ARTICLE 27
by
Stephen M. Phillips
Flat 4, Oakwood
House, 117119 West Hill Road. Bournemouth. Dorset BH2 5PH. England.
Website: http://smphillips.mysite.com
Abstract
The Pythagorean
tetractys of ten points (‘yods’) symbolises the geometrical sequence: point,
line, triangle and tetrahedron. Its next higherorder differentiation has 85
yods that denote the 85 geometrical elements needed to construct the tetrahedron
from tetractyses. The 15 Archimedean solids and their duals — the 15 Catalan
solids — can likewise be constructed from this template. Starting from its
centre, the construction of the truncated tetrahedron requires 248 geometrical
elements. The simplest Archimedean solid therefore embodies the dimension 248 of
the superstring gauge symmetry group E_{8}. Its dual, the triakis
tetrahedron, has 168 geometrical elements surrounding the axis passing through
two opposite vertices. The simplest Catalan solid therefore embodies the
superstring structural parameter 168 discussed by the author in many previous
articles. Its 78 edges and 90 vertices and triangles correspond to the gematria
number values of the Hebrew words ‘Cholem’ and ‘Yesodoth’ in the Kabbalistic
Mundane Chakra of Malkuth. This geometrical composition has its exact
counterpart in the yod population of the first six regular polygons of the inner
Tree of Life. When its faces are constructed from three tetractyses, the triakis
tetrahedron has 240 elements surrounding its axis, i.e., 72 extra ones.
E_{8} has 240 nonzero roots, of which 72 are also nonzero roots of its
exceptional subgroup E_{6}. The disdyakis triacontahedron has 1680
geometrical elements surrounding its axis. This is the number of circularly
polarised oscillations in each of the ten whorls of the ‘ultimate physical
atom’, the basic unit of matter described paranormally 105 years ago by Annie
Besant and C.W. Leadbeater and identified in earlier work by the author as the
spin½ subquark state of the E_{8}×E_{8} heterotic superstring.
The most complex Catalan solid therefore embodies in its geometry the number
characterising the oscillatory form of the superstring. The geometrical
composition of this solid is represented by the yod population of the first six
polygons enfolded in ten Trees of Life. The disdyakis triacontahedron is their
polyhedral counterpart. The numbers of geometrical elements in the triakis
tetrahedron are the numbers of nonzero roots of E_{8}, E_{7}
and E_{6}. The disdyakis triacontahedron has corresponding numbers that
are ten times as large, demonstrating its tenfold Tree of Life
nature.

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, Order 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.

2
1.
Introduction
The Pythagorean tetractys is a triangular array of
ten dots:
These dots will be called ‘yods.*The six yods at the
corners of a hexagon and the yod at its centre will be called ‘hexagonal yods.’ They are
shown above as coloured circles.
The four rows of points in the tetractys symbolise the sequence of what
mathematicians call the first four “simplexes”:
As the 3simplex, the tetrahedron consists of 4 vertices (0simplexes) and 6
edges (1simplexes) (i.e., ten simplexes), 4 triangles and one tetrahedron (i.e., 5
simplexes), totalling 15 simplexes. The simplest regular polyhedron is
prescribed by the Godname YAH (YH) assigned to Chokmah in the Tree of Life (Table 1) with number value 15,^{1} the value 10 of Y (yod) denoting the number of 0simplexes and
1simplexes and the value 5 of H (he) denoting the number of 2simplexes and 3simplexes.
The complete Godname YAHWEH (YHVH) prescribes the sequence of the first four simplexes
because it contains 26 vertices, edges, triangles and tetrahedra:
The letter values of Y and H are as above, the value 6 of V (vav) is the
number of 0simplexes in the first three simplexes (shown encircled above) and the value 5
of the second letter H is the number of 1 and 2simplexes in the 1simplex and the
2simplex.
Any polyhedron can be constructed from tetractyses by dividing its polygonal
faces into triangular sectors and then changing the latter into tetractyses — it does not
matter whether these sectors are equilateral or isosceles triangles. The edges of the
polyhedron are also sides of internal triangles formed by joining its centre to two adjacent
vertices. These triangles can be constructed from three tetractyses. For consistency, a
polyhedron that has both triangular faces and faces of other shapes must have the former
divided into three tetractyses rather than be considered as single ones. However, if all its
faces are triangular, there are two possible constructions: either each face is a tetractys
(case A) or each face is divided into three tetractyses (case B).
With its faces and internal triangles formed by its edges single tetractyses
(case C), the tetrahedron contains also one internal corner (its centre) and 4 internal
sides of triangles (totalling 5 elements) and 6 triangles, a total of 11 geometrical
elements. This is the same as the number of elements in the first three simplexes, as shown
above. Starting from a point (centre), 24 more points, lines and triangles have to be put in
place before the interior and exterior of the tetrahedron can be constructed from
triangles.
____________________________________________
* The tenth letter of the Hebrew alphabet is yod ( י), which looks somewhat like a dot.
3
With its faces and internal triangles divided into three tetractyses (case
B), the geometrical composition of the tetrahedron is:

Corners


Sides


Triangles


Total

faces:

8

+ 
18

+ 
12

= 
38

interior:

7

+ 
22

+ 
18

= 
47

Total =

15

+ 
40

+ 
30

= 
85

It has 15 vertices prescribed by YAH with number value 15 and
26 corners & sides in its faces defined by YAHWEH with number value 26. It
has 70 sides & triangles and 85 corners, sides & triangles. This property proves its
wholeness and perfection as the fundamental building block of solid geometry, for the next
higher order tetractys — the 2ndorder tetractys:
has 85 yods. Each yod symbolises a geometrical element needed to construct
the tetrahedron. The 15 (white) yods at the corners of the ten tetractyses denote the
15 vertices of the tetractyses forming the tetrahedron and the 70 hexagonal
(coloured) yods denote the 70 sides & triangles. The Godname ELOHA with number value
36 prescribes this symbol of the simplest regular polyhedron because 36 yods lie
on its boundary. It is prescribed by the Godname EL CHAI assigned to Yesod with number value
49 because 49 yods are inside its boundary.
The Tetrad Principle^{2} expresses this measure of a holistic system as
The number of geometrical elements surrounding the centre of the tetrahedron
and symbolised by the 84 yods surrounding the central one is 84, where
i.e., 84 is the sum of the squares of the first four odd integers.
The tetrahedron is the polyhedral realisation of the 2ndorder tetractys.
2. The
Archimedean & Catalan solids
There are 13 Archimedean solids (polyhedra with two or more types of regular polygons as
faces). Two have different mirror images (chiral partners). Their duals (vertices replaced by
faces and faced replaced by vertices) are the Catalan solids. The two sets have 26
members, showing how YAHWEH prescribes this family of polyhedra, whilst YAH with number value
15 prescribes each set with the enantiomorphic members included. Their properties will
now be calculated and then discussed.
Definitions: C = number of vertices of polyhedron.
E = number of polyhedral edges.
4
F = number of polyhedral faces.
C, E and F are related by Euler’s equation for a convex polyhedron:
C – E + F = 2.

m =

number of triangular sectors of a face with m edges. 
n(m) =

number of faces with m sides in polyhedron. 
L ≡

Σmn(m) = number of triangular sectors of
faces. 

^{ }^{m} 
The internal triangles formed by the centre of the polyhedron and two ends
of an edge are divided into three tetractyses; any triangular faces are either considered as
single tetractyses (case A) or divided into three tetractyses (case B).
Surface
Number of corners in faces = C (case A)
= C + F = 2 + E (case B).
Number of sides in faces = E (case A)
=
E + Σmn(m) = E + L (case B).
^{m}
Number of triangles in faces = F (case A)
= L (case B).
Number of corners, sides & triangles = C + E + F = 2E + 2 (case A)
= C + F + E + 2L = 2E + 2L + 2 (case
B).
Table 2. Population of geometrical elements in the Archimedean & Catalan
solids.
N'

