ARTICLE 48
by
Stephen M. Phillips
Flat 4, Oakwood House, 117119 West Hill Road. Bournemouth. Dorset
BH2 5PH. England.
Website: http://www.smphillips.mysite.com
Abstract
As a development of earlier work
showing that certain subsets of the (7+7) polygons of the inner form of the
Tree of Life have analogous properties to the latter, this article proves that
the first (4+4) polygons also constitute such a subset. It analyses their
geometrical composition and their yod populations generated from tetractyses of
10 points (“yods”) by regarding them as either Type A or Type B polygons.
Through their gematria number values, the Godnames of the 10 Sephiroth of the
Tree of Life prescribe the first (4+4) polygons. The Decad determines the 120
geometrical elements outside the shared, root edge of the (4+4) enfolded Type A
polygons because 120 is the sum of the first 10 odd integers after 1. Previous
articles have established that the number 90 is a parameter of holistic
systems, e.g., the five Platonic solids have 90 edges and the sum of the
integers in the Platonic Lambda Tetractys is 90. It appears in the first four
separate Type B polygons as the 90 sides of their 54 triangles and as their 90
corners & triangles. The first four enfolded Type A polygons also have 90
yods outside their root edge. The 108 sectors of the (4+4) separate Type B
polygons have 260 corners & sides. Their counterparts in the inner Tree of
Life are the 260 yods in the seven enfolded Type A polygons outside their root
edge. The first four separate Type B polygons embody the 248 roots of the
superstring gauge symmetry group E_{8} as the 248 yods other than
corners that are outside their root edges. Enfolded, these polygons have 247
yods. Their counterparts in the inner Tree of Life are the 247 yods lining
sides of tetractys sectors of the seven separate Type A polygons. 192 yods
outside the root edge line the sides of the tetractyses making up each set of
the first four enfolded Type B polygons. This 192:192 division is
characteristic of representations of holistic systems, being found, for
example, in the 192 lines and 192 broken lines that make up the 64 hexagrams in
the I Ching table. The (248+248) yods outside the root edge of two joined, Type
C hexagons symbolise the (248+248) roots of the heterotic superstring gauge
symmetry group E_{8}×E_{8}'. With 2ndorder tetractyses as
sectors, two joined hexagons have 840 yods outside the root edge. This is the
number of circular turns in the inner or outer halves of each helical whorl of
the basic unit of matter described by the Theosophists Annie Besant and C.W.
Leadbeater. The 10 pairs of Type A hexagons enfolded in 10 overlapping Trees of
Life have (840+840) yods that symbolise the (840+840) turns in a whorl. The 80
corners of the 108 triangles in the first (4+4) separate Type B polygons
correspond to the 80 corners of the 94 sectors of the (7+7) enfolded Type A
polygons of the inner Tree of Life. They also correspond to the 80 yods in the
lowest Tree of Life. These and other correspondences demonstrate that the first
(4+4) polygons constitute a holistic structure that is isomorphic to the
greater whole of the inner Tree of Life.

1
Table 1. Gematria number values of the 10 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 
The Sephiroth exist in the four
Worlds of Atziluth, Beriah, Yetzirah and Assiyah. Corresponding to them are the
Godnames, Archangels, Order of Angels and Mundane Chakras (their physical
manifestation, traditionally symbolised by celestial bodies). 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 to a word a
number value that is the sum of the numbers of its letters.

(All numbers in this table referred to in the article are written
in boldface).
2
1. Geometrical properties of the first four polygons
Many previous articles have established the role of the
Tetrad Principle in determining the properties of holistic systems embodying the universal design or pattern found in the sacred geometries of some major
religions, as well as in objects wellknown to mathematicians. They have revealed numerous
examples of how the Tetrad (4) and the integers 1, 2, 3 & 4 symbolized by the four rows of dots in
Pythagoras’ tetractys define or express the attributes of these systems. In view of this, the
first four regular polygons of the inner Tree of Life and the hexagon — the fourth polygon —
might therefore be expected to display defining parameters of these systems, such as the numbers
384, 248 & 168. This expectation will be confirmed by examining their
geometrical and yod composition. The appearance of so many of the 50 gematria number values
(shown in boldface) of the ancient Hebrew names of the 10 Sephiroth of the Tree of Life and
their manifestation in the four Kabbalistic Worlds of Atziluth, Beriah, Yetzirah & Assiyah
(Table 1) cannot, plausible, be explained away in terms of coincidence because
their occurrence is too frequent to be due to chance. Instead, these numbers are signatures of
the underlying sacred geometrical nature of these polygons. Previous articles have shown that
various subsets of the (7+7) regular polygons making up the inner Tree of Life are counterparts
of the whole set in the sense that their properties are analogous, being measured by the same
set of parameters. The first (4+4) regular polygons are another example of this amazing
fractallike property of the inner Tree of Life, wherein smaller and smaller subsets of the 14
polygons display analogous properties quantified by the same set of numbers, namely, the
number values of the 10 Sephiroth, their Godnames, Archangels, Orders of Angels & Mundane
Chakras.
A polygon with n sides is known in mathematics as an
‘ngon.’ It is transformed into a Type A polygon by dividing it up into its n sectors, which may
then be further turned into tetractyses. Figure 1a shows the Type A triangle, or 3gon with its three sectors turned into
tetractyses. When each sector of the ngon is divided into three sectors and each new sector
turned into a tetractys, it becomes a Type B ngon (Fig. 1b shows the Type B triangle). This transformation may be extended
indefinitely, creating Type C ngons, Type D ngons, etc. This article will consider only Type
A, Type B & Type C ngons.
Instead of dividing sectors into ever smaller triangles/tetractyses, they may be
regarded as members of an infinite sequence of nthorder tetractyses as ever greater
differentiations of Unity (the Monad):
We shall consider only the 1st and 2ndorder tetractyses. In this section, we
shall examine the geometrical properties of the triangle, square, pentagon & hexagon,
considering, firstly, the case when they are Type A polygons and then when they are Type B. The
next section will analyse their construction from tetractyses in each case.
The following formulae are useful for examining geometrical properties of
ngons:

Type A ngon

Type B ngon

Number of corners =
Number of sides =
Number of triangles =
Number of corners & sides =
Number of corners & triangles =
Number of sides & triangles =
Number of geometrical elements =

n+1
2n
n
3n+1
2n+1
3n
4n+1

2n+1
5n
3n
7n+1
5n+1
8n
10n+1

3
First four Type A polygons
The separate polygons will be considered first.
First four separate
polygons
Table 2 shows the geometrical composition of each Type A polygon:
Table 2. The geometrical composition of
the first four separate Type A polygons.

triangle 
square 
pentagon 
hexagon 
Total 
Number of corners of sectors =

3+1 
4+1 
5+1 
6+1 
18+4 
Number of sides =

6 
8 
10 
12 
36 
Number of triangles =

3 
4 
5 
6 
18 
Number of corners & sides =

9+1 
12+1 
15+1 
18+1 
54+4 
Number of corners & triangles =

6+1 
8+1 
10+1 
12+1 
36+4 
Number of sides & triangles =

9 
12 
15 
18 
54 
Number of corners,
sides & triangles = 
12+1 
16+1 
20+1 
24+1 
72+4 
First (4+4) separate polygons
 Number of corners of polygons = 2×18 = 36.
 Number of sides of polygons = 2×18 = 36.
 Number of corners & sides of polygons = 2×36 =
72.
 Number of triangles = 2×18 = 36.
 Number of corners of triangles = 2×22 = 44.
 Number of sides of triangles = 2×36 =
72.
 Number of corners & sides of triangles = 2×58 = 116.
 Numbers of corners & triangles = 2×40 = 80.
 Number of sides & triangles = 2×54 = 108.
 Number of corners, sides & triangles = 2×76 = 152.
There are (152 – 4×3 = 140) geometrical elements outside the sides that become
the root edge when the polygons are enfolded.
Comments

72 corners, sides & triangles surround the
centres of the four polygons. They have 76 geometrical elements. The
two sets of polygons separated by the root edge have 2×76 + 3 =
155 geometrical elements. ADONAI MELEKH, the Godname of Malkuth with
number value 155 and YAHWEH ELOHIM, the Godname of Tiphareth with
number value 76, prescribe the geometrical composition of the
polygons. 140 elements are outside the root edges of the (4+4)
polygons, where 140 is the number value of Malachim, the Order
of Angels assigned to Tiphareth. ELOHA, the Godname of Geburah with number value
36, prescribes the number of sides of triangles in each set of four
polygons, as well as the number of triangles in both sets. It also prescribes the total
number of geometrical elements surrounding the centres of each set because
72 is the 36th even integer.

(2×72=144) geometrical elements surround the centres of the
(4+4) polygons, where
144
=

1^{0}
2^{0} 3^{0}
4^{0}
1^{1} 2^{1}
3^{1} 4^{1}
1^{2} 2^{2}
3^{2} 4^{2}
1^{3} 2^{3}
3^{3} 4^{3}

It demonstrates how the integers 1, 2, 3 & 4 express
properties of holistic systems. The (4+4) polygons have 36 corners, where
36 = (1+3+5+7) + (2+4+6+8),
i.e., 36 is the sum of the first four odd integers and
the first four even integers, showing how the Tetrad determines the number of corners of
the (4+4) polygons. The 18 sectors of the first four polygons have 22 corners, where
22 = 1^{4} + 2^{3} +
3^{2} + 4^{1}.
 The (4+4) polygons have 80 corners & triangles, where
80 is the number value of Yesod.
First four enfolded polygons

Number of corners of polygons = 3 + (4–2=2) + (5–2=3) + (6–2=4)
= 12; (12–2=10) corners are outside their root edges (“external”).
 Number of sides of polygons = 3 + (4–1=3) + (5–1=4) + (6–1=5) =
15; (15–1=14) sides are external.
 Number of corners & sides of polygons = 12 + 15 = 27;
(27–3=24) corners & sides are external.
 Number of triangles = 18 – 1 = 17 (the triangle fills one sector of
the hexagon).
4
 Number of corners of triangles = (3+1=4) +
(2+1=3) + (3+1=4) + 4 = 15; (15–2=13) are external.
 Number of sides of triangles = (3+3=6) +
(3+4=7) + (4+5=9) + (5+4=9) = 31; (31–1=30) are external.
 Number of corners & sides of triangles
= 15 + 31 = 46; (46–3=43) are external.
 Number of triangles & their sides = 17
+ 31 = 48.
 Number of corners, sides & triangles =
46 + 17 = 63; (63–3=60) are external.
First (4+4) enfolded polygons
 Number of corners of polygons = 2×10 + 2 = 22; (22–2=20) are
external.
 Number of sides of polygons = 2×14 + 1 = 29; (29–1=28) are
external.
 Number of corners & sides of polygons = 22 + 29 = 51;
(51–3=48) are external.
 Number of corners of triangles = 2×13 + 2 = 28; (28–2=26)
are external.
 Number of sides of triangles = 2×30 + 1 = 61; (61–1=60) are
external.
 Number of triangles = 2×17 = 34.
 Number of corners & sides of triangles = 28 + 61 = 89;
(89–3=86) are external.
 Number of corners & triangles = 28 + 34 = 62;
(62–2=60) are external.
 Number of triangles & their sides = 34 + 61 =
95.
 Number of corners, sides & triangles = 89 + 34 = 123;
(123–3=120) are external.
Comments

