs of yods (light
red & light blue) in its lower half. They correspond to, respectively, the 84
geometrical elements surrounding centres of polygons in S_{1} and in S_{2}
and their mirror images in S_{1}′ and in S_{2}′. Each pair of yods in one
half of the Sri Yantra corresponds to a geometrical element in S_{1} and a
geometrical element in S_{2}. Each pair of yods in the other half corresponds to a
geometrical element in S_{1}′ and a geometrical element in S_{2}′. Unlike in
the case of the inner Tree of Life, where each element has a chiral counterpart in the other
set of seven enfolded polygons generated by reflection in the mirror plane containing the
root edge that is perpendicular to the plane containing all the polygons, the 336 yods
consist of 168 yods and their 168 point reflections, although some are not
true inversions, as the 2dimensional Sri Yantra is not centrally symmetric. Despite the
lack of exact mirror symmetry in this yantra, it is symmetric enough for the counterparts of
its yods and geometrical elements to be identifiable in an unambiguous way.
Geometrical composition of the Sri Yantra
Table 9 shows
the numbers of corners, sides & triangles in the 3dimensional Sri Yantra:
Table 9.
Geometrical composition of the Sri Yantra.
(The bindu is
a point, like the corners of the triangles, and may be included in the count of corners
because what is really being counted here are points, lines & triangles). The 42
triangles in the four levels have 84 corners and 126 sides. The Godname EHYEH with number
value 21 (see Table 1) prescribes the number of triangles because 42 is the
21st even integer. It prescribes the number of sides because 126 is the sum of the
values of all the possible combinations of the three letters A, H & I in
AHIH:
A = 1, H = 5,
I = 10
21
1. 
A +
H + I 
= 
16

2. 
AH
+ HI + AI + HH 
= 
42

3. 
AHI
+ HIH + AHH 
= 
47

4. 
AHIH 
= 
21


Total

= 
126

The number of
geometrical elements is 252, which is the 126th even integer. The first three levels have 84
sides, 56 corners and 28 triangles, that is, 84 corners & triangles, totalling
168 geometrical elements. This 84:84 division of the number 168 was
encountered in Section 2 (p. 9) as the 84 geometrical elements surrounding the centres of
S_{1} and of S_{2}. The fourth layer has 84 geometrical elements, whilst all
four levels have 84 corners and 168 sides & triangles (84 in each half of the Sri
Yantra). We do not yet see in its geometry the appearance of the holistic parameter
336 as four sets of 84 (that must await the division of the triangles into their sectors —
to be discussed next). That there are, according to Table 9, 260 geometrical elements in all
43 triangles when the bindu is included is a clue to the more appropriate comparison,
namely, with the yod composition of S_{1} & S_{2} outside the root edge.
We see from Table 5 that S_{1} has 132 yods and that S_{2} has 128 yods
outside the root edge. The centre of the triangle in S_{1} coincides with a
hexagonal yod on the line joining Chesed and Geburah in the outer Tree of Life, whilst seven
yods (four hexagonal yods & three corners) on the internal vertical sides of tetractyses
in the hexagon in S_{2} are shared with it. Therefore, 252 yods are intrinsic to
S_{1} & S_{2} and (1+7=8) yods are shared. Compare this with the 252
geometrical elements in the 42 triangles of the Sri Yantra and the eight geometrical
elements (one triangle, four points & three lines) that make up the central triangle
& bindu (see Table 9). All these correspondences cannot be accidental. It is clear that
the geometry of the Sri Yantra manifests in a onetoone way information abstractly
symbolised by the yods in the inner Tree of Life because it is a different representation of
the same archetypal blueprint.
Table 10
shows the geometrical composition of the 3d Sri Yantra when its triangles are Type
A:
Table 10.
Geometrical composition of the 3d Sri Yantra with Type A triangles.
Whereas
before, the first three layers had 168 geometrical elements, they now have 336
elements, whilst the fourth now has 168 geometrical elements instead of 84. The 336
elements comprise 168 sides, 84 corners and 84 triangles, i.e., 168 corners
& triangles, reproducing the holistic division: 336 = 168 + 168, whilst
the 168 elements in the fourth layer comprise 84 sides and 84 corners &
triangles, reproducing the holistic division: 168 = 84 + 84. The division 48 =
24 + 24, which appeared in the separate polygons of S_{1} and S_{2} as their
24 corners, reappears in the first layer of triangles as the 24 sides of the four triangles
in each half, or, alternatively, as the 24 corners & triangles in each half of the Sri
Yantra.
According to
Table 10, the 3d Sri is composed of 518 geometrical elements. It has 131 corners,
where 131 is the number value of Samael, the Archangel of Geburah. The 43
triangles have 130 corners, 258 sides and 129 sectors, where 130 is the 129th
integer after 1, 258 is the 129th even integer and 129 is the number value of
YAHWEH SABAOTH, the Godname of Netzach. This number appears in the 43 triangles because
129 = 3×43. The (7+7) enfolded polygons of the inner Tree of Life contain 524 yods,
of which four make up the root edge and two are the topmost corners of the two hexagons that
coincide with the lowest corners of the hexagons enfolded in the next higher Tree. Hence,
(524–4–2=518) yods outside the root edge belong exclusively to the set of 14 polygons
enfolded in each Tree (Fig. 38). Each of these yods corresponds to a geometrical element
making up the Sri Yantra with Type A triangles. Surrounding the centres of the 14 polygons
are (518–14=504) such yods. They symbolise the 504 geometrical
22

a

b

Figure 38. The 518
geometrical elements composing the 3d Sri Yantra correspond to the 518 yods
in the (7+7) enfolded polygons outside their root edge that are unshared
with polygons enfolded in the next higher Tree. The 504 geometrical elements
in the four layers of triangles correspond to either (a) the 504 intrinsic
yods that surround the centres of the 14 polygons, or (b) the 504 intrinsic
yods unshared with the outer Tree of Life.

elements in
the four layers of triangles. Six yods in S_{1}, which has 132 yods outside the root
edge, are centres of polygons and one yod in S_{2}, which has 128 yods, is a centre.
Hence, 126 yods in S_{1} are not centres and 127 yods in S_{2} are not
centres. One of the latter yods is the topmost corner of the hexagon that is shared with the
hexagon enfolded in the next higher Tree. Therefore, S_{2} has 126 intrinsic
yods that are not centres. Compare this with the 126 corners, 126 triangles & 252
triangles in the 3dimensional Sri Yantra, that is, with the 126 corners & triangles and
the 126 sides in each half of it. Just as four sets of 126 yods in S_{1},
S_{2}, S_{1}′ & S_{2}′ surround the (7+7) centres in the
polygons of the inner Tree of Life, so, too, four sets of 126 geometrical elements make up
each half of the Sri Yantra. The 14 centres correspond to the 14
geometrical elements making up the central triangle and the bindu. An alternative
possibility needs to be considered. The 518 yods include six yods in each hexagon that are
shared with the outer Tree because they lie on its side pillars and the centre of each
triangle that coincides with a hexagonal yod belonging to the outer Tree because it lies on
the side of the triangle connecting Chesed to Geburah. Hence, 14 yods in the set of 518 yods
are shared with the outer Tree, leaving 504 yods outside the root edge that are unshared.
Each set of seven enfolded polygons contains 252 such yods. S_{1}, then, has two
shared yods (centre & corner of triangle), so that it has (132–2=130) unshared yods,
whilst S_{2} has six shared yods, so that it has (128–6=122) unshared yods.
Comparing these two numbers with the 126 corners & triangles and 126 sides in each half
of the Sri Yantra, we conclude that, although the total number of yods is the same as the
number of geometrical elements, S_{1} and S_{2} do not have the
same
23
numbers of
unshared yods. The alternative view in which the 14 yods contributed by the bindu and
central triangle correspond to the 14 shared yods is, therefore, incorrect. These 14 yods
correspond only to the centres of the 14 polygons making up the inner Tree of
Life.
Comparison with the Type C dodecagon The number of yods in a Type C ngon =
42n + 1, where “1” denotes its centre. A Type C dodecagon (n=12) has
(12×42=504) yods
surrounding its centre. It was pointed out in Section 4 that this regular polygon is the
single, polygonal representation of the Tree of Life (see Fig. 27). The Type A version has
60 hexagonal yods, which is the number of hexagonal yods in the outer of Life, 264 yods line
the 72 tetractyses in
two joined Type B dodecagons and this is the number of yods in the seven enfolded polygons,
whilst the 504 yods surrounding the centre of the Type C dodecagon is the number of yods intrinsic to
the
(7+7) enfolded polygons that are outside the root edge. As 168 = 12×14
and 42 =
3×14, these 504
yods consist of three sets of 168 yods, the 42 yods per sector consisting of three
sets of 14 yods. One set comprises 14 red yods that are either corners & centres of
tetractyses (Fig. 39). The 168 such yods in the Type C dodecagon correspond to the
168 geometrical elements in the fourth layer of 14 red triangles in the
3dimensional Sri Yantra. The 28 yods remaining in each sector comprise 10 green,
hexagonal yods, 10 blue, hexagonal yods and 8 violet, hexagonal yods — the same numbers
as the number of triangles in the first three layers. The 120 green yods, the 120
blue yods and the 96 violet yods in the dodecagon symbolise the geometrical elements in
respectively, the third, second and first layers of the Sri Yantra with Type A triangles.
In other words, the Type C dodecagon is an exact representation of the geometrical
composition of the Sri Yantra. This is because they are both equivalent
representations of the divine paradigm.

