ARTICLE 60
by
Stephen M. Phillips
Flat 4, Oakwood House, 117119 West Hill Road. Bournemouth.
Dorset BH2 5PH. England.
Website: http://smphillips.mysite.com
Abstract
According to the Tetrad Principle, the seven regular
polygons making up the inner Tree of Life should acquire a special status when
they are Type C because this variety of polygon is the fourth, starting from a
bare polygon. This article analyses each Type C polygon and the sets of seven
Type C polygons when separate and when enfolded. The Type C triangles embody the
superstring structural parameter 168 and the dimension 248 of E_{8}, the
rank8, exceptional Lie group describing the symmetry of the forces of one of
the five types of superstrings. The two Type C squares embody the superstring
structural parameter 336. The Type C pentagon embodies both the number 137
determining the finestructure constant and the number of bones in the adult
human skeleton. The two Type C hexagons embody the dimension 496 of
E_{8}×E_{8}. A single Type C octagon embodies the
number 336. The two Type C decagons embody the superstring structural parameter
840. The Type C dodecagon embodies the number 336. The (7+7) Type C polygons
separated by the root edge have 496 corners of triangles. The superstring
structural parameter 1680 is the number of hexagonal yods inside the seven
separate Type C polygons. The Godnames ADONAI, YAHWEH, ELOHIM & ELOHIM
SABAOTH prescribe the inner Tree of Life when its polygons are Type C.
Representing by a Tree of Life each of the five revolutions made by each of the
10 whorls of the UPA/subquark superstring, the triangles making up the first
(6+6) Type C polygons enfolded in 50 Trees have 16800 corners that correspond to
the 16800 turns in the 10 helical whorls. The inner form of a single Tree of
Life embodies the number 1680 as the number of corners & sides of triangles
outside the root edge that are not shared with its outer form and which are not
pure corners of polygons.

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 Yesodoth
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, 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. Properties of the seven separate Type C
polygons A polygon is Type C when its sectors are Type B triangles with seven
corners & 15 sides of nine triangles, i.e., 31 geometrical elements, where
15 is the number value of YAH, the Godname of Chokmah, and 31 is the number
value of EL,
the Godname of Chesed (Fig. 1). This means that each sector contributes 28
geometrical elements (5 corners, 14 sides & 9 triangles). The numbers of geometrical
elements in a Type C ngon (note that it need not be regular) are:
Number of corners ≡ C = 5n + 1
Number of sides ≡ S = 14n
Number of triangles ≡ T = 9n
Total = 28n + 1
("+1" denotes the centre of the polygon). The number of hexagonal yods ≡ H =
2S + T = 37n. The number of corners of the 9n tetractyses = C = 5n + 1. The number of yods ≡
Y = C + H = 42n + 1. The number of yods on sides of tetractyses ≡ B = C + 2S = 33n + 1.
Table 2 and Table 3 below tabulate the geometrical and yod compositions of the seven
separate Type C polygons that make up the inner form of the Tree of Life ("+7" refers to
their seven centres):
Table 2. Number of geometrical elements in the 7 separate Type C
polygons.

Triangle
(n=3)

Square
(n=4)

Pentagon
(n=5)

Hexagon
(n=6)

Octagon
(n=8)

Decagon
(n=10)

Dodecagon
(n=12)

Total

Corners

16=15+1

21=20+1

26=25+1

31=30+1

41=40+1

51=50+1

61=60+1

247=240+7

Sides

42

56

70

84

112

140

168

672

Triangles

27

36

45

54

72

90

108

432

Total

85=84+1

113=112+1

141=140+1

169=168+1

225=224+1

281=280+1

337=336+1

1351=1344+7

Table 3. Number of yods in the 7 separate Type C polygons.