The first four enfolded polygons have 15 sides and
15 corners of 17 sectors with 31 sides, showing how YAH, the Godname of
Chokmah with number value 15, and EL, the Godname of Chesed with number value
31, prescribes the shape of the four polygons. They have 63 geometrical
elements. The top corner of the hexagon coincides with the bottom corner of the hexagon
enfolded in the next higher, overlapping Tree of Life. This means that 62
elements are intrinsic to each set of the first four enfolded polygons. The number
62 is the 31st even integer and the number value of Tzadkiel, the
Archangel of Chesed. The first (4+4) enfolded polygons also have 62 corners
& sectors. YAHWEH with number value 26 prescribes the 26 corners of
sectors outside the root edge.

120 geometrical elements are outside the root edge of the first
(4+4) enfolded polygons, where
120 = 2^{2} + 4^{2} +
6^{2} + 8^{2}.
Two of the 123 geometrical elements (top corners of the
two hexagons) in the first (4+4) enfolded polygons are shared with the corresponding polygons
enfolded in the next higher Tree of Life. The number of intrinsic geometrical elements = 123 – 2
= 121 = 11^{2}. i.e., it is the square of the tenth integer after 1, 120 is the
sum of the first 10 odd integers after 1, showing how the Decad determines the geometrical
composition of the first (4+4) polygons. 30 sides of sectors and 30 corners & sectors lie on
either side of the root edge, where
30 = 1^{2} + 2^{2} + 3^{2} +
4^{2}.

The 34 sectors of the first (4+4) enfolded polygons have 89
corners & sides. As the top corners of both hexagons coincide with the bottom
corners of the hexagons enfolded in the next higher Tree of Life, there are 87
corners & sides that are intrinsic to the first (4+4) enfolded polygons. 87
is the number value of Levanah, the Mundane Chakra of Yesod.

The first (4+4) enfolded polygons have 95 sectors &
sides, where 95 is the number value of Madim, the Mundane Chakra of
Geburah.

Outside the root edge are 48 corners & sides of the
first (4+4) enfolded polygons. 48 is the number value of Kokab, the
Mundane Chakra of Hod, and the number of triangles & sides in the four enfolded
polygons. The first (4+4) enfolded polygons have 51 corners & sides. 51 is the
50th integer after 1. They have 49 intrinsic corners & sides.
49 is the number value of EL ChAI, the Godname of Yesod.
First four Type B polygons
We will next consider the first four Type B polygons,
starting with the separate polygons.
First four separate polygons
Table 3 displays the properties of each separate Type B polygon:
Table 3. Geometrical composition of the
first four, separate Type B polygons.

triangle 
square 
pentagon 
hexagon 
Total 
Number of corners of sectors = 
6+1

8+1

10+1

12+1

36+4

Number of sides of sectors =

15

20 
25 
30 
90 
Number of sectors =

9

12 
15 
18 
54 
Number of corners & sides of sectors = 
21+1

28+1 
35+1 
42+1 
126+4 
5
Number of corners & sectors = 
15+1

20+1

25+1

30+1

90+4

Number of sides & sectors = 
24

32

40

48

144

Number of corners, sides & sectors = 
30+1

40+1

50+1

60+1

180+4

Comments

The first four polygons have 40 corners, where
40 = 4 + 8 + 12 + 16. Both sets of polygons have 80 corners, where 80 is the
number value of Yesod. 36 corners surround the centres of each set of polygons.
72 corners surround the centres of both sets.

The first four polygons have 90 sides. 25 sides
belong to the pentagon and 65 belong to the other three polygons. This 25:65
division has its counterpart in the Lambda Tetractys because the sum of its uppermost six
integers is 25 and the sum of the four integers in its base is 65:
The 90 corners & triangles surround their centres
comprise 36 corners and 54 triangles. Many previous articles have illustrated how this
36:54 division is characteristic of holistic systems, the sum of the integers at the
corners of the Lambda Tetractys being 36 and the sum of the seven remaining
integers being 54. This conformity to the archetypal Lambda Tetractys
is a clear indication of the holistic nature of the first four polygons. The counterparts of
this division in the five Platonic solids are the 36 sides of the tetrahedron and
dodecahedron and the 54 sides of the octahedron, cube & icosahedron.
Figure 2. The 180 yods
surrounding the centre of the Type B
dodecagon symbolise
the 180 geometrical elements that
surround the centres of the first four
polygons.
 The first four polygons have 144
sides & triangles, where
144 =

1^{0} 2^{0}
3^{0} 4^{0}
1^{1 } 2^{1} 3^{1}
4^{1}
1^{2} 2^{2} 3^{2}
4^{2}
1^{3} 2^{3} 3^{3}
4^{3 .}

The first (4+4) polygons have 288 sides & triangles, where
288 = 1^{1} + 2^{2} + 3^{3} + 4^{4}.
36 corners surround the centres of each
set of four polygons, where 36 = (1+3+5+7) + (2+4+6+8), i.e., this number is the sum of
the first four odd integers and the first four even integers. 360 geometrical
elements surround the centres of both sets. This is the sum of the Decad (10) assigned to each
of the 36 yods that surround the centre of the Type A hexagon (fourth polygon).
These properties demonstrate how the Tetrad and the integers 1, 2, 3 & 4 symbolised by the
tetractys express properties of this set of polygons.