Figure 40. The 64
hexagrams.

This isomorphism between two
sacred geometries and its embodiment of the holistic parameters 84, 168 & 336 are
unmistakable, strong indications of the objective nature of the centuryold, micropsi
observations of the UPA by Annie Besant and C.W. Leadbeater. Had they been otherwise, these
numbers would not manifest so frequently in the geometrical and yod compositions of these and
other sacred geometries to be discussed later in this article. Nor, indeed, would the gematria
number values of the 10 Sephiroth and their Godnames, etc. This is not — as a sceptic might
suggest — because their appearance has been contrived by some kind of cherrypicking of
properties. The parameters with a paranormal provenance have entirely
24
natural meanings in the context of
these sacred geometries that cannot result from random chance or deliberate selection. To
suggest the latter is to beg the question that it is always possible to reproduce all
the levels of correlation between two geometries as soon as some global choice of yods or
geometrical properties has been made. It clearly is not possible do this
deliberately. The fact that these levels exist at all can only be sensibly understood as due
neither to chance, which is highly implausible, nor to cherrypicking, because the
correspondence runs too deep to make that possible. Instead, it can be due only to an
inherent, mathematical design that is revealing a transcendental intelligence within
it.
6. I Ching table of
hexagrams Used in China for thousands of years as a
form of fortune telling, the system of 64 hexagrams (Fig. 40), upon which the
commentaries in the famous I Ching (“Book of Changes”) are based, consists of an 8×8
square array of the (8^{2}=64) possible pairings of eight trigrams. Each
trigram (Fig. 41) is a
combination
of

Figure 41.
The eight trigrams. 
three lines.
A line can be either ‘yin,’ the female power, or gentleness: − − , or ‘yang,’ the male
power, or strength: —. Fig. 41 shows the eight
combinations (2×2×2=8) of yin and yang in the three lines of a trigram. They will be called
‘lines’ if they denote yang and ‘broken lines’ if they denote yin. Each pairing of trigrams is
called a ‘hexagram.’ More details can be found in Article 18 [4].
That there are (2^{6}=64) hexagrams is a direct consequence of the Taoist belief
in a fundamental duality in Nature, where active (yang) and reactive, or passive (yin) forces
create a balance, or harmony. This cosmic polarity is represented in the Pillar of Mercy and
the Pillar of Judgement of the Tree of Life, the middle Pillar of Equilibrium representing the
homeostasis resulting from the dynamic interplay of these two complementary, cosmic principles.
The number 64 is the gematria number value of Nogah, the Mundane Chakra of
Netzach. The number value 148 of this Sephirah is the number of yods in the four
separate polygons in S_{1} (see Fig. 18), so that it determines the division in half of
the seven types of polygons with 48 corners that make up the inner form of the Tree of
Life, this number being the number value of Kokab, the Mundane Chakra of Hod, which
follows Netzach in the Tree of
Life.

Figure 42.
The 14 enfolded polygons of the inner Tree of Life with 70 corners share six
corners (black yods) with the outer Tree, which has 70 yods. 64 yods or
corners are intrinsic to the outer and inner Trees. They correspond to the
64 yods outside the root edge on the boundaries of the two dodecagons
whose sectors are tetractyses. 
The holistic
parameter 64 is revealed
by superimposing the outer form of the Tree onto the plane of its inner form (Fig. 42).
When the 16 triangles in the former are tetractyses, three of their corners located at the
positions of Sephiroth (the black yods in Fig. 42) lie on each side pillar and are shared with
each hexagon as corners of its sectors. Of the 70 yods in the outer Tree of Life,
(70−6=64) yods are not corners of triangles in the inner Tree. Similarly, 64 of
the 70 corners in the inner Tree are intrinsic to it. None of the four yods in the root edge
are shared because they lie in the plane formed by the two side pillars and this does not
contain any yods belonging to triangles in the outer Tree of Life. The 94 sectors of the (7+7)
enfolded polygons have 80 corners, (80−6=74) of which are unshared with the outer
Tree, so that the (94+16=110) triangles in the combined Trees of Life have (10+74=84) corners.
This embodiment of the holistic parameter 84 demonstrates par excellence their holistic
character. Their counterpart in the Sri Yantra are the 84 corners of the 42 triangles
surrounding the central one. The pair of dodecagons also display the holistic parameter
64 because 32 yods outside the root edge line each one, so that 64 yods line
both.
According to
Table 10, the 42 Type A triangles in the 3dimensional Sri Yantra have 126 corners & 252
sides. The corners of the central triangle symbolise the Hindu trinity,
or trimȗrti, of Siva, Vishnu & Brahma. If we
regard it as an ordinary triangle without sectors, so that it has three corners and three
sides, 129 corners and 255 sides surround the centre of the Sri Yantra, that is,
384 corners & sides. This is the
25
counterpart
of the polygonal geometry of the inner Tree of Life because 384 corners, sides & triangles
surround the centres of the 14 polygons, each half of the Sri Yantra having 192 corners &
sides and each of the inner Tree of Life having 192 corners, sides & triangles. The 384
corners & sides include the 24 corners & 24 sides of the eight
triangles in the first layer. This corresponds to the 24 corners, sides &
triangles surrounding the
centre of each hexagon.
Surrounding the centres of the six other polygons in each half of the inner Tree of Life are
168 geometrical elements. Similarly, the Sri Yantra has 336 geometrical elements
(168 elements in each half) other than the 48 corners & external sides of the
eight Type A triangles. The counterparts of these in the table of hexagrams are the 48
lines & broken lines in the eight diagonal hexagrams and the 168 lines & broken
lines in the 28 offdiagonal hexagrams in the upper and lower halves of the
8×8 square array of
64 hexagrams (Fig. 43). It is no coincidence that the first layer in the Sri
Yantra has eight triangles. They correspond to the eight diagonal hexagrams, each
trigram corresponding to either three corners (two external, one internal) of each Type A
triangle or its three external
sides.

Figure 45. 168 is the sum
of the 12 odd integers after 1 forming the edges of a square. It is the
number of lines and broken lines in the 28 hexagrams on the four sides of
the 8×8 square array.

26
The 84:84 division
The 168 lines &
broken line in the 28 hexagrams on either side of the diagonal of the
8×
8 array consist of 84
lines and 84 broken lines. The yang lines are coloured dark or light red and the yin
lines are dark or light blue. This colour scheme is applied in order to correspond to
that used earlier to distinguish the two halves of the inner Tree and the two halves of
each half (Fig. 44). As both yang and yin lines exist in each diagonal half of the square array, the two halves

Figure 46. The sum of the first 28
odd integers arranged eight to a side of a square is 840. This is the number
of circularly polarised oscillations in the inner or outer half of a whorl
of the UPA. It is also the number of such oscillations in a halfrevolution
of all 10 whorls. The Tetrad symbolised by the square determines holistic
parameters like this structural parameter of the E_{8}×E_{8}
heterotic superstring.

of the inner
Tree should not be seen as the manifestation of the cosmic polarisation or duality of yang and
yin. Rather, as each half is a whole in itself, it has a yang sector and a yin sector that
correspond to, respectively, S_{1} & S_{2} (and to
S_{1}′
&
S_{2}′). What is remarkable that each sector should
embody the
same