Triangle
(n=3)

Square
(n=4)

Pentagon
(n=5)

Hexagon
(n=6)

Octagon
(n=8)

Decagon
(n=10)

Dodecagon
(n=12)

Total

C

16=15+1

21=20+1

26=25+1

31=30+1

41=40+1

51=50+1

61=60+1

247=240+7

H

111

148

185

222

296

370

444

1776

B

100=99+1

133=132+1

166=165+1

199=198+1

265=264+1

331=330+1

397=396+1

1591=1584+7

Y

127=126+1

169=168+1

211=210+1

253=252+1

337=336+1

421=420+1

505=504+1

2023=2016+7

Comments
Triangle
The Type C triangle has 85 geometrical elements (84 surrounding its centre),
where
85 = 4^{0} + 4^{1} + 4^{2} +
4^{3}
and
84 = 1^{2} + 3^{2} + 5^{2} +
7^{2}.
It has 15 corners and 69 sides & triangles surrounding its
centre (Fig. 2). Compare this with the 2ndorder tetractys (Fig. 3). It has 85 yods
consisting of 15 corners and 69 hexagonal yods that surround its centre:
3
As the first stage in the sequence of development of the seven regular
polygons making up the inner Tree of Life, the triangle embodies the pattern of this
Pythagorean representation of holistic systems. The two separate Type C triangles
have 168 geometrical elements surrounding their centres (Fig. 4). They
embody the number value 168 of Cholem Yesodoth, the Mundane Chakra of
Malkuth. This is a structural parameter of the subquark superstring, being the number of turns
of each
halfrevolution of a whorl of the UPA. The Type C triangle has 127
yods, where 127 is the 31st prime number and 31 is the number of EL, the Godname
of Chesed. The two joined Type C triangles have 246 yods outside their shared side,
where 246 is the number value of Gabriel, the Archangel of Yesod. 248 yods
surround their centres, where 248 is both the number value of Raziel, the
Archangel of Chokmah, and the dimension of E_{8}, the rank8 exceptional Lie group.
Alternatively, they have 248 yods that are intrinsic to them in the sense that they are
unshared with the outer Tree of Life, their lefthand and righthand corners coinciding with,
respectively Geburah and Chesed (Fig. 5). Embodied in the Type C triangle are both the
structural and the dynamical parameters of a superstring!
Square The two joined Type C squares have 221 geometrical elements
surrounding their centres. This is the
4
number of hexagonal yods in the 1tree and in the seven
enfolded Type A polygons outside their shared root edge (Fig. 6).
Figure 7.
168 yods surround the centre of the Type C square.
Surrounding the centre of each separate Type C square are
168 yods (Fig. 7). Whereas both triangles are needed to embody this superstring
structural parameter, the square on its own achieves this.
Pentagon A Type A pentagon has 31 yods (Fig. 8). It is
a representation of EL ("God"), the Godname of Chesed with number value 31. The yod
at its centre A denotes the letter aleph א (E) with gematria value 1 and the 30 yods surrounding it express
the value
30 of the letter lamed ל (L). The Type B pentagon has 76 yods (Fig. 9). This
is the number value of YAHWEH ELOHIM, the Godname of Tiphareth. The number
26 of YAHWEH is the number of red yods lining the sides of the five basic
sectors of the Type C pentagon. The number 50 of ELOHIM is the number of
black yods. These embodiments of two wellknown Godnames are described here to set the
stage for the later discussion of how the pentagon embodies the number of bones in the
adult human skeleton and the number 137 that determines the finestructure constant at
the heart of quantum electrodynamics, the highly successful theory of electromagnetic
interactions.
The Type C pentagon (Fig. 10) has
26 corners & 70 sides of 45 triangles, i.e., 141 geometrical
elements. Two separate Type C pentagons have 282 geometrical
elements, of which 50 are corners surrounding
5
their centres. 282 is the number value of
Aralim, the Order of Angels
assigned to Binah, whose Godname ELOHIM has the number value 50. The number 70 is a parameter of
holistic systems, being the number of yods in the Tree of Life when its 16 triangles are
tetractyses. The five black sides of the pentagon and the five black sides of its sectors
correspond to the (5+5) Sephirothic corners of these triangles. The 60 remaining red sides
of the 45 triangles correspond to the 60 hexagonal yods in the Tree of Life constructed from
tetractyses. This 10:60 pattern expresses the difference between Malkuth and the six higher
Sephiroth of Construction in the outer Tree of Life. Its counterpart in its inner form is
the dodecagon and the six regular polygons that precede it.
140 geometrical elements in the Type C pentagon
surround its centre, where 140 is the number
of Masloth, the Mundane Chakra of Chokmah, whose Godname YAHWEH has the
number 26. 137 geometrical elements outside the root edge surround its
centre (Fig. 11). We see that the geometrical composition of the Type C pentagon embodies the
number 137 determining the finestructure constant α = e^{2}/ħc ≈
1/137 wellknown to physicists, whilst the number of bones in the adult human skeleton is
embodied in its yods other than its corners (see here).
ELOHIM, the Godname of Binah, prescribes the pair of joined Type C pentagons
because its number value 50 is the number of corners of their
90 triangles, whilst YAHWEH, the Godname of Chokmah with number value 26, prescribes
the 26 corners
of the 45 triangles in each pentagon. The two joined Type C pentagons have 229 lines &
triangles, where 229 is the 50th prime number. ELOHIM prescribes both
the number of corners of the triangles in the two pentagons and the number of their sides
& triangles!