180 corners, sides & triangles surround the
centres of the first four polygons. They comprise 36 corners and 144 sides
& triangles. The counterpart of this in the Type B dodecagon is the 180 yods that
surround its centre. They comprise 36 yods that line its boundary and 144
yods in its interior (Fig. 2). Alternatively, there are 36 corners & sides
shaping the polygons and 144 interior, geometrical elements. They are symbolised by the
corresponding 36 yods on the boundary of the dodecagon and by the 144
yods inside it.
6

The 54 sectors of the first four polygons have
130 corners & sides, where 130 is the 65th even integer and the 129th
integer after 1. The (4+4) polygons have 260 (=26×10) corners & sides. The seven
enfolded polygons with their 47 sectors turned into tetractyses have 260 yods outside their
root edge (Fig. 3a), whilst 260 yods other than the corners and centres of two
joined, Type B dodecagons line the sides of their 72 tetractyses
outside their shared edge (Fig. 3b).
Figure
3. The Godname YAHWEH with number value 26
prescribes the 260 yods outside the root edge of
the seven enfolded Type A polygons of
the inner Tree of Life and the 260 yods other than
corners & centres of the Type B dodecagons that
are outside the root
edge on sides of tetractyses.
It is evidence that the first (4+4) polygons constitute
a holistic system having properties analogous to those of the inner Tree of Life.
First four enfolded polygons
One of
the sectors of the hexagon is replaced by the three sectors of the triangle when they become
enfolded in one another. Table 4 displays the properties of the first four enfolded
polygons:
Table 4. Geometrical composition of the first four enfolded
Type B polygons.

4 enfolded polygons 
(4+4) enfolded polygons 
Number of corners of sectors
=

40 – 3×2 – 2 = 32

2×30 + 2 =
62

Number of sides of sectors
=

90 – 3 – 5 = 82

2×81 + 1 = 163

Number of sectors =

54 – 3 = 51

2×51 = 102

Number of corners & sides of
sectors =

32 + 82 = 114

62 + 163 =
225

Number of corners & sectors
=

32 + 51 = 83

62 + 102 =
164

Number of sides & sectors
=

82 + 51 = 133

163 + 102 = 265

Number of corners, sides & triangles
= 
32 + 82 + 51 = 165

62 + 163 +
102 = 327

Comments

The four polygons have 32 corners, where 32 is
the 31st integer after 1. The (4+4) polygons have 62 corners, where 62
is the number value of Tzadkiel, the Archangel of Chesed, which is the fourth
Sephirah from the top of the Tree of Life. 31 corners are associated with each set,
where 31 is the number value of EL, the Godname of Chesed. One is an endpoint of the
root edge and there are 30 corners outside it. This reflects the number values E = 1 and L
= 30.

The four polygons have 51 sectors, where 51 is the 50th integer
after 1. The 114 corners & sides of their sectors comprise 12 corners and 15
sides of polygons, leaving 87 corners and sides generated by their transformation,
where 87 is the number value of Levanah, the Mundane Chakra of Yesod.

The (4+4) polygons have 265 sides & triangles, that is, 264 sides
& triangles are outside the root edge. This is the number of yods in the seven
enfolded, Type A polygons (see Fig. 3).

There are 165 corners, sides & triangles in the four polygons,
where
165 = 1^{2} + 3^{2} + 5^{2} + 7^{2} +
9^{2}.
Such a beautiful property is no accident but, instead, the manifestation of
perfect mathematical design. They include the 12 polygonal corners, leaving 153 geometrical
elements, where 153 is the number value of ELOHIM SABAOTH, the Godname of Hod. The (4+4)
polygons have 327 geometrical elements. They include the three corners and two sides on each side
pillar of the outer Tree of Life. This means that (327–5–5=317) geometrical elements are
unshared with it. 317 is the number value of Shabathai, the Mundane Chakra of
Binah.
2. Yod composition of the first four polygons
The following formulae are useful for studying the yod composition of ngons:
7

Type A

Type B

Number of yods at corners of
tetractyses =
Number of hexagonal yods =
Number of yods =



First four Type A polygons
The first four separate polygons are considered first.
First four separate polygons
Table 5 shows the numbers of yods in the first four Type A polygons:
Table 5. Yod composition of the first four separate Type A
polygons.

triangle

square 
pentagon 
hexagon 
Total 
Number of
corners of tetractyses =

3+1

4+1

5+1

6+1

18+4

Number of
hexagonal yods =

15

20

25

30

90

Number of
yods =

18+1

24+1

30+1

36+1

108+4

Number of yods outside
root edge = 
14+1

20+1

26+1

32+1

92+4


Number of yods at corners of 18 tetractyses =
22.

Number of hexagonal yods = 15 + 20 + 25 + 30
= 90.

Number of yods = 19 + 25 + 31 + 37 =
112; 108 yods surround their centres.

Number of yods on boundaries of polygons = 18 + 2×18 = 54; (2×18=36)
are hexagonal.

Number of yods on boundaries of tetractyses = 54 + 2×18 + 4 = 94;
(94–22=72) are hexagonal.
First (4+4) separate polygons

Number of yods at corners of tetractyses = 2×22 =
44.

Number of hexagonal yods = 2×90 = 180.

Number of yods = 2×112 = 224.
(2×108=216) yods surround their centres.