Figure 47.
Two sets of 84 yods surround the centre of a Type C square. The (4×84=336) yods
in two separate Type C squares are the counterpart of the 336 geometrical
elements in the pair of two sets of polygons [(S_{1}, S_{2})
& (S_{1}′, S_{2}′)] that make up the inner Tree of
Life. 
parameter 84 as both the Sri Yantra and the square array of 64 hexagrams —
extraordinary, certainly, but not surprising, in view of the fact that they are all isomorphic
representations of the same archetypal pattern.
The boundary
of the 8×8 array of hexagrams contains 28 hexagrams with 168 lines & broken lines
made up of 84 lines and 84 broken lines (Fig. 45). Arithmetically, this can be expressed as
the sum of the first 12 odd integers after 1 lining the sides of a square, four to each
side, because:
13^{2}
– 1 = 3 + 5 + 7 + 9 + 11 + 13 + 15 = 168.
As
29^{2} – 1 = 3 + 5 + 7 +… +
57 = 840,
the number
840 is the sum of the first 28 odd integers after 1 that can be assigned to the sides of a
square, eight to each side (Fig. 46). That the square, either four units long or
eight units wide, where 8 is the fourth, even integer, should determine both the
superstring parameters 168 and 840 is a remarkable illustration of the Tetrad
Principle at work [5]. The square even embodies the number 168 as the number of yods
surrounding the centre of a Type C square (Fig. 47).
The
archetypal divisions:
384 = 192 +
192,
84 =
48 + 336,
48 = 24 + 24,
336 =
168 + 168,
etc,
displayed by
holistic systems are hidden within them when they are sacred geometries. They are
most
27
explicit in
the 8×8 array of hexagrams because they can be recognised by simple counting. Moreover, the
yin/yang nature of its lines and broken lines is explicit. But this duality does not
correspond to the chiral nature of the geometrical elements in the two mirrorsymmetric
halves of sacred geometries (or almost so, in the case of the Sri Yantra). The 192 lines
& broken lines in each diagonal half of the square array of hexagrams consist of 96
lines and 96 broken lines that form eight copies of the basic set of eight trigrams. The
168 lines & broken lines in each set of 28 offdiagonal hexagrams are seven
copies of the 12 lines & 12 broken lines making up the basic set of eight trigrams. The
two sets of 28 hexagrams are not mere mirror images of each other, reflected across the
diagonal. Nor is the “reflection” of each hexagram the result of replacing each line by a
broken line and vice versa. Rather, its reflection inverts the order of its pair of
trigrams. This is like what happens in the intervals between the eight notes in a musical
scale. The rising interval between two notes with tone ratios m & n (m>n) is m/n. The
falling interval between them is n/m. It is as though the array of 56 offdiagonal hexagrams
is representing the 28 rising intervals between the eight notes in an octave (represented by
the eight trigrams) and their 28 falling intervals, whilst the eight diagonal hexagrams
represent the unit interval between a note and itself. In fact, it is doing so, as the
eightnote diatonic scale is a holistic system, the 36 possible intervals being
symbolised by the 36 corners of the seven enfolded polygons of the inner Tree of
Life. But a more fundamental level of interpretation exists for the trigrams, lines &
broken lines: the 64 hexagrams also express the intervallic composition of the seven
types of diatonic scales [6]. As explained in Article 39 [7], the 192 lines & broken
lines in the upper diagonal half of the square array denote the 192 rising intervals between
the notes in the seven types of diatonic musical scales, whilst the 192 lines & broken
lines in the lower half denote their falling counterparts with reciprocal tone ratios. How
the rising and falling intervals and the types of intervals correspond to the 84 geometrical
elements in S_{1} and the 84 geometrical elements in S_{2} will be discussed
in Section 10. The following correspondences exist between the 64 hexagrams and inner
form of the Tree of Life:
28 upper,
offdiagonal hexagrams
84 yang lines
→ 84 geometrical elements surrounding centres of polygons in S_{1}.
84 yin lines
→ 84 geometrical elements surrounding centres of polygons in S_{2}.
eight
diagonal hexagrams → root edge.
28 lower,
offdiagonal hexagrams
84 yang lines
→ 84 geometrical elements surrounding centres of polygons in S_{1}′.
84 yin lines
→ 84 geometrical elements surrounding centres of polygons in S_{2}′.
The
connection to the UPA
Each whorl of the UPA makes
168 circularly polarised oscillations in every halfrevolution around its axis. The five
inner halfrevolutions and the five outer halfrevolutions of each whorl correspond in the Tree
of Life to, respectively, the five Sephiroth belonging to its Upper Face and the five Sephiroth
spanning

Figure 48. Each halfrevolution of
a whorl of the UPA/superstring comprises 168 circularly polarised
oscillations. Each oscillation has a positive phase (yang) and a negative
phase (yin). The 168 positive phases correspond to a yang line in the
56 offdiagonal hexagrams. The 168 negative phases correspond to a
yin line. They are split into 84 oscillations belonging to an
outer ¼ revolution and 84
oscillations belonging to an inner ¼revolution. The two sets of 28
offdiagonal hexagrams, each with 84 yang lines & 84 yin lines,
represent these two sets of 84 oscillations, each having a positive and a
negative phase.

28
its Lower
Face. This tenfold nature of a whorl is, simply, the manifestation of its Tree of Life
nature, each one corresponding to a Sephirah because the UPA with 10 whorls is a microscopic
manifestation of the Tree of Life blueprint. As a Tree of Life structure in its own right, a
halfrevolution may be divided into an “Upper Face,” namely, a quarterrevolution comprising
84 oscillations that belong to the inner half of the UPA, and a “Lower Face”, a
quarterrevolution with 84 oscillations that belongs to its outer half. Being a whole, it is
represented by a 2ndorder tetractys, the 84 yods surrounding its centre symbolising these
oscillations.
A sinusoidal
oscillation has a positive phase and a negative phase. The upper diagonal half of the 8×8
array with 28 offdiagonal hexagrams composed of 84 yang lines and 84 yin lines represents
the 84 positive and 84 negative phases of 84 oscillations making up the outer half of the
UPA (Fig. 48); the lower diagonal half with 28 offdiagonal composed of 84 yang lines and 84
yin lines represents the 84 positive and 84 negative phases of 84 oscillations making up its
inner half.
Notice that
in all cases of sacred geometries considered so far (including the 64 hexagrams), a
onetoone correspondence exists between certain subsets and their elements in each one.
There is not necessarily the same number of elements in all cases, only the same numbers of
corresponding elements in certain subsets of structural elements. The differences between
these cases arises from the different forms taken by source. It may be a mathematical point,
as in the case of the dodecagon, the root edge in the case of the inner Tree of Life, the
central triangle and bindu in the case of the Sri Yantra and the generic set of eight
trigrams whose pairing with themselves forms the eight hexagrams in the diagonal of
the square array of
64 hexagrams. They all display
a

Figure
49. The five Platonic solids with Type A polygons as their
faces.

symmetry
(point reflection symmetry in the inner Tree of Life, imperfect mirror symmetry of the Sri
Yantra, inversion symmetry for the dodecagon and symmetry between yang and yin lines in the
table of hexagrams). This symmetry exists in them because they are different, but
equivalent, representations of the Whole, which, by definition, is mathematically
complete.
7. The five Platonic
solids Let us consider the construction of the five
regular polyhedra from the tetractys. The simplest case is when their regular faces are Type A
polygons (Fig. 49). To be consistent, the internal triangles generated by joining their centres
to their vertices must be Type A as well. The yod populations of individual Platonic solids
need not be discussed here. Details may be found in Article 3 [8]. For the present purpose,
only the classes of yods and their numbers needed to construct the mathematically complete set
of five Platonic solids need be considered here. Table 11 lists the numbers of yods &
hexagonal yods in the faces and interior of each polyhedron.
Table 11. Yod
populations of the five Platonic solids.
(V = number of
vertices; E = number of edges; F = number of faces; m = number of sectors of a
face).
Polyhedron

Face

V

E

F

m

Surface

Interior

Total







Number
of yods

Number of hexagonal
yods

Number
of yods

Number of hexagonal
yods

Number of
yods

Number of hexagonal
yods

Tetrahedron

Triangle

4

6

4

3

56

48

68 + 1

62

124 + 1

110

Octahedron

Triangle

6

12

8

3

110

96

132 + 1

120

242 + 1

216

Cube

Square

8

12

6

4

110

96

136 + 1

124

246 +
1

220

Icosahedron

Triangle

12

30

20

3

272

240

324 + 1

294

596 + 1

534

Dodecahedron

Pentagon

20

30

12

5

272

240

340 + 1

310

612 + 1

550

Total

50

90

50

–

820

720

1000 +
5

910

1820 + 5

1630

(“1” denotes
the centre of the polyhedron).
1820 yods
surround the centres of the five regular polyhedra. Of these, 50 yods are centres of
faces and 90 yods are centres of internal triangles. Hence, (1820−50−90=1680) yods surround these
(90+50=140) centres. 140 is the number value of Malachim, the
Order of Angels assigned to Tiphareth. As we shall see, these centres, which are the
starting points for the construction of faces and internal triangles, play
29
the same role
as the centres of the (7+7=14) regular polygons of the inner Tree of Life, surrounding which
are (168+168=336) yods that are intrinsic to them. In a remarkable, analogous
way, (1680/5=336) yods on average surround the centres of faces and internal triangles in
the five Platonic solids, 168 yods in each half. These famous polyhedra are the
3dimensional representation of the 1680 turns of each helical whorl of the UPA, there being
168 turns in eah half of one revolution of a whorl. It is unequivocal evidence for C.W. Leadbeater’s paranormally derived number
1680 being embodied in the five Platonic solids constructed from tetractyses. The
average yod population 336 is the number of turns in one revolution of a whorl. The 840 yods
in each half of all five Platonic solids correspond to the 840 turns in each half of a
whorl. There are, of course, not 168 yods in the half of every Platonic solid.
That number represents only an average, made possible by the fact that the number of
Platonic solids is the same as the number of revolutions of each whorl, so that the 10
halves of the former correspond to the 10 halfrevolutions of the latter.
Let us now
examine the classes of yods making up the tetractyses in the faces and interiors of the
Platonic solids, as well as the numbers of each type. There are seven types, excluding the
centres of faces and internal triangles:
Faces
1. vertices
(V);
2. hexagonal yods
on edges (2E);
3. hexagonal yods
on sides of sectors (2mF);
4. hexagonal yods
at centres of sectors (mF);
Interior
5. hexagonal yods
on sides joining centre of polyhedron to vertex (2V);
6. hexagonal yods
on sides of sectors of internal triangles (6E);
7. hexagonal yods
at centres of sectors of internal triangles (3E).
Total number
of yods = 3V + 11E + 3mF = 6 + 14E + 3(m−1)F,
where Euler’s
polyhedral formula for a convex polyhedron:
V – E + F =
2
has been
used. The number “6” measure the number of yods on the two semiaxes passing through two
opposite vertices and the centre of a polyhedron. Excluding their centres, 30 yods line the
10 semiaxes of the five Platonic solids, so we see from Table 11 that 1790 yods surround
these axes. On average, a Platonic solid has 384 yods (192 in each half), of which six yods
line its two semiaxes and 378 yods surround them (189 in each half). These divisions define
all sacred geometries. For example, the 3dimensional Sri Yantra has six hexagonal yods on
the sides of its central triangle/tetractys and 336 yods line the 126 sides of its
surrounding 42 tetractyses (Fig. 36), at the centres of which are 42 hexagonal yods, so that
it has 384 yods. On average, it takes as many yods to build a Platonic solid as it does
to construct the 3dimensional Sri Yantra from tetractyses. This, of course, is not a
coincidence, for they are both expressions of a universal, archetypal pattern. The number
that shapes the 42 triangles is 336, and this is the average number of yods in a Platonic
solid that surround the centres of its faces and interior triangles. The question to be
answered now is whether the 168 yods in an average half of a Platonic solid divide
evenly into two sets of 84 yods, just as we have seen for the 168 yods or geometrical
elements in other sacred geometries.
Table 12
displays the yod populations of half a Platonic solid for the seven classes of yods other
than centres of faces and internal triangles:
Table 12. Yod
populations in half a Platonic solid for the seven classes of yods.