F

E

C

Archimedean solid

244

8

18

12

truncated tetrahedron 
322

14

24

12

cuboctahedron 
490

14

36

24

truncated cube 
490

14

36

24

truncated octahedron 
646

26

48

24

rhombicuboctahedron 
802

38

60

24

snub cube 
802

38

60

24

snub cube (chiral partner)

808

32

60

30

icosidodecahedron 
982

26

72

48

truncated cuboctahedron 
1228

32

90

60

truncated icosahedron 
1228

32

90

60

truncated dodecahedron 
1618

62

120

60

rhombicosidodecahedron 
2008

92 
150

60

snub dodecahedron 
2008

92

150 
60

snub dodecahedron
(chiral partner) 
2458

62 
180 
120 
truncated icosidodecahedron 


Catalan solid

F

E

C

N

N'

triakis tetrahedron 
12

18

8

168

240

rhombic dodecahedron 
12

24

14



324

triakis octahedron 
24

36

14

336

480

tetrakis hexahedron 
24

36

14 
336

480

deltoidal icositetrahedron 
24

48

26



648

pentagonal icositetrahedron 
24

60

38 


816

pentagonal icositetrahedron
(chiral partner) 
24

60

38



816

rhombic triacontahedron 
30 
60

32 


810

disdyakis dodecahedron 
48

72

26

672

960

triakis icosahedron 
60

90

32 
840

1200

pentakis dodecahedron 
60

90

32 
840

1200

deltoidal hexacontahedron 
60

120 
62 


1620

pentagonal hexacontahedron 
60

150 
92 


2040

pentagonal hexacontahedron
(chiral partner) 
60

150 
92 


2040

disdyakis
triacontahedron 
120

180 
62 
1680

2400


Interior
Number of corners = E + 1.
Number of sides = C + 3E.
Number of triangles = 3E.
Number of corners, sides & triangles = C + 7E + 1.
5
Number of corners, sides & triangles = 2C + 8E + F + 1 = C + 9E + 3
(case A)
= 2C + 8E + F + 2L +
1 = C + 9E + 2L + 3 (case B)
The axis is made up of five geometrical elements (three corners, two sides).
In case A, the number of elements surrounding the axis ≡ N = 2C + 8E + F + 1 – 5
= 2C + 8E + F – 4 = C + 9E – 2.
In case B, the number ≡ N' = 2C + 8E + F + 2L + 1 – 5 = C + 9E + 2L –
2.
Table 2 gives values of N and N' for the Archimedean and Catalan solids (as
none of the former have just triangular faces, only values of N' can be listed for them). We
find that the simplest Catalan solid — the triakis tetrahedron — has 168
geometrical elements surrounding an axis through two vertices, when its triangular faces are
tetractyses (case A), and 240 elements if they are divided into three tetractyses (case B),
that is, 72 more elements are needed to construct the faces from three
tetractyses instead of from one. We also see that the most complex Catalan solid — the
disdyakis triacontahedron — has 1680 (=168×10) geometrical elements in case
A and 2400 (=240×10) elements in case B, i.e., 720 (=72×10) more elements
are needed. Numbers of elements for the last Catalan solid are exactly ten times their
counterparts for the first one. The first and last Catalan solids are equivalent in
this sense. The division:
240 = 168 + 72
corresponds in the group mathematics of the Lie group E_{8}
appearing in superstring theory to the fact that, of its 240 nonzero roots,
72 are nonzero roots of its exceptional subgroup E_{6} leaving
168 simple roots.
Table 3. Number values of the Sephiroth in the four Kabbalistic Worlds.
Sephirah 
Title