Number of yods on boundaries of polygons = 2×54 =
108; (2×36=72) are hexagonal.
 Number of yods on boundaries of tetractyses = 2×94 = 188; (2×72=144) are
hexagonal.
First four enfolded
polygons

Number of yods at corners of 17 tetractyses = 22 –
3×2 – 1 = 15 (13 outside root edge).
 Number of hexagonal yods = 90 – 3×2 – 2 – 2 – 1 = 79 (77 outside
root edge).
 Number of yods = 15 + 79 = 94
(90 outside root edge).
First (4+4) enfolded
polygons

Number of yods at corners of 34 tetractyses = 2 +
2×13 = 28.

Number of hexagonal yods = 2 + 2×77 = 156;
(156–2=154) are external.

Number of yods = 28 + 156 = 184; (184–4=180) are
external.

Number of yods on 29 sides of polygons with 22
corners = 2×29 + 22 = 80; (80–4=76) are external.

Number of yods on sides of 34 tetractyses = 184 – 34 = 150 =
15×10; (150–4=146) are external, 73 on each side. Number of yods on sides
of tetractyses other than 20 corners of polygons outside root edge = 150 – 20 =
130.
First four Type B polygons
The separate polygons are considered first.
First four separate polygons
Table 6 shows the numbers of yods in the four separate polygons:
Table 6. Yod composition of the first
four separate Type B polygons.

triangle

square 
pentagon 
hexagon 
Total 
Number of
corners of tetractyses =

6+1

8+1

10+1

12+1

36+4

Number of
hexagonal yods =

39

52

65

78

234

Number of
yods =

45+1

60+1

75+1

90+1

270+4

Number of yods
outside root edge = 
41+1

56+1

71+1

86+1

254+4

Comments

The first four polygons have 274 yods (18 corners and 256 other yods,
where 256 = 4^{4}). This is a remarkable example of how the Tetrad expresses
properties of holistic systems like these four polygons.
 There are 258 yods outside their sides that become the root edge when they are enfolded in
one
8
another. The hexagon (fourth polygon) has 87 such
yods, where 87 is the number value of Levanah, the Mundane Chakra of Yesod. Number of
yods in the polygons outside their ‘root edges’ other than their 10 corners = 258 – 10 =
248. This is the number value of Raziel, the Archangel of Binah. It is the dimension
of the superstring gauge symmetry group E_{8}, the rank8 exceptional Lie group. The
polygons contribute to this number as follows:

triangle

square

pentagon

hexagon

248 = 
41 
55 
69 
83 
= 22 nonpolygonal corners + 226 hexagonal
yods. 
There are 54 hexagonal yods at centres of tetractyses,
so that there are (54+22=76) yods that are either centres of tetractyses or their
nonpolygonal corners outside their root edges. This is how the Godname YAHWEH ELOHIM with number
value 76 prescribes the embodiment of the 248 roots of E_{8} in the four
separate Type B polygons. As the hexagon has 78 hexagonal yods, it has 76 hexagonal yods outside
its root edge.
Figure 4. The first four enfolded Type B
polygons have 240 yods outside the root edge surrounding their four centres. This is the
number of yods on sides of tetractyses that surround the centres of the seven polygons of
the inner Tree of Life. The 72 yods in the hexagon correspond to
the 72 yods that either lie on the sides of the dodecagon or are corners of the
first six polygons. The 168 yods outside the root edge surrounding the centres of
the first three enfolded polygons correspond to the 168 yods surrounding the centres
of the seven polygons that are either inside the dodecagon or are not corners of the first
six polygons.
First four enfolded polygons
There are 19 yods (15
outside root edge) in the sector of the hexagon which coincides with the triangle. Four corners
of tetractyses in the hexagon disappear when the polygons become enfolded. Table 7 shows the numbers of yods in the four enfolded
polygons.
Table 7. Yod composition of
the first four enfolded Type B polygons.

triangle

square

pentagon 
hexagon 
Total 
Number of corners of tetractyses =

7

9–2=7

11–2=9

13–4=9

32

Number of hexagonal yods =

39

52–2=50

65–2=63

78–2–13=63

215

Number of yods =

46

57

72

72

247

Number of yods outside root edge = 
46–4–1=41

57–1=56

72–1=71

72

240

Comments

The first four enfolded polygons have 247 yods.
This is the number of yods on the sides of the 48 sectors of the 7 separate polygons
of the inner Tree of Life when the sectors are tetractyses (Fig. 4). 240 yods outside the root edge surround the centres of the
polygons. 168 such yods are in the first
9
three and 72 yods are in the fourth. As
illustrated in many previous articles, this 72:168 division is characteristic of
holistic systems. It signifies the distinction between the 72 roots of E_{6}, an
exceptional subgroup of E_{8}, and the remaining 168 roots of E_{8}. It
manifests in the E_{8}×E_{8} heterotic superstring itself (Fig. 5) as the distinction between its three major whorls and its seven minor
whorls, each of which carries 24 gauge charges of E_{8}, according to the author’s work.
There are also 240 yods in the first 4 enfolded polygons outside the root edge that
do not coincide with the 3 Sephiroth on either side pillar of the Tree of Life.
Figure 6. The first (4+4)
enfolded Type B polygons have the same
number (490) of yods as there are
outside the
root edge of the (7+5) enfolded Type A
polygons which, together with the root
edge, separately have 550 yods.