Faces

Interior

Total


Vertices

Hexagonal yods on
edges

Hexagonal yods on
sides

Hexagonal yods at
centres

Hexagonal yods on
sides

Hexagonal yods on sides
of sectors

Hexagonal yods at
centres


Tetrahedron

2

6

12

6

4

18

9

57

Octahedron

3

12

24

12

6

36

18

111

Cube

4

12

24

12

8

36

18

114

Icosahedron

6

30

60

30

12

90

45

273

Dodecahedron

10

30

60

30

20

90

45

285

Total

25

90

180

90

50

270

135

840

En passant, it is worth noting that
each half of the tetrahedron — the building block of solid geometry —
30
has 26
yods in its faces of each half and 31 yods in its interior, showing how YAHWEH, the
Godname of Chokmah with number value 26, and EL, the Godname of Chesed below it with
number value 31, prescribe this Platonic solid. Also, the 12 faces of the complete
dodecahedron contain 260 (=26×10) yods, showing how it is prescribed by YAHWEH. Of
its 285 yods in each half, three yods lie on the semiaxis, leaving 282 yods, where
282 is the number value of Aralim, the Order of Angels assigned to
Binah.
The following
analysis applies to either half of a Platonic solid: according to Table 12, there are
(30+45=75) hexagonal yods at the centres of the 75 tetractyses in the dodecahedron, leaving
(285−75=210) yods lining their sides. The corresponding numbers for the first four Platonic
solids are tabulated below:
Table 13.
Boundary & central hexagonal yods in the first four halfPlatonic solids.
Platonic solid

Central hexagon yods

Boundary yods

Tetrahedron

6 + 9 = 15

57 –
15 = 42

Octahedron

12 + 18 = 30

111 –
30 = 81

Cube

12 + 18 = 30

114 –
30 = 84

Icosahedron

30 + 45 = 75

273 –
75 = 198

Total

150

405

The number of
yods in the tetrahedron and on the sides of tetractyses in the octahedron, cube &
icosahedron = 405 + 15 = 420. This leaves (150−15=135) central hexagonal yods
in these three solids. The octahedron, cube, icosahedron & dodecahedron have
(135+75=210) central hexagonal yods. The number of yods either at centres of tetractyses in
these four solids or lining the sides of tetractyses in the dodecahedron = 210 + 210 =
420.
We find that
the 840 yods in each half of the five Platonic solids comprise two sets of 420 yods. On
average, the 168 yods in half a Platonic solid consist of two sets of (420/5=84)
yods. On average, there are 336 yods in a complete Platonic solid that comprise four sets
of 84 yods surrounding the centres of their faces or internal triangles. This is the
polyhedral counterpart of the 84:84:84:84 division of the holistic parameter 336 found in
previous sections for the inner Tree of Life, the dodecagon and the Sri Yantra.
According to Table 12, the
number of yods in the faces of the five halfsolids is 385. This the sum of the squares of the
first 10 integers:
It is a
beautiful illustration of how the Decad expresses properties of holistic systems. The
dodecahedron has 285 yods in each half surrounding centres of faces and internal triangles,
so the first four Platonic solids have (840−285=555) such yods in each half, i.e., 1110
altogether, where
1110 =
10^{1} + 10^{2} + 10^{3}.
If we imagine
them as concentric and orientated so that none of their yods coincide, the total number of
yods, including their shared centre = 1111, where
1111 =
10^{0} + 10^{1} + 10^{2} + 10^{3}.
This is the
sum of the first four powers of the fourth triangular number. It demonstrates
in a remarkable way how the Tetrad expresses the properties of holistic systems.
Table 14. Yod
and geometrical composition of the five complete Platonic solids.
Polyhedron

Vertices

Hexagonal yods on sides

Central hexagonal yods

Total number of yods

Sides

Triangles

Total number of geometrical
elements

Tetrahedron

4

80

30

114

40

30

74

Octahedron

6

156

60

222

78

60

144

Cube

8

160

60

228

80

60

148

Icosahedron

12

384

150

546

192

150

354

Dodecahedron

20

400

150

570

200

150

370

Total

50

1180

450

1680

590

450

1090

31
It is
instructive and revealing to calculate the average yod and geometrical composition of
a Platonic solid. Table 14 shows how many yods & geometrical elements surround the
centres of faces and internal triangles of each whole Platonic solid (the centres of the
solids are excluded for the moment).
Comments
1. (1180+50=1230) yods
line the 450 tetractyses. Including the 140 corners that are centres of faces or
internal triangles, 1370 yods lines their sides. This property is stunning, for it is the
number of yods in the inner form of the Tree of Life with Type B polygons (Fig. 50)! It
means that the number of yods that shape all the tetractyses needed to construct the five
Platonic solids is precisely the number of yods in the inner Tree of
Life. Moreover,
it means that,
on

Figure 50.
The (7+7) enfolded Type B polygons of the inner Tree of Life contain 1370 yods.
This is the number of yods that line the 450 tetractyses needed to build the
five Platonic solids.

average, 137
yods line the 45 tetractyses in a halfPlatonic solid, showing how the Platonic solids
embody the mysterious number 137 at the heart of atomic and particle physics whose
reciprocal is the approximate value of the finestructure number 137.
2. The number
of vertices & sides is 640. This is the number value of Shemesh, the
Mundane Chakra of Tiphareth, whose Order of Angels is Malachim, with number value
140! If the 140 centres of faces and internal triangles are included, it is
also the number of corners & central hexagonal yods.
3. The 25
vertices of the five halfPlatonic solids are joined by 25 sides of 225 tetractyses, where
225 = 15^{2} and 15 is the number value of YAH, the shorter form of
the Godname assigned to Chokmah. Each side has two hexagonal yods, and therefore the number
of yods (including centres of solids) that line these sides in the five halfsolids = 2×25 +
25 + 5 = 80. This is the number value of Yesod. The number of yods lining the sides
joining centres to the 50 vertices of the Platonic solids = 2×50 + 50 +
5 = 155. This is the number value of ADONAI MELEKH, the complete Godname of
Malkuth.
4. According
to Table 12, the number of hexagonal yods lining tetractyses in the faces of the five
halfPlatonic solids = 90 + 180 = 270. Their number of sides = 135. The 50 faces of
the five Platonic solids have 270 sides of 180 triangles with 50 vertices and
50 more corners that are the centres of faces. The total number of geometrical
elements in the faces = 550, that is, 50 vertices and (50×10) more geometrical
elements. This demonstrates how ELOHIM, the Godname of Binah with number value 50,
prescribes the shapes of the Platonic solids. It is determined by the Decad because
550 = 10×55 and 55 is the tenth triangular number:
55 = 1 + 2 + 3
+ 4 + 5 + 6 + 7 + 8 + 9 + 10.
5. Including
the 140 centres of faces & internal triangles, the total number of geometrical
elements = 1090 + 140 = 1230. This is also the number of boundary yods (excluding the
140 corners) in the 450 tetractyses. The average number is 246. This is the
number value of Gabriel, the Archangel of Yesod. There are, on average, 123
geometrical elements per halfPlatonic solid. 123 is the tenth Lucas number in the
Lucas series:
2, 1, 3, 4,
7, 11, 18, 29, 47, 76, 123, 199, 322, ...
32
named after
French mathematician, François Édouard Anatole Lucas (18421891), who gave the name of
“Fibonacci numbers” to the closely related type of number. The number of yods in a Type D
ngon = 123n + 1. Therefore, the Type D decagon has 1230 yods surrounding its centre (Fig.
51).

Figure 51. 1230 yods
surround the centre of the Type D decagon. This is the number of geometrical
elements in the 5 Platonic solids whose faces & internal triangles are
Type A. The 123 yods per sector denote the 123 geometrical elements on
average in each halfPlatonic solid.