The pair of joined Type C pentagons has 189 corners & sides, i.e.,
187 corners
& sides surround their two centres. 187 is the number of
Auphanim, the Order of
Angels assigned to Chokmah. 95 corners & sides in a Type
C pentagon surround its centre, where 95 is the number value of
Madim, the Mundane Chakra
of Geburah. A corner of each pentagon coincides with the centre of the decagon when it is
part of the inner Tree of Life. This means that surrounding the centre of each pentagon
are 136 geometrical elements outside the shared root edge that are intrinsic to this
polygon in the sense that they are not part of any other polygon in the inner Tree of
Life. The pair of pentagons has 272 such geometrical elements,
where 272 is
the number value of Cherubim, the Order of Angels assigned to
Yesod.
The Type C pentagon has 211 yods, that is, 210 (=21×10) yods surround
its centre. 21 is the number
6
value of EHYEH, the Godname of Kether. It contains 206 yods other than its
five defining corners (Fig. 12). In other words, given a pentagon, 206 more yods in 45
tetractyses (comprising 21 corners and 185 hexagonal yods) are needed to
transform it into a Type C pentagon. This is how the pentagon, whose diagonal width a and
side b are in the proportion of the Golden Ratio: a/b = Φ = (1+√5)/2 = 1.618..., embodies
the number of bones in the adult human skeleton. Many artists through the centuries have
believed that this famous ratio determines the relative proportions of parts of the ideal
human body. Whatever the truth of this controversial claim, here is an indisputable, exact
mathematical connection between its bone composition and the Golden Proportion Φ. Article 32 & Article 33 explain how the outer & inner Trees of Life
embody the human axial and appendicular skeletons (see also Human skeleton). Article 36 shows how the disdyakis triacontahedron — the
polyhedral form of the inner Tree of Life — embodies 206 yods symbolizing the 206 bones (see
also here).
Hexagon
168 geometrical elements surround the centre of the Type C hexagon. They comprise
84 corners & triangles and 84 sides. Its geometrical composition displays the 84:84
division of this number that is characteristic of holistic systems. The Type C hexagon contains
222 hexagonal yods. This is the number of hexagonal yods associated with each set of seven
enfolded
Type A polygons. 248 yods
outside the shared root edge surround the centre of each hexagon (Fig. 13). This is the
dimension of E_{8}, the rank8, exceptional Lie
group. The two joined Type C hexagons embody the dimension 496 of E_{8}×E_{8}′. This
is highly significant in view of the Tetrad Principle formulated in Article 1, for the Type C hexagon is the fourth in the
series of this polygon, which is the fourth type of regular polygon:
Triangle → square → pentagon → hexagon → heptagon →...
Hexagon → Type A hexagon → Type B hexagon → Type C hexagon → Type D hexagon
→....
7
Octagon
The octagon has 225 geometrical elements, so that the two joined octagons have 444 geometrical
elements outside their shared side, 222 in each polygon. This is the number of hexagonal yods
in the inner Tree of Life, 222 hexagonal yods being
associated with each set of seven enfolded polygons (Fig. 14). 336 yods
surround the centre of the octagon (Fig. 15). This is the number of turns in one revolution
of each whorl of the UPA. The octagon embodies this major structural parameter of the
subquark superstring. Surrounding its centre are 264 yods lining the sides
of its 72 tetractyses. This is the yod population of
the seven enfolded Type A polygons (Fig. 16).
Decagon
280 geometrical elements comprising 50 corners,
140 sides and 90 triangles surround the centre of the decagon. 840
yods surround the centres of two separate decagons. This is the number of turns
in an outer or inner half of a whorl of the UPA.
Dodecagon
336 geometrical elements (168 corners & triangles, 168
sides) surround the centre of a dodecagon. This is the number of yods other than corners in
two joined Type B dodecagons (Fig. 17). The number 168 factorises in the geometrical
case as 12×14 — exactly as it does for the yods in each Type B dodecagon,
8
each sector contributing 14 yods other than corners. The holistic division
336 = 168 + 168 arises because the number of corners & triangles in any Type
C polygon is equal to the number of sides (as is the case for Type A & Type B
polygons). The dodecagon has 444 hexagonal yods — exactly the same as for the inner Tree of
Life. 504 yods surround its centre (Fig. 18). They comprise 168 black yods (the number
of yods in a Type B dodecagon other than the corners of its sectors) and
(2×168=336) red yods. The superstring/UPA significance of this is
discussed here under the heading "TYPE C DODECAGON", here and here. The way in which the heptagon and outer
Tree of Life embody the number 504 is discussed here. The fact that the dodecagon is the first
single polygon to embody superstring structural parameters in terms of both its geometrical
and yod compositions (the hexagon does the same but only as two separate polygons) confirms
its unique status as the polygonal version of the outer Tree of Life. 500 (=50×10)
yods outside the root edge surround its centre. This is how ELOHIM, the Godname of Binah
with number 50, prescribes the yods that construct the dodecagon. Surrounding the
centres of the two joined dodecagons (the tenth regular polygon) are (1000=10^{3})
yods outside the root edge. This illustrates the power of the Decad.
Properties of the seven separate polygons