There are (247–4=243) yods outside the root
edge. Number of yods in the (4+4) enfolded polygons = 4 + 2×243 = 490 =
49×10, showing how EL CHAI, the Godname of Yesod with number value
49, prescribes the (4+4) enfolded polygons. This is also the number of
yods outside the root edge of the (7+5) enfolded polygons which, separately and together
with the four yods of the root edge, have 550 yods that symbolize the 550 SLs of CTOL
(Fig. 6) [1]. The 490 yods in the (4+4) enfolded polygons comprise
62 corners (31 corners associated with each set), 326
hexagonal yods on 163 sides of 102 tetractyses & 102 hexagonal yods at their
centres, i.e., (326+102=428) hexagonal yods. (490−102=388) yods line
their sides, 194 yods per set. 31 is the number value of EL, Godname of
Chesed, 62 is the number value of its Archangel Tzadkiel,
428 is the number value of Chasmalim, its Order of Angels and
194 is the number value of Tzadekh, the Mundane Chakra of
Chesed. Here is a remarkable example of how the number values of the same
Sephirah in the four Kabbalistic Worlds of Atziluth, Beriah, Yetzirah and Assiyah
measure properties of an object possessing sacred geometry.
The (7+7) enfolded Type B polygons of the inner Tree of
Life have 1370 yods [2]. The last (3+3) enfolded polygons have (1370−490=880) yods outside the root
edge. 880 = 88×10, where 88 is the 87th integer after 1 and 87
is the number value of Levanah, the Mundane Chakra of Yesod. They have
50 corners (48 outside the root edge), where
50 is the number of ELOHIM and 48 is the number of
Kokab, the Mundane Chakra of Hod. The last 3 enfolded polygons have 26
corners, where 26 is the number value of YAHWEH.
10

As the (4+4) enfolded Type B polygons have 102 tetractyses,
each with a hexagonal yod at its centre, there are (490–102=388) yods lining their 163
sides. (388–4=384) yods outside the root edge line 162 sides, 192 in each set
(Fig. 7). This 192:192 division is another characteristic of holistic
systems, being found, for example, in the 192 lines and 192 broken lines making up
the 64 hexagrams used in the ancient Chinese system of divination
known as I Ching and in the 192 yods associated with each set of the first six
enfolded polygons and unshared with polygons enfolded in the next higher Tree of
Life (Fig. 8). 24 yods line the nine sides of the triangle, square &
pentagon. They correspond to the 24 lines & broken lines in the upper or lower
trigrams in the eight hexagrams forming the diagonal of the 8×8 square array of
hexagrams and to the 24 corners associated with each set of the first six enfolded
polygons unshared with the polygons enfolded in the next higher Tree of Life. The
remaining 168 yods correspond to the 168 lines & broken lines in
the 28 offdiagonal hexagrams in either diagonal half of the array and to the
168 yods in the first six enfolded polygons that are not corners of
polygons.
The number 384 is the sum of the first nine squares
after 1:
384 = 2^{2} + 3^{2} + 4^{2} +
5^{2} + 6^{2} + 7^{2} + 8^{2} + 9^{2} +
10^{2}.
It illustrates the fundamental connection between the
Pythagorean Decad and this measure of holistic systems represented by sacred geometries.
3. How the Godnames prescribe the first four polygons
Set out below is a summary of the way in which the Godnames
of the 10 Sephiroth prescribe the first four polygons and the (4+4) polygons:
Sephirah 
Godname


Kether 
EHYEH = 21 
22 corners of first (4+4) enfolded
polygons, where 22 = 21st integer after 1.

Chokmah 
YAH = 15 
15 sides of first 4
enfolded polygons; 15 corners of 17 sectors of first 4 enfolded Type A
polygons. 

YAHWEH = 26 
26 corners outside root
edge of 34 sectors of first (4+4) enfolded Type A polygons. 
Binah 
ELOHIM = 50 
51 corners & sides of 34 sectors of
first (4+4) enfolded Type A polygons, where 51 = 50th integer after
1. 51 sectors of first 4 enfolded Type B
polygons.

Chesed 
EL = 31 
31 sides of 17 sectors of first 4 enfolded Type A
polygons.
62 corners & triangles in first (4+4) enfolded Type A polygons,
where 62 = 31st even integer.
62 intrinsic elements in first 4 enfolded Type A
polygons.

Geburah 
ELOHA = 36 
36 corners/sides of
first 4 separate Type A polygons.
36 corners surround centres of sectors in first 4 separate Type B
polygons.
72 geometrical elements surround centres of first 4 separate Type A
polygons, where 72 = 36th even integer.
72 corners & sides of first (4+4) separate Type A polygons.

11


360 (=36×10) geometrical
elements
separate Type B polygons.

Tiphareth 
YAHWEH ELOHIM = 76 
76 geometrical
elements in first 4 separate Type A polygons.
76 nonpolygonal corners & hexagonal yods at centres of 54 tetractyses
outside root edges of first 4 separate Type B polygons.

Netzach 
YAHWEH SABAOTH = 129 
130 corners & sides of 54 sectors
of first 4 separate Type B polygons, where 130 = 129th integer after
1.
130 yods on sides of 34 tetractyses in first (4+4) enfolded Type A polygons
other than external corners of polygons.

Hod 
EL ChAI = 49 
153 geometrical
elements in first 4 enfolded Type B polygons other than their corners.
154 hexagonal yods outside root edge of first (4+4) enfolded Type A polygons,
where 154 = 153rd integer after 1.

Yesod 
EL ChAI = 49 
49 intrinsic corners & sides of
first (4+4) enfolded Type A polygons. 
Malkuth 
ADONAI = 65 
130 corners & sides of 54 sectors
of first 4 separate Type B polygons, where 130 = 65th even integer.
130 yods on sides of 34 tetractyses in first (4+4) enfolded Type A polygons
other than external corners of polygons.