Once again, the
Decad determines the geometrical composition of the five
Platonic solids. It is a remarkable example of the Tetrad Principle at work in
determining holistic parameters, for the decagon has 10 corners, where 10 is the
fourth triangular number, and the Type D decagon is the fourth in the
series of types of division of the decagon:
Type A decagon
→ Type B decagon → Type C decagon → Type D decagon.
6. The five
Platonic solids have 90 edges and 500 (=50×10) more sides of 450
triangles.
The average
Platonic solid has:
1. (50/5=10) vertices, (90/5=18) edges, (1180/5=236) hexagonal yods lining the
(590/5=118) sides of (450/5=90) tetractyses. There are (450/5=90) hexagonal yods at their
centres. The average number of boundary yods excluding vertices (236) is the same as the
number of yods that line the sides of the (7+7) enfolded polygons of the inner Tree of Life
(Fig. 52).

Figure
52. 236 yods line the sides of the (7+7) enfolded polygons of the inner Tree
of Life.

2. 55
geometrical elements in its faces, where 55 is the tenth triangular
number.
3.
(10+236+90=336) yods and (1090/5=218) geometrical elements surrounding centres of faces and
internal triangles. Including, on average, the 28 centres of faces & internal triangles
per solid, (336+28=364) yods and (218+28=246) geometrical elements surround the
centre of the solid, whilst (236+28=264) yods other than vertices line the 90 tetractyses.
This is the number of yods in the seven enfolded polygons of the inner Tree of Life (Fig.
8). Including vertices, (10+264=274) yods on average line the 90 tetractyses, 137 in each
half of a Platonic solid, which also contains 168 yods other than the 14 centres of
faces & internal triangles. As a check, according to Table 11, the five Platonic solids
have 1820 yods, so that the average number of yods per halfsolid = 182 = 137 + 45, where 45
is the average number of hexagonal yods at the centres of the 45 tetractyses in half a
Platonic solid, all five complete Platonic solids having 450 tetractyses.
33
8. Disdyakis
triacontahedron There are only five Platonic solids, each
having regular polygons, that can exist in 3dimensional space.If a polyhedron has two or
more faces that are different polygons, then 13 socalled “Archimedean solids” exist
(15, including two chiral varieties). Their duals are called the Catalan solids (Fig.
53). The Catalan solid with the most faces is the disdyakis triacontahedron. It has
62 vertices (31 in each half), where 62 is the number value of
Tzadkiel, the
Archangel
of

Figure 53. The disdyakis
triacontahedron has the most faces of all the Catalan
solids.

Figure 54. 360 (=36×10) geometrical elements
surround the axis of the disdyakis triacontahedron.

Chesed, and 31 is the number value of EL, the Godname of this Sephirah.
It has 180 edges and 120 faces that are scalene triangles. The disdyakis triacontahedron is
formed by sticking rhombic pyramids on the 30 faces of the rhombic triacontahedron (another
Catalan solid that is the compound of the icosahedron and the dodecahedron). Figure 54 shows
the three types of vertices: 30 A vertices (peaks of the pyramids), 12 B vertices (vertices of
the icosahedron) and 20 C vertices (vertices of the dodecahedron). There is the same number
(60) of the three types of edges.
Many previous
articles by the author have accumulated evidence that the disdyakis triacontahedron is the
single, polyhedral form of the inner Tree of Life (or, more generally, the universal
blueprint governing holistic systems). The five Platonic solids, too, are its polyhedral
form, but only collectively, not in any individual sense. The disdyakis
triacontahedron should be seen as their apotheosis, containing all those types of
information that exist separately in individual members of the mathematically complete set
of regular polyhedra. In confirmation of this conclusion about its status as sacred
geometry, further evidence will now be presented proving that the holistic
divisions:
336 =
168 + 168
and
168 =
84 + 84
found in
earlier sections to be present in the inner Tree of Life, the dodecagon, the Sri Yantra
& the 64 hexagrams are embodied as well in the disdyakis
triacontahedron.
Let us,
firstly, consider the case where all faces and internal triangles formed by joining vertices
to the centre of the polyhedron are simple triangles; later, we shall consider them divided
into their sectors. Table 15 shows the geometrical composition of the faces and interior of
the disdyakis triacontahedron.
Table 15.
Geometrical composition of the disdyakis triacontahedron.

Corners

Sides

Triangles

Total

Faces

2 + 60

180

120

2 + 360

Interior

1

2 + 60

180

3 + 240

Total

3 + 60

2 + 240

300

5 + 600

600
geometrical elements surround the axis, which comprises five elements (three vertices, two
sides).
34
Including the
centre, there are 363 corners & triangles, where 363 is the number value
of SHADDAI EL CHAI, the complete Godname of Yesod. There are 150 (=15×10) corners
& sides surrounding the semiaxis in each half of the polyhedron, where 15 is the
number value of YAH. Including this semiaxis, each half has (150+3=153) corners
& sides, where 153 is the number value of ELOHIM SABAOTH, the Godname of Hod.
Including both semiaxes, there are (150+5=155) corners & sides in the axis and
faces of half the polyhedron. This is the number value of ADONAI MELEKH, the complete
Godname of Malkuth. They comprise one centre, (2+30=32) vertices, 90 edges and 32 internal
sides, i.e., 90 edges and 65 corners & sides. 90 is the number value of MELEKH
and 65 is the number value of ADONAI.
Surrounding
the axis are 24 vertices, 84 edges & 60 triangular faces either above or below the
central 12gon. They comprise (24+60=84) corners & triangles and 84 sides. These
168 geometrical elements consist of two sets of 84 elements, so that the 336
geometrical elements surrounding the axis in both halves of the polyhedron comprise four
sets of 84 elements. This is the polyhedral counterpart of the four sets of 84 structural
elements found in previous sections for the inner Tree of Life, the dodecagon, the
64 hexagrams and the Platonic solids. This pattern is part of the larger
holistic pattern:
384 =
48 + 336 = (24 + 168) + (24 + 168),
which is
revealed particularly clearly in the 64 hexagrams as the 48 lines & broken
lines in the eight diagonal hexagrams and as the 336 lines & broken lines in the 56
offdiagonal hexagrams (168 in each half). The counterpart in the disdyakis
triacontahedron of the central set of 48 lines & broken lines (24 lines, 24
broken lines) are the 24 corners & triangular sectors of the central 12gon and their 24
sides.
Next,
consider each face and internal triangle as a Type A triangle. Table 16 lists the
geometrical composition of the disdyakis triacontahedron:
Table 16.
Geometrical composition of the disdyakis triacontahedron with Type A triangles.

Corners

Sides

Triangles

Total

Faces

2 + 60 + 120 =
2 + 180

180 +
3×120 = 540

3×120 =
360

2 + 1080

Interior

1 + 180

2 + 60 +
3×180 = 2 + 600

3×180 =
540

3 + 1320

Total

2 + 360

2 + 1140

900

5 + 2400

2400
geometrical elements surround the five geometrical elements making up the axis. The 900
(=90×10) triangles comprise 360
(=36×10) triangles in the faces and 540
(=54×10) internal triangles. This simple
36:54 division of the holistic parameter is characteristic of holistic systems, being
embodied in Plato’s Lambda Tetractys [9] as the difference between the sum (36) of
the integers at its corners and the sum (54) of the seven remaining integers at the centre
and corners of a hexagon (Fig. 55). Comparing Tables 14 & 15, the intermediate stage of
transformation of the disdyakis triacontahedron in which only internal triangles are Type A
adds 1320 internal geometrical elements (180 corners, 600 sides, 540 triangles) to the 360
elements (60 corners, 180 sides, 120 triangles) that surround the axis The 1680 geometrical
elements surrounding the axis comprise 840 in each half. This should be compared with the
1680 turns
35
in each whorl
of the UPA described paranormally by C.W. Leadbeater, its outer or inner half having 840
turns. It is additional evidence for the disdyakis triacontahedron being the polyhedral
version of the universal blueprint that governs the form of the basic constituent of matter
that he claimed to remoteview. Notice that the upper or lower half of the polyhedron is now
composed of (600+180=780=78×10) sides and (180+540+60+120=900=90×10) corners &
triangles. As well as embodying the number value 168 of Cholem Yesodeth, its
geometry also embodies the gematria number value 78 of Cholem and the number 90 of
Yesodeth! Evidence that this particle really was (and is) a superstring is the fact
that 720 geometrical elements are added by dividing the 120 faces of the disdyakis
triacontahedron into their 360 sectors, so that:
2400 = 720 +
1680.
Compare this
with the mathematical fact that the rank8 exceptional Lie group E_{8} in
E_{8}×E_{8} heterotic superstring theory has 240 roots, whilst
E_{6}, its rank6 exceptional subgroup, has 72 roots, leaving 168
other roots. We see that there is a correlation between the geometrical
composition of the disdyakis triacontahedron and the root composition of
E_{8} and its subgroups (it is analysed in detail on the author’s website
(10)). This would not exist unless:
1. the UPA is
a state of the E_{8}×E_{8} heterotic superstring, making its “observed”
structural parameter 168 assume grouptheoretical significance;
2. the
disdyakis triacontahedron is the polyhedral representation of holistic systems in which the
group E_{8} plays a fundamental role.
The central
plane of the disdyakis triacontahedron is a 12gon whose corners are three A vertices, three
B vertices & three C vertices. As the number of geometrical elements surrounding the
centre of a Type A ngon = 10n, the centre of the 12gon is surrounded by 120 geometrical
elements. (2400−120=2280=228×10) geometrical elements above and below the central plane
surround the axis. 228 is the 227th integer after 1, where 227 is the 49th prime
number. This shows how EL CHAI, the Godname of Yesod with number value 49, prescribes
the disdyakis triacontahedron.
Figure 56
depicts the types of yods in a face and an internal triangle when both are either
tetractyses or Type A triangles. In the former case, in each half of the disdyakis
triacontahedron (24+60+84=168) corners or centres of tetractyses, (2×84=168)
hexagonal yods on edges and (2×24=48) hexagonal yods on internal sides surround the
axis. In other words, the holistic parameter 384 manifests in each half of the
disdyakis triacontahedron. It embodies the characteristic pattern:
384 =
48 + 336 = 48 + 168 + 168
that we find
in the 64 hexagrams and sacred geometries [11]. As the 24 vertices consist of 12 A
vertices, four B vertices & eight C vertices, i.e., all even numbers, and all the other
sets of yods are exactly divisible by 2, the 384 yods divide naturally into two sets of 192
yods that display the characteristic pattern:
192 = 24 +
168.
The number of
yods surrounding the axis above or below the 12gon that line faces = 24 + 2×84 = 192. The
same pattern therefore appears in just the yods that line the 120 tetractyses in the
disdyakis triacontahedron. The division:
384 = 192 +
192
divides the
polyhedron into two equal halves above or below the central 12gon, each half being composed
of 60 tetractyses bounded by 192 yods.
Let us now
regard as a tetractys every sector of the 300 triangles making up the disdyakis
triacontahedron. For simplicity, the yods on the axis will be excluded from the calculation.
Table 17 shows the number of yods in the faces and interior that surround an axis
perpendicular to the 12gon passing through two diametrically opposite A vertices. Figure 56
depicts the different types of yods.
Table 17. Yod
composition of the disdyakis triacontahedron with Type A triangles.