The 240 corners of 432 triangles with 672 sides surround the centres of the seven
polygons. The number 240 is a parameter of holistic systems (Fig. 19). For example,
when the 19 triangles in the 1tree are Type A, it contains 240 yods other than
their 11 corners, whilst the seven separate Type A triangles contain 240 hexagonal
yods. There are (240+240=480) corners of 864 triangles surrounding the 14 centres
of the (7+7) Type C polygons, i.e., 494 corners in total. If we imagine the two
sets of seven polygons separated by a straight line representing the root edge,
whose endpoints count as corners,
9
there are 496 corners present. Associated with each set are
248 corners (240 corners of triangles, 7 centres & one endpoint). This
248:248 pattern represents the (248+248=496)
roots of E_{8}×E_{8}′, one of the two symmetry groups of heterotic
superstrings.

The seven separate polygons have 672 sides and 672 corners & triangles
surround their centres. This number was encountered in the discussion
here of the first four Platonic solids when
their faces and interiors are constructed from tetractyses. It was found that
they contain 672 yods, making an average of 168 yods per Platonic
solid.

The 1344 geometrical elements surround the centres of the seven polygons. This
number is embodied in the (7+7) enfolded Type B polygons in the following way: when
the outer and inner Trees of Life are superposed, the two side pillars lie in the
plane of the polygons, coinciding with the vertical axes of the two hexagons. Each
axis contains seven yods shared with a hexagon. One of these is its centre, leaving
six other centres. Each set of seven enfolded polygons contain (7+6=13) yods that
are either shared yods or centres. Both sets of enfolded polygons have 1370 yods,
of which 26 are such yods, leaving 1344 yods intrinsic to the polygons that
surround their centres.