ADONAI MELEKH = 155 
155 geometrical
elements in root edge and in first (4+4) separate Type A polygons.
156 hexagonal yods in first (4+4) enfolded Type A polygons, where 156 =
155th integer after 1.

4. The hexagon embodies the dimension 248 of E_{8}
The pair of joined Type A hexagons has 70 yods when
their 12 sectors are transformed into tetractyses. This is the number of yods in the Tree of
Life when its 16 triangles are tetractyses (Fig. 9). The 10 corners of the two hexagons correspond to the 10 Sephirothic
corners of the triangles making up the
Tree of Life. Their 60 hexagonal yods are the counterpart of the
60 hexagonal yods in the latter. This correspondence
suggests that the two joined hexagons are the single polygonal counterpart of the Tree of
Life.
It should therefore not be surprising that they embody
parameters of holistic systems, as we now show. The next level of transformation of the Type B
polygon is the Type C polygon in which each triangle of the former is
12
replaced by a Type A triangle. The Type C ngon has
(42n+1) yods. A Type C hexagon (n=6) has 252 yods surrounding its centre. (252–4=248) yods
in each hexagon outside their root edge surround their centres (Fig. 10). 500 (=50×10) yods surround the two centres, 496 yods
being outside the root edge, where 496 is the number value of Malkuth. This is how
ELOHIM, the Godname of Binah with number value 50, prescribes the polygonal
representation of the dimension 496 of E_{8}×E_{8}, one of the two gauge
symmetry groups governing the unified interaction of heterotic superstrings.
E_{6} subgroup
Inside each sector of a Type C ngon are 13 yods that
are either corners of tetractyses or hexagonal yods at their centres. A Type C hexagon has
(6×13=78) such yods. 78 is the dimension of E_{6}, the rank6 exceptional subgroup of
E_{8}. The six yods at the centres of each sector denote the six simple root of
E_{6} and the 72 other yods denote its 72 roots. The top and bottom
corners of the hexagon (those which are shared with the hexagons enfolded in the adjacent Trees
of Life) denote the remaining simple roots of E_{8}. The number 248
divides
into 80 yods that denote the eight simple roots
of E_{8} and the 72 roots of E_{6} and 168 yods denoting the
168 roots of E_{8} that are not roots of E_{6}. This 80:168
division of the number 248, which is essentially the 72:168 division of the
holistic parameter 240 discussed earlier, manifests in the context of the outer Tree of Life as
follows: there are 80 yods in the 1tree and 168 yods above it up to Chesed of the
5tree, which is the 31st SL prescribed by the Godname EL of Chesed (Fig. 11a). It also appears in the dodecagons because joined Type A and Type B
dodecagons have 80 yods at corners and centres of their 48 tetractyses and
168 yods that are either the endpoints of the root edge or hexagonal yods on the 83 sides
of tetractyses (Fig 11b). The 72:168 division appears in a pair of
separate Type B dodecagons
as their 72 triangles and as the 168
corners & sides that surround their centres (Fig. 11c). It exists because, as the tenth type of polygon determined by the
Decad, the dodecagon is a holistic object
13
that displays the characteristics of holistic systems.
For example, 24 corners of triangles surround the centre of each separate Type B dodecagon,
comparing with the 24 corners of the first six enfolded polygons on either side of the root edge
with its two endpoints, the properties of these two sets of polygons being shown in Article 4
[3] to be prescribed by the Godnames of the 10 Sephiroth and to display properties
characteristic of such systems. Another example is the fact that 168 more yods are needed
to transform the 12 sectors of
the dodecagon with 13 corners into a Type B dodecagon with 181
yods (Fig. 12).
5. The hexagon embodies
superstring structural parameters
The number of yods in an ngon with 2ndorder
tetractyses as sectors is 72n + 1 [4]. A hexagon has 433 yods. Thirteen yods lie on an side, leaving 420 yods outside
it. Two joined hexagons have 840
yods outside their root edge (Fig. 13). The two hexagons embody not only
the number of gauge bosons mediating the unified superstring force but
also the number of coils in an outer or inner half of a helical whorl of the UPA/heterotic
superstring (Fig. 14) described by Annie Besant & C.W. Leadbeater [5]. This information about the structure of each stringlike whorl of the UPA
is embodied in the pair of joined hexagons with Type A triangles as sectors. Each such
hexagon has 91 yods, that is, 85 yods other than their six corners, where
85 = 4^{0} + 4^{1} + 4^{2} +
4^{3}.
This demonstrates how the Tetrad expresses the extra number of
yods needed to construct a Type B
14
15
hexagon from 18 tetractyses. The two joined Type B
hexagons therefore have 85 + 85 – 2 = 168 yods other than corners. 84 such yods are
associated with each hexagon. Enfolded in 10 overlapping Trees of Life are 20 hexagons with
(840+840=1680) yods other than corners (Fig. 15a). The same information is embodied in the first (6+6) polygons enfolded
in 10 overlapping Trees of Life (Fig. 15b) because 1680 yods line their sides outside their root edges. It is
also embodied in the 10 Type B dodecagons enfolded in 10 Trees of Life as the 1680 yods other
than corners that surround their centres (Fig. 15c). Here are equivalent, holistic objects embodying the same information,
namely, the number of circular polarized oscillations in each vibrating whorl. This structural
parameter of the heterotic superstring is
prescribed by ADONAI, the Godname of Malkuth, because
the lowest 10 overlapping Trees of Life have 65 SLs. This Sephirah is appropriate, as
Malkuth signifies the outer, physical form of any holistic object.
The 840 yods other than corners associated with each set
of hexagons enfolded in 10 overlapping Trees of Life have their counterpart in the polyhedral Tree
of Life [6]. Its outer form is the disdyakis triacontahedron (Fig. 16), which has 62 vertices, 180 edges & 120 triangular faces.
Surrounding any axis passing through two diametrically opposite vertices are 840 vertices, edges
& triangles in each half of the polyhedron when the 180 interior triangles formed by its
edges and centre are Type A triangles:
Moreover, when its exterior and interior triangles are
turned into tetractyses, there are 840 yods surrounding the axis of the disdyakis
triacontahedron:
As the hexagon is the fourth regular polygon, this is
confirmation of the Tetrad Principle, which states that the fourth member of a class of
mathematical object embodies numbers that are characteristic of holistic systems. It is the fourth
member of a set of polygons that constitute a holistic system displaying properties
16
that are analogous to the Tree of Life. The most obvious
example of this is that, as was found in the analysis of the yod composition of the first (4+4)
enfolded Type A polygons, 80 yods line the sides of these polygons. This is the number value
of Yesod and the number of yods in the 1tree (Fig. 17) when its 19 triangles are turned into tetractyses, as well as the
number of corners of the 94 sectors of the (7+7) enfolded polygons of the inner Tree of Life.
The 35 red yods making up its ‘trunk’:
Point (Kether)
Line (Path connecting Chokmah & Binah)
Triangle ((triangle with Chesed, Geburah & Tiphareth at its corners)
Tetrahedron (tetrahedron with Netzach, Hod, Yesod & Malkuth at its
vertices)
have their counterpart in the 35 red corners associated with one set of seven
enfolded polygons and in the 35 red yods lining the outer sides of one set of the first four
enfolded polygons, apart from the corner of the triangle and the top and bottom corners of the
hexagon, which coincide with Sephiroth on the Pillar of Mercy of the Tree of Life, i.e., the 35
red yods are intrinsic to the four polygons themselves). The 35 blue yods making up the
‘branches’ of the 1tree (the region outside its trunk) correspond to the 35 blue corners
associated with the other set of seven enfolded polygons and to the 35 blue yods lining the
outer sides of the other set of the first four enfolded polygons, apart from the corner of the
triangle and the top and bottom corners of the hexagon, which coincide with Sephiroth on the
Pillar of Judgement. The 10 green yods added by conversion of the Tree of Life with 70 yods into
the 1tree with 80 yods correspond to the 10 green centres of the (7+7) enfolded polygons that do not coincide with Sephiroth and to
the 10 green yods in the first (4+4) enfolded
polygons that either coincide with Sephiroth or are in
the root edge. Such natural, onetoone correspondence with both the outer and inner forms of the
Tree of Life is clear evidence that the first (4+4) enfolded polygons constitute what in previous
articles has been called a “Tree of Life pattern” that embodies properties of holistic systems,
such as the 248 gauge bosons that transmit the E_{8}×E_{8}'invariant
interaction between heterotic superstrings and the structural parameters 840 and 1680 of the
E_{8}×E_{8}' heterotic superstring.
This conclusion about the holistic character of the first (4+4) polygons is
given further support by the fact that the 108 triangles making up the first (4+4) separate Type B
polygons have 80 corners (see comment 1 on page 6). They correspond to the 80 corners
of the 94 sectors of the (7+7) enfolded Type A polygons (Fig. 18). The eight centres of the two sets of the first four polygons
correspond to the eight corners of the latter that coincide with the positions of Daath,
Tiphareth and the Sephiroth on the side pillars of the Tree of Life. The first (4+4) separate
Type A polygons have 80 corners & sectors (see comment 8 on page 4). Truly, the
Tetrad reveals here its power in defining members of classes of mathematical objects that have a
holistic character because they are equivalent to, or embody properties of, the universal
blueprint of the Tree of Life and other sacred geometries.
References