Faces

Interior

Total


Corners

Hexagonal
yods

Corners

Hexagonal yods




Sides

Central


Sides

Central


Above

24 + 60 = 84

2×84 +
2×3×60 =
528

3×60 =
180

84

2×24 +
2×3×84 =
552

3×84 =
252

1680

12gon

12

2×12 = 24

–

12

2×4×12 = 96

3×12 =
36

180

Below

24 + 60 = 84

2×84 +
2×3×60 =
528

3×60 =
180

84

2×24 +
2×3×84 +
552

3×84 =
252

1680

Total

180

1080

360

180

1200

540

3540

36
Above or
below the central plane are 1680 yods surrounding the axis that include 24 vertices. The
24:24 division of the 48 vertices surrounding the axis above and below the central
plane defines the division:
3360 = 1680 +
1680,
which (apart
from the tetractys/Tree of Life factor of 10) was proved in earlier sections to exist in
certain sacred geometries. The question is whether each yod population 1680 has the natural
division:
1680 = 840 +
840
which (apart,
again, from the factor of 10) was found for these sacred geometries. Figure 56 shows that 10
yods are inside an internal Type A triangle having an edge of the polyhedron as its base.
But there are 84 such edges above the central plane containing the 12gon and 84 edges below
it. Hence, the 1680 yods in each half contain (10×84=840) yods inside 84 internal triangles,
leaving 840 yods either in the faces or on sides of internal triangles. The 3360 yods
outside the 12gon that surround the axis divide into four sets of 840 yods. This conforms
to the pattern:
336 =
4×84
found for
sacred geometries discussed in previous sections.
By exchanging
the 84 blue yods at the centres of internal Type A triangles with 24 black vertices, one set
of 840 yods decreases by 60, becoming 780 (=78×10) yods, and the other set increases by 60,
becoming 900 (=90×10) yods (see Fig. 56). With its yods regrouped this way, the disdyakis
triacontahedron is seen to embody in a natural way the gematria number values 78 and 90 of,
respectively, the words Cholem and Yesodeth making up the Kabbalistic name of
the Mundane Chakra of Malkuth.
Including the
central 12gon, 3540 yods surround the axis of the disdyakis triacontahedron, of which 360
yods are yods at the centres of tetractyses in faces and 540 yods are at centres of
tetractyses in the interior. The number of yods lining the 900 tetractyses = 3540 – 360 –
540 = 2640 = 264×10. The number of yods in the seven enfolded polygons of the inner Tree of
Life is 264 (Fig. 8). Assigning the Decad to all their yods generates the number of yods on
the sides of the tetractyses in the disdyakis triacontahedron. Including just its centre,
the polyhedron has 3541 yods. Amazingly, this is the 496th prime number, where
496 is the number of Malkuth and the dimension of the two symmetry groups SO(32) and
E_{8}×E_{8} providing superstring interactions that are free of quantum
anomalies! [12]. Alternatively, including the seven yods spaced evenly along its axis, the
disdyakis triacontahedron constructed from Type A triangles contains 3547 yods in total,
where 3547 is the 496th prime number after 2. There must be a profound connection
between the prime numbers and the structural parameters of holistic systems, for it is too
improbable for these two properties to be due to chance.
9.
3torus
The German mathematician Felix
Klein (18491925) is wellknown to mathematicians for his work in group theory, complex
analysis, nonEuclidean geometry and on the connections between geometry and group theory. In
1878 he discovered that the quartic equation
u^{3}v
+ v^{3}w + w^{3}u = 0
(henceforth
called the “Klein quartic” [13]) relating three complex variables u, v, w that are the
homogenous coordinates of a point on its Riemann surface has the 336 symmetries of the group
SL(2,7). Klein showed that it is mapped onto itself by 168 analytic transformations,
or orientationpreserving “automorphisms.” The remaining 168
antiautomorphisms” reverse orientation, causing reflection. Hurwitz [14] showed that a
compact Riemann surface of genus g≥2 has at most 84(g−1) automorphisms and the same number
of antiautomorphisms. A Riemann surface with the maximum number of automorphisms for a given
g is called a “Hurwitz curve” of genus g (g≥3). A Hurwitz curve of genus 3 has 168
automorphisms, so that the Klein curve is the Hurwitz curve with the smallest genus g = 3.
In plain English: it has the most symmetries of any surface that is topologically equivalent to a
sphere

Figure
57. The 3torus.

with three
holes, and there is no sphere with fewer holes that has the most symmetries for that number
of holes. Macbeath [15] showed that there is an infinite number of Hurwitz curves. But none
other than the Klein curve with g = 3 and one with genus 7 have equations currently known to
mathematicians [16]. Because the Klein Quartic is a Riemann surface of genus 3, it can be
realised as a regular map in an orientated 2manifold of genus 3, i.e., the 3torus (Fig.
57).
SL(2,7) is
the double cover of the Klein quartic symmetry group PSL(2,7). Its order 336 is the number
of
37

Figure 58. The {7,3} &
{3,7} tiling on the Poincaré disk of the 168 symmetries of the Klein quartic. 
automorphisms
and antiautomorphisms of the Klein quartic. The 168 elements of PSL(2,7) are
represented either by the {7,3} tiling [17] of the Poincaré disc with 24 heptagons having
168 vertices, three heptagons meeting at each of the 56 vertices, or by its {3,7}
tiling with 56 triangles having 168 vertices, seven triangles meeting at each of the
24 vertices (Fig. 58). The Riemann surface of the Klein quartic is a 168sheeted
covering of the sphere, branched over the following three points of the sphere:

Above one of these
points, the 168 sheets join in groups of seven to give 24 points of the
surface. These are the points of inflection of the curve described by the Klein
quartic. They are also the Weirstrass points. The 24 orbits of subgroups of order 7
correspond to these points.

Above another branch
point, there are 84 points of the surface, where the sheets join in twos. These are
the sextactic points through which pass a conic section that has sixfold contact
with the curve. The 84 orbits of subgroups of order 2 correspond to
them.

Above the third
branch point, the sheets join in threes to give 56 points of the surface. These 56
points are the points of contact of the Klein quartic with the 28 bitangents, or
lines that are tangent to the curve at two points. The 56 orbits of subgroups of
order 3 correspond to them.
The author’s
website and articles has provided illustrations of not only how the numbers 336 &
168 are characteristic parameters of sacred geometries but also how they naturally
display the factorisations:
336 =
2×168
168 =
84×2 = 56×3 = 24×7.
If we dismiss as implausible
that these properties appear merely by coincidence, it implies that the geometry underlying the
Klein quartic conforms to the archetypal patterns of sacred geometries. This was explored in
earlier articles ([18], [19]). To avoid repetition, we shall focus here on showing how the
{3,7} tessellation of the 3torus by triangles that are either tetractyses or Type A triangles
displays the 168:168


Figure 59. The 3torus
constructed from 4 triangular prisms and 6 square antiprisms.

Figure 60. The triangular
prism and the square antiprism constructed from tetractyses.