Godname

Archangel

Order of
Angels

Mundane
Chakra

Kether 
620

21

314

833

636

Chokmah 
73

15, 26

248

187

140

Binah 
67

50

311

282

317

Chesed 
72

31

62

428

194

Geburah 
216

36

131

630

95

Tiphareth 
1081

76

101

140

640

Netzach 
148

129

97

1260

64

Hod 
15

153

311

112

48

Yesod 
80

49

246

272

87

Malkuth 
496

65,155

280

351

168

Notice that 168, 240, 1680 and 2400 do not appear in the
table of numbers for the Archimedean solids. Notice also that the tenth Catalan solid — the
triakis icosahedron — has 840 geometrical elements surrounding its axis, whilst the
15th solid has 1680 such elements. This corresponds to the fact that the 1680
circularly polarized oscillations of each whorl of a subquark superstring comprise 840
oscillations making 2½ outer revolutions around its axis and 840 oscillations making 2½
revolutions in the core of the superstring (see Fig. 6). This division is prescribed by the Godname YAH (YH) of Chokmah. Its
number 15 (see the
6
upper orange cell in Table 2) prescribes the disdyakis triacontahedron as the
15th Catalan solid, whilst the letter value Y = 10 in YH defines that
Catalan solid which has 840 elements surrounding its axis. 168 is the
number value of Cholem Yesodoth, the Mundane Chakra of Malkuth (see the last cell
in Table 2).
The twelve faces of the triakis tetrahedron are isosceles triangles with one
long side and two short sides. They are arranged as threesided pyramids of height √6/15
constructed on the four faces of a regular tetrahedron with unit edge length (Fig. 1).
The geometrical composition of the triakis tetrahedron in case A is given
below:

Corners

Sides

Triangles

surface: 
8

18


minus: 
–2 (poles)



Subtotal =

6



interior: 
18

8 + 3×18 = 62

3×18 = 54

minus: 

–2 (two
sides)



__

Subtotal = 60

__

Total =

24

+
78
+

66 =
168

Compare these numbers with the geometrical composition of the disdyakis
triacontahedron in case A:

Corners

Sides

Triangles

surface: 
62

180

120

minus: 
–2 (poles)



Subtotal =

60



interior: 
180

62 + 3×180 = 602

3×180 = 540

minus: 

–2 (two sides)