A Type C ngon contains 37n hexagonal yods. 2n hexagonal yods line its n
sides, inside which are 35n hexagonal yods. The number of hexagonal yods inside the
seven separate, Type C polygons = ∑35n = 35∑n = 35×48 = 1680. This is the
number of turns in each helical whorl of the UPA. Embodied, therefore, inside the
seven separate polygons is this major structural parameter of the subquark
superstring. 840 hexagonal yods are inside the dodecagon and the first three
polygons (triangles, square & pentagon) and 840 hexagonal yods are inside the
next three polygons (hexagon, octagon & decagon). The characteristic 24:24
division of the holistic parameter 48 manifests in the inner Tree of Life as
the 840 hexagonal yods inside each of these sets of polygons with 24 corners. This
manifests in the subquark superstring as the 840 turns in the 2½ revolutions of the
outer half of each whorl and the 840 turns in the 2½ revolutions of its inner half.
2. Properties of the seven
enfolded polygons
When the seven regular polygons of the inner Tree of Life are enfolded in
one another, a corner of the triangle coincides with the centre of the hexagon and a corner
of the pentagon coincides with the centre of the decagon. All the 31 geometrical
elements in the sector of the hexagon replaced by the triangle disappear, so that, instead
of 169 geometrical elements (see table above). the hexagon is now left with five sectors
containing (169–31=138) elements. 46 yods also disappear from this sector because it
is a Type A triangle with 46 yods, so that, instead of 253 yods, the hexagon is left with
(252–46=207) yods.
Tabulated below in Tables 4 & 5 are the geometrical and yod compositions
of the seven enfolded Type C polygons outside their shared root edge.
Table 4. Number of geometrical elements outside the root edge in the 7
enfolded Type C polygons.

Triangle
(n=3)

Square
(n=4)

Pentagon
(n=5)

Hexagon
(n=6)

Octagon
(n=8)

Decagon
(n=10)

Dodecagon
(n=12)

Total

Corners

14

19

24

24

39

48

59

227

Sides

41

55

69

69

111

139

167

651

Triangles

27

36

45

45

72

90

108

423

Total

82

110

138

138

222

277

334

1301

Table 5. Number of yods outside the root edge in the 7
enfolded Type C polygons.

Triangle
(n=3)

Square
(n=4)

Pentagon
(n=5)

Hexagon
(n=6)

Octagon
(n=8)

Decagon
(n=10)

Dodecagon
(n=12)

Total

C

14

19

24

24

39

48

59

227

H

109

146

183

183

294

368

442

1725

B

96

129

162

162

261

326

393

1529

Y

123

165

207

207

333

416

501

1952

Comments
10
49th prime number, showing how EL ChAI, the Godname of Yesod with number value
49, prescribes the skeletal shape of the seven enfolded Type C polygons.
Including the two endpoints of the shared root edge, there are 229 corners. The Godname
ELOHIM with number value 50 prescribes the number of corners because 229 is the
50th prime number. This is a spectacular illustration of the power of Godnames
at work in mathematically determining the geometry of the inner Tree of Life.

The seven enfolded polygons have 650 (= 65×10) corners & triangles and
651 sides outside the root edge. This shows how ADONAI, the Godname of Malkuth with
number value 65, prescribes the seven enfolded polygons. As the topmost
corner of the hexagon coincides with the lowest corner of the hexagon belonging to
the seven polygons enfolded in the next higher, overlapping Tree of Life, 1300
geometrical elements are intrinsic to the set of seven polygons enfolded in
adjacent Trees of Life, where



1^{4} 



2^{4} 
2^{4} 

1300 = 1^{5} + 2^{5} +
3^{5} + 4^{5} =


3^{4} 
3^{4} 
3^{4 } 

4^{4}

4^{4} 
4^{4} 
4^{4} 
This means that 2600 geometrical elements are intrinsic to both sets of seven
enfolded polygons, where
2600 = 51^{2} − 1 = 3 + 5 + 7 + ... + 101
is the sum of the first 50 odd integers after 1. This shows how both ELOHIM with
number value 50 and YAHWEH with number value 26 prescribes how
many intrinsic geometrical elements are needed to construct the (7+7) enfolded
polygons, starting with the root edge. Such beautiful harmony between number and
geometry is powerful evidence of the archetypal status of this sacred geometry.