38
division of
336 and the 84:84 division of 168, thereby providing more evidence that the geometry
underlying the Klein quartic conforms to the patterns characteristic of sacred
geometries.
Figure 59
shows the {3,7} tessellation onto the 3torus of the 56 orbits of order 3 that correspond to
the 56 bitangents of the Klein quartic. The 3torus is topologically equivalent
to six square antiprisms (Fig. 60) with their 12 square faces stuck to the 12 square faces
of four triangular prisms in a tetrahedral formation, each antiprism being given a twist.
This leaves 56 triangular faces. Each triangular prism has six vertices, nine edges &
two triangular faces; each square antiprism has eight vertices, 16 edges, of which eight
edges coincide with edges of the former when they are stuck together, and eight triangular
faces. This leaves 24 vertices and (4×9 + 6×8 = 84) edges. Compare this with the 24 vertices
and 84 edges of 60 triangles that surround the axis of the disdyakis triacontahedron in its
upper or lower halves (Fig. 61). In fact, just as it has 144 edges & faces, so the six
square antiprisms have 144 edges & triangular faces. Of itself, this does not prove some
kind of isomorphism exists between the polyhedron and the 3torus constructed from prisms
and antiprisms, because it is illegitimate to compare an intact system with the separate
components of another — whether holistic or not! However, the division:
168 =
24 + 144
is
characteristic of such systems because it is part of the tetractys representation of the
holistic parameter 240:
For example,
the 168 yods needed to transform the sectors of a dodecagon into Type A triangles
comprise 24 hexagonal yods on its sides and 144 yods inside it (Fig. 62), whilst the
168 corners & sides in two separate Type B dodecagons consist of their 24 corners
and 144 sides & other corners (Fig. 63). This


Figure 62. The 168
yods needed to turn the sectors of a dodecagon into Type A triangles consist of
24 red hexagonal yods lining its sides and 144 black yods. 
Figure 63. 240 geometrical
elements surround the centres of the two Type B dodecagons. They comprise
72 blue triangles and their 168 corners & sides. The
latter comprise 24 red corners of the dodecagons and 144 black corners &
sides.

division
reflects the fundamental Kabbalistic distinction between Malkuth and the six Sephiroth of
Construction preceding it:
7 = 1 +
6
168 = 24×7 =
24×1 + 24×6 = 24 + 144.
39
The 24
vertices surrounding the axis above or below the equator of the disdyakis triacontahedron
minimally mark out the skeletal shape of each half of the polyhedron, as expressed by
its 144 edges & faces (see Fig. 61); similarly, the 24 vertices of the four triangular
prisms minimally define the prismatic construction of the 3torus, the 48 vertices of
the six square antiprisms not adding to its shape because they coincide with these vertices
when the 12 square faces of the triangular prisms are stuck to the 12 square faces of the
square antiprisms.
To
investigate other analogous properties, we will analyse the geometrical and yod composition
of the prisms and antiprisms when separate and when assembled. Table 18 shows the
populations of geometrical elements and yods in the separate polyhedra when their faces are
simple triangles/tetractyses and when they are Type A triangles.
Table 18.
Geometrical/yod composition of the four triangular prisms & the six square
antiprisms.

Tetractys

Type A
triangle


Triangular
Prism

Square
Antiprism

Total

Triangular
Prism

Square
Antiprism

Total

Number of vertices

4×6 = 24

6×8 = 48

72

24 + 4×2 = 32

48 +
48×1 = 96

128

Number of edges

4×9 = 36

6×16 = 96

132

36 + 8×3 =
60

96 + 48×3 =
240

300

Number of triangles

4×2 = 8

6×8 = 48

56

8×3 = 24

48×3 =
144

168

Number of geometrical
elements

68

192

260

116

480

596

Number of hexagonal yods on
sides of tetractyses

2×36 =
72

2×96 = 192

264

2×60 = 120

2×240 = 480

600

Number of hexagonal
yods

8 + 72 =
80

48 + 192 =
240

320

24 + 120 = 144

144 + 480 = 624

768

Number of yods on sides of
tetractyses

24 + 72 =
96

48 + 192 =
240

336

32 + 120 = 152

96 + 480 = 576

728

Number of yods

8 + 96 = 104

48 + 240 =
288

392

24 + 152 = 176

144 + 576 = 720

896

Comments
1. The 10 separate
prisms & antiprisms have 72 vertices. This is the 36th even integer,
showing how ELOHA, the Godname of Geburah with number value 36, prescribes the
geometry of the 3torus in a minimal way. They have 132 edges, where 132 is the 65th
even integer after 2. This shows how ADONAI, the Godname of Malkuth with number value
65, prescribes the form of the 10 prisms & antiprisms through their edges. The
two types of structural units have 14 vertices, 25 edges & 10 triangular faces, i.e.,
49 geometrical elements, showing how they are prescribed by EL CHAI, the Godname of
Yesod with number value 49.
2. There are 260
(=26×10) geometrical elements in the 10 separate prisms & antiprisms. This is
strong evidence for the holistic nature of the 3torus prescribed by the Godname YAHWEH with
number value 26 because there are 260 yods outside the root edge of the seven
enfolded polygons, whilst 260 geometrical elements make up the 3dimensional Sri Yantra
(Fig. 64).


Figure 64. The Godname YAHWEH with number value 26 prescribes the yod
population of the inner Tree of Life and the number of geometrical elements in both
the 3d Sri Yantra & the 10 separate prisms & antiprisms that form the
3torus. 
3. 264 hexagonal yods
line the 132 edges of the 10 prisms & antiprisms. This is the number of yods in the
seven enfolded polygons of the inner Tree of Life when their 47 sectors are tetractyses. The
132 hexagonal yods in half the complete set of prisms & antiprisms have their
counterparts in the 132 yods
40
making up
S_{1} and the 132 yods in S_{2} — the two halves of the set of seven
polygons. This correlation indicates that the 3torus is the counterpart of one set
of seven enfolded polygons. The counterpart of the set in the mirrorimage half of
the inner Tree of Life will be discussed shortly.
4. 336 yods line the 132
sides of the 56 tetractyses (Fig. 65). The number 336 is the order of SL(2,7), the group
describing the 336 automorphisms and antiautomorphisms of the Klein quartic whose Riemann
surface is topologically equivalent to a 3torus. It is remarkable that this parameter of
holistic

Figure 65. 336 yods line
the edges of the 10 separate prisms and antiprisms

systems
should be embodied in the separate building blocks needed for the {3,7}
tessellation of the 3torus. Not that it is unique, for 336 yods are needed to turn the
separate polygons in either S_{1} or
S_{2} with 24 sectors into Type B
polygons, whilst the seven separate Type B polygons have 336 corners & sides of 144
triangles [20]. Furthermore, compare this property with the fact that 336 yods line the
sides of the 42 triangles in the Sri Yantra (see Fig. 35), whilst 336 yods other than
corners surround the centres of two joined, Type B dodecagons (see Fig. 31). This number
appears in the building blocks of the {3,7} tessellation of the 3torus not by accident but
because the geometry underlying the Klein quartic conforms to the universal patterns of
sacred geometry.
5. The
168 sectors of the 56 Type A triangles contain 896 yods, of which 56 are their
centres, leaving 840 yods that surround these centres. When the triangles are tetractyses,
there are 336 yods surrounding these same centres. Here is the meaning of these two numbers
in the context of the structure of the UPA: one revolution of each helical whorl with 1680
turns contain 336 turns and the 2½ revolutions constituting an outer or inner half of a
whorl contain 840 turns. It is remarkable that the ten prisms & antiprisms embody three
structural parameters (168, 336 & 840) of the basic unit of matter paranormally
described over a century ago by Annie Besant & C.W. Leadbeater. Far from being
coincidental, it points towards the Klein quartic and its connection to the octonions
playing a leading role in the stillsought unification of superstrings and supergravity by
Mtheory.
Let us now
consider the prisms and antiprisms joined together at their square faces to form the {3,7}
tessellation of the 3torus. Table 19 lists their geometrical and yod
composition.
Table 19.
Geometrical/yod composition of the 3torus constructed from tetractyses & Type A
triangles.