For a discussion of CTOL, see Article 5 (WEB, PDF).

Phillips, Stephen M. Article 36: ”The Sri Yantralike
pattern of the 15 layers of vertices in the disdyakis triacontahedron and its scientific
meaning,” (WEB, PDF), p. 10. Here is the proof that the (7+7) enfolded
Type B polygons have 1370 yods:

triangle

square 
pentagon 
hexagon 
octagon 
decagon 
dodecagon 
Number of yods = 
46 
61 
76 
91 
121 
151 
181 
17
Of the 727 yods in the 7 separate polygons, the centre of the
decagon coincides with a corner of the pentagon when they become enfolded. Fifteen yods in a
sector of the hexagon disappear when it is replaced by the triangle in the enfolded set of
polygons. (6×4=24) yods in the root edges of 6 polygons disappear when they become enfolded.
Number of yods in the 7 enfolded polygons = 727 – 1 – 15 – 24 = 687. Number of yods
in the (7+7) enfolded polygons = 687 + 687 – 4 = 1370.

Phillips, Stephen M. Article 4: “The Godnames prescribe the inner
Tree of Life,” (WEB, PDF), p. 4.

Proof: a 2ndorder tetractys contains 85 yods. 13 yods lie along
each side. Number of yods in an ngon with its n sectors turned into 2ndorder
tetractyses = (85–13=72)n + 1.

Besant, A, and Leadbeater, C.W. “Occult Chemistry,” 3rd ed.,
Theosophical Publishing House, Adyar, Madras, India, 1951.
 Phillips, Stephen M. Article 23: “The polyhedral Tree of
Life,” (WEB, PDF).
18