There are 1529 yods outside the root edge lining the 423 tetractyses in the seven
enfolded polygons. Hence, (2×1529 + 4 = 3062) yods line the 846 tetractyses in the
(7+7) enfolded polygons. The topmost corners of the two hexagons coincide with the
lowest corners of the hexagons enfolded in the next higher Tree. This means that
3060 (=306×10) boundary yods are intrinsic to each set of 14 enfolded polygons.
1530 (=153×10) intrinsic, boundary yods are associated with each set. This
shows how ELOHIM SABAOTH, the Godname of Hod with number value 153,
prescribes the shape of the inner Tree of Life whose polygons are Type C.

Outside the root edge are 227 corners of the 423 triangles in the
seven enfolded polygons. The dodecagon has 59 such corners, so that the first six
enfolded polygons have 168 such corners of 315 triangles. Both sets of the
first six enfolded polygons, which have 50 polygonal corners, have
(168+168=336) corners of 630 triangles outside the root edge,
where 630 is the number value of Seraphim, the Order of Angels
assigned to Geburah. ELOHIM, the Godname of Binah with number value 50,
prescribes the superstring structural parameter 336, which is the number of turns in
each of the 50 revolutions of the ten whorls of the UPA about its spin axis.
Including the two endpoints of the root edge, the first (6+6) enfolded polygons have
338 corners. But
11
the topmost corners of their two hexagons coincide with the lowest corners of the
hexagons enfolded in the next higher Tree of Life. This means that 336 corners are
intrinsic to each set of (6+6) enfolded polygons. The number of intrinsic corners of
the first (6+6) polygons enfolded in n Trees of Life = 336n. The 120 such polygons
enfolded in 10 overlapping Trees have 3360 intrinsic corners. This is the number of
turns per revolution in all ten whorls of the UPA. Each revolution of a whorl can be
represented by a Tree of Life, so that 50 overlapping Trees represent the
50 revolutions of the whorls of the UPA. The 600 polygons of the first six types
enfolded in them have (336× 50=16800) intrinsic corners. This is the number of
turns in the ten whorls of the UPA. The five sets of 10 Trees making up
50 Trees correspond to the five revolutions of 10 whorls. These Type C
polygons provide a geometrical representation of the UPA/subquark superstring because
the first six polygons constitute a holistic subset of the complete set of seven
polygons, as has been illustrated many times elsewhere in this website. The fact that
the first six polygons of the inner Tree of Life have 36 corners when separate
and 26 corners when enfolded, both enfolded sets having 50 corners serves
to illustrate how they are prescribed by the Godnames of the ten Sephiroth (see
Article 8 for more details).

The 423 triangles in the seven enfolded polygons have 881 corners
& sides. There are five corners & sides on the vertical axis of the hexagon
that are shared with the outer Tree of Life as its Pillar of Mercy (they are
coloured green in Figure 20). Similarly for the hexagon in the other set of
polygons, five corners & sides coincide with the Pillar of
Judgement. One of these corners is
the centre of the hexagon, so that (881−5−6=870=87×10) intrinsic corners
& sides surround the centres of the seven enfolded polygons. 87 is the
number value of Levanah, the Mundane Chakra of Yesod. 878 corners & sides
in each set of seven enfolded polygons are outside the root edge. Of these, five
corners & sides are shared with the outer Tree, leaving 873 intrinsic corners
& sides. Of these, five are centres of polygons that are not also corners and 28
corners are pure corners in the sense that none of them is also the centre of a
polygon. Hence, the centres of each set of polygons are surrounded by (873−5−28=840)
intrinsic corners & sides other than their corners. Outside the root edge of
both sets of enfolded polygons are (840+840=1680) intrinsic corners & sides that
are not pure corners of polygons. Alternatively, each set of seven enfolded polygons
has (including the root edge) 881 corners & sides, of which 41 corners are
corners of their 47 sectors, leaving 840 corners & sides, so that both
separate sets have 1680 such corners & sides other than corners of
sectors. Each set separately embodies the superstring structural parameter 840,
whilst both sets of enfolded polygons embody the superstring structural parameter
1680. Ten overlapping Trees mapping the 10 whorls of the UPA have an inner form
containing 16800 intrinsic corners & sides that are not pure corners of their
(70+70) enfolded Type C polygons. This is
the inner Tree of Life representation of the subquark state of the
E_{8}×E_{8} heterotic superstring.
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