Tetractys

Type A
triangle


Triangular
Prism

Square
Antiprism

Total

Triangular
Prism

Square
Antiprism

Total

Number of vertices

4×6 = 24

–

24

24 + 4×2 = 32

48×1 = 48

80

Number of edges

4×9 = 36

6×8 = 48

84

36 + 8×3 =
60

48 +
48×3 = 192

252

Number of triangles

4×2 = 8

6×8 = 48

56

8×3 = 24

48×3 = 144

168

Number of geometrical
elements

68

96

164

116

384

500

Number of hexagonal yods on sides
of tetractyses

2×36 =
72

2×48 = 96

168

2×60 = 120

2×192 = 384

504

Number of hexagonal
yods

8 + 72 =
80

48 + 96 =
144

224

24 + 120 = 144

144 + 384 = 528

672

Number of yods on sides of
tetractyses

24 + 72 = 96

96

192

32 + 120 = 152

48 + 384 =
432

584

Number of yods

8 + 96 = 104

48 + 96 =
144

248

24 + 152 = 176

144 + 432 = 576

752

Comments
1. When each face
is a tetractys, the 56 tetractyses contain 248 yods. Given the
connection to UPAs as superstrings indicated by the presence of three of their structural
parameters, it is very remarkable that the dimension 248 of the
very Lie group E_{8} that describes the unified force of one of the two types
of heterotic superstrings should manifest as the number of yods needed to construct the
prisms and antiprisms from tetractyses. Once again, it is highly implausible to be due to
chance. Rather, the presence of the holistic parameter 248 is yet
again an indication of the sacred geometrical nature of the {3,7} tessellation of the
3torus whose 56 triangles represent the 168 automorphisms of the
Klein quartic. But it is more than this, for, if each yod corresponds to one of the
roots of E_{8}, then we should expect to see the geometry of the prisms and
antiprisms dividing this group up into certain subgroups. Indeed, we do.
The 248 yods comprise eight yods at the centres of the faces of
the four triangular prisms, leaving 240 yods that are made up of their 24
vertices, 48 hexagonal yods at the centres of the faces of the
six
41
antiprisms,
i.e., 72 yods, and 168 hexagonal yods on the 84
sides of the 56 tetractyses. The division:
248 = 8 + 240
corresponds to the eight simple roots of
E_{8} and its 240 roots. The division:
240 = 72 + 168
corresponds to
the 72 roots of E_{6}, an exceptional subgroup
of E_{8}, and to the 168 remaining roots of
E_{8}.
We see that the tetractys construction of the tessellated 3torus reveals some of the
root structure of E_{8} (this will not be explored further here).
The division:
168 = 2×84
that was shown in previous sections to appear
in sacred geometries manifests trivially in the pair of hexagonal yods lying on each of the
84 sides of the tetractyses in the six square antiprisms. It denotes the 84 sextactic points
where the 168 sheets of the Riemann surface of the Klein quartic join in 84 pairs.
It corresponds in the 64 hexagrams as the 84
solid lines (yang) and the 84 broken lines (yin) that make up the 56 trigrams in the
upper or lower set of 28 offdiagonal hexagrams. The division:
336 = 2×168
that previous sections showed resulted from
the mirror symmetry of the sacred geometry (or near mirror symmetry, in the case of the Sri
Yantra) expresses the fact that the number of antiautomorphisms of the Klein quartic is
equal to the number of its automorphisms. The former can be represented by a version of the
3torus which is turned inside out, reversing the orientations of all triangles (Fig. 66).
Their counterparts in the inner Tree of Life are the 168 geometrical

Figure
66. The 496 yods in the (56+56) tetractyses tiling the 3‑torus and
its turnedout version symbolise the 496 roots of
E_{8}×E_{8}.

elements outside the root edge that are
intrinsic to one set of seven enfolded polygons and their 168 chiral counterparts in
the other set of seven enfolded polygons. The pair of 3tori contains 336 hexagonal yods on
sides of the two sets of 56 Type A triangles, there being (248+248=496) yods altogether. The
number 496 is the crucial dimension of any gauge symmetry group that describes
superstring forces free of
quantum anomalies [12]. The conjunction
of

Figure 67. Correspondence between the 1st (6+6) enfolded polygons of the inner
Tree of Life and the {3,7} tessellation of the 3torus & its version turned
inside out.

structural parameters of a particle claimed to have been
remoteviewed over a century ago and this theoretical requirement for a certain type of
superstring is undeniable. These numbers have not been concocted by cherrypicking. Instead,
they quantify natural, mathematical features of the tessellated 3torus. It is evidence that
UPAs are E_{8}×E_{8} heterotic
superstrings.
2. The
number of yods on the 84 sides of the 56 tetractyses is the sum of the 24 vertices of the
four triangular prisms and the 168 (=84×2) hexagonal
yods on these sides:
42
24 + 168 = 192.
This is a
characteristic division of holistic systems. For example, Figure 67 shows that, associated
with each set of the first six enfolded polygons of the inner Tree of Life, are 24 corners
and 168 other yods. The pair of chiral 3tori exhibit the
complete division displayed by holistic systems:
384 = 192 + 192 =
48 + 336,
where
48 = 24 + 24
is the number
of vertices of their eight triangular prisms and
336 =
168 + 168.
is the number
of hexagonal yods on the sides of their 112 tetractyses. The number
48 is the number value of Kokab, the Mundane Chakra of Hod, and
112 is the number value of Beni Elohim, the Order of Angels assigned to this
Sephirah. Figure 31 in Article 43 compares how the holistic parameter 384 is embodied in
various sacred geometries, including the {3,7} tessellation of the 3torus.
3. The 10 prisms &
antiprisms constructed from Type A triangles form a 3torus with 500 (=50×10)
geometrical elements. This shows how ELOHIM, the Godname of Binah with number value 50,
prescribes the formdetermining geometrical composition of the 3torus constructed from Type A
triangles. It is a clear demonstration of the archetypal power of Godnames in their
prescription of holistic systems. It is the counterpart of the 500 geometrical elements needed,
starting from their 50 vertices, to shape the faces of the five Platonic solids
(see comment 4, p. 33).

Figure 68.
The 672 hexagonal yods in the 168 tetractyses of the 56 Type A triangles
of the 3torus comprise two sets of 336 yods (336 red or blue yods; 336 green
yods). 
4. The
168 triangles making up the 3torus when the triangular faces of the prisms
&antiprisms are Type A consist of the 24 triangles in the former and the 144 triangles in
the latter. This is another example of the pattern displayed by holistic systems that was
discussed on p. 39. The 56 sets of three triangles and their counterparts in the 3torus turned
inside out are the geometrical analogue of the 56 trigrams in the 28 offdiagonal hexagrams
that make up the upper and lower diagonal halves of the 8×8 square array of
hexagrams.
5. The
168 tetractyses in the 3torus tessellated with 56 Type A triangles contain
168 hexagonal yods on the 84 edges of the prisms & antiprisms,
168 hexagonal yods at the centres of these tetractyses and 168 pairs of
hexagonal yods on sides inside the Type A triangles (Fig. 68). The 672 hexagonal yods comprise
two sets of 336 yods. This division conforms to the patterns of sacred geometry. For example,
the first four Platonic solids contain 672 yods (336 yods in each half) when they are
constructed from tetractyses [22], whilst each set of seven separate Type B polygons in the
inner form of the Tree of Life has 336 corners & sides surrounding their centres [23]. The
3torus and its version turned inside out have (672+672=1344) hexagonal yods. This is the
number of yods surrounding the centres of the 14 enfolded Type B polygons of the inner Tree of
Life that are unshared with the outer Tree of Life [24]. This demonstrates that the 3torus
with Type A triangles is the counterpart of one half of the inner Tree of Life and that its
turned insideout version is the counterpart of its other half. The turning inside out of the
3torus, which reverses the orientation of all triangles, is analogous to the reflection of all
geometrical elements/yods in the seven enfolded polygons across the root edge that joins them
to their chiral images.
For further
analysis of the remarkable correspondences between the 3torus and sacred geometries, see
reference 19.
10. The seven diatonic
musical scales Two divisions characteristic of holistic
systems:
336 =
168 + 168
and
168 = 84 + 84
are embodied
not only in sacred geometries but also in the seven ancient, diatonic musical scales
that
43
were known to
the ancient Greeks as “harmoniae” and which entered the plainsong of the Roman Catholic
Church. That these patterns exist in music will be demonstrated in this section in two
ways.
The
Pythagorean musical scale consists of five tone intervals of 9/8 (T) and two leimmas of
256/243 (L):
By starting with successive notes to generate scales of eight notes and then normalising the
tone ratios so that the tonic in each scale has a tone ratio of 1, the pattern of intervals is
repeated at the eighth sequence, thereby generating seven distinct, octave species:
8. 







T 
L 
T 
T 
T 
L 
T 
Hypomixolydian
(Mode 8, D'
scale) 
7. 






T 
T 
L 
T 
T 
T 
L 

Hypolydian
(Mode 6, C scale) 
6. 





L 
T 
T 
L 
T 
T 
T 


Hypophrygian
(Mode 4, B scale) 
5. 




T 
L 
T 
T 
L 
T 
T 



Hypodorian
(Mode 2, A scale) 
4. 



T 
T 
L 
T 
T 
L 
T 




Mixolydian
(Mode 7, G scale) 
3. 


T 
T 
T 
L 
T 
T 
L 





Lydian
(Mode 5, F scale) 
2. 

L 
T 
T 
T 
L 
T 
T 






Phrygian (Mode 3, E
scale) 
1. 
T 
L 
T 
T 
T 
L 
T 







Dorian (Mode 1, D scale) 
These are the
musical modes used for hundreds of years in the plainsong of the Roman Catholic Church. Mode
8 (Hypomixolydian) is the D scale with different ‘dominant’ and ‘finalis’ notes. Table 20
displays the values of the tone ratios of their notes with Pythagorean tuning.
Table 20. Tone
ratios of the seven diatonic scales.
Musical scale

Tone
ratio

B
scale

1

256/243

32/27

4/3

1024/729

128/81

16/9

2

A
scale

1

9/8

32/27

4/3

3/2

128/81

16/9

2

G
scale

1

9/8

81/64

4/3

3/2

27/16

16/9

2

F
scale

1

9/8

81/64

729/512

3/2

27/16

243/128

2

E
scale

1

256/243

32/27

4/3

3/2

128/81

16//9

2

D
scale

1

9/8

32/27

4/3

3/2

27/16

16/9

2