___

Subtotal
= 600

___

Total = 
240

+
780
+

660 = 1680

7
Every number of elements of a given type in the exterior and interior of the
disdyakis triacontahedron is exactly ten times the corresponding number in the same colour
in the triakis tetrahedron. The triakis tetrahedron is made up of 78 sides and 90 corners
& triangles, whilst the disdyakis triacontahedron is composed of 780 sides and 900
corners & triangles. This division of the geometrical elements comprising the basic
Catalan solid reflects the number value of Cholem Yesodoth, the Hebrew name of the
manifestation in Assiyah, the fourth Kabbalistic World, of the Sephirah Malkuth
(Fig. 2).
The tenfold difference between the composition of the first and last
Catalan solids can be represented by a star with ten points, each one a parallelogram formed
from 32 tetractyses with 168 yods below its point at the centre of the star
(Fig. 3). The division:
168 = 90 + 78
reflects the 90 yods in one half of the parallelogram and the 78 yods in the
other half. It is encoded in the inner form of the Tree of Life (Fig. 4).
Associated with each set of its first six enfolded, regular polygons are
168 yods other than corners of the 35 tetractyses into which the six
polygons can be divided. 90 of these representing the number of Yesodoth are in the square,
hexagon and decagon, and 78 yods representing the number value of Cholem are in the
triangle, pentagon and octagon.
8
The fact that the numbers of geometrical elements in the faces and interior
of the last Catalan solid are ten times the numbers of corresponding elements in the faces
and interior of the first Catalan solid is further evidence of the unique status of the
disdyakis triacontahedron as the completion of the Catalan solids. For the Pythagorean
tetractys symbolises holistic (that is, whole) systems — systems that are complete
in themselves rather than mere intermediate members of some developing sequence. The triakis
tetrahedron is the unit, or yod, of a tetractys representing the disdyakis
triacontahedron:
Viewed as part of the complete sequence of 35 solids:
5 Platonic solids → 15 Archimedean
solids → 15 Catalan
solids
the triakis tetrahedron is the 21st member, whilst the
disdyakis triacontahedron is the 31st member when the enantiomorphic
versions of solids are excluded. This shows how the Godname EHYEH of Kether with number
value 21 prescribes the first Catalan solid with
168 geometrical elements surrounding the axis joining two opposite
vertices and how the Godname EL of Chesed with number value
31 prescribes the last Catalan solid with 1680 geometrical elements
surround its axis. The Godname YAHWEH with number 26 prescribes the
number of Archimedean solids and their duals, the Catalan solids, when enantiomorphs are
excluded, whilst its older version, YAH, with number value 15, is the
number of each type when their enantiomorphs are included.
Figure 5. The pentagram represents the Godname EL with number value 31 because
31 yods define its boundary and centre. The 30 yods on the former divide into two
alternating sets of 15 yods, one set symbolising 15
Platonic/Archimedean solids and the other set symbolising their duals. The central yod
denotes the selfdual tetrahedron.
The 31 solids comprise the tetrahedron,
15 Platonic/Archimedean solids and their duals. This 1:30 pattern is
reflected in the name EL, where E = 1 and L = 30. It is also represented by the pentagram
because 30 yods lie on its edge when each side of the star's point is made up of four yods
(Fig. 5). The central yod denotes the selfdual tetrahedron and the three yods
on each side of a point symbolise Platonic or Archimedean solids and their duals.
9
The tetractys pattern of ten triakis tetrahedra that is equivalent to the
disdyakis triacontahedron has its counterpart in the subquark superstring, which consists of
ten closed standing waves, each with 1680 circularly polarised oscillations (Fig. 6). These vibrations are the subatomic counterpart of the 1680
geometrical elements surrounding the axis of the disdyakis triacontahedron. The
correspondence exists because the former is the manifestation of the Tree of Life in the
10dimensional spacetime of superstrings and the latter is its polyhedral manifestation in
the 4dimensional spacetime of the largescale universe.
The fact that the triakis tetrahedron has 168 geometrical
elements surrounding its axis in case A and 240 elements in case B, i.e.,
72 extra elements, itself reflects the tetractys arrangement of the number
24, where 24 = 1×2×3×4:
The first Catalan solid is constructed by what mathematicians call
“cumulation” of a tetrahedron by a pyramid. Constructed from tetractyses, the tetrahedron in
case B has 15 corners, 40 sides & 30 triangles (70 sides &
triangles), totalling 85 geometrical elements. It conforms to the next higher
differentiation of the tetractys:
because this consists of 85 yods, of which 15 are corners
of the ten tetractyses and 70 are hexagonal yods (shown coloured). These 85 yods
signify degrees of freedom. For the tetrahedron, the 15 corners symbolise
the 15 corners of its 30 triangles and the 70 hexagonal yods denote its 70
sides & triangles. The geometrical sequence:
10
point →
line → triangle → tetrahedron
has its Pythagorean counterpart:
So the 2ndorder tetractys symbolises the geometrical composition of the
tetrahedron — the simplest regular polyhedron.
Ignoring enantiomorphic Archimedean solids, the 21st solid
in the sequence:
5 Platonic solids →
15 Archimedean solids →
15 Catalan solids
is the triakis octahedron. It has F = 24, E = 36, C = 14
and N = 336. Comparing these values with the values F = 120, E = 180, C = 62
and N = 1680 for the disdyakis triacontahedron, we see that the latter has five times as many
faces and five times as many edges, whilst the number of polyhedral vertices surrounding an
axis passing through two opposite ones = 62 – 2 = 60, comparing with (14–2=12)
such vertices for the former, that is, five times as many vertices of this type. The disdyakis
triacontahedron has therefore five times as many elements of each variety surrounding its axis
as the triakis octahedron has surrounding its axis. This difference of a factor of 5 appears in
the subquark superstring described by Annie Besant and C.W. Leadbeater (Fig. 6). Each of its ten whorls make five revolutions about its spin axis,
the number of oscillations made in one revolution being 1680/5 = 336, which is the number of
geometrical elements surrounding the axis of the triakis octahedron.
Three Platonic solids (tetrahedron, octahedron & icosahedron) have all
triangular faces. None of the Archimedean solids has this property, whilst seven of the
15 Catalan solids have all triangular faces: the triakis tetrahedron, the
triakis octahedron, the tetrakis hexahedron, the disdyakis dodecahedron, the triakis
icosahedron, the pentakis dodecahedron and the disdyakis triacontahedron. Ten of the
31 different solids have faces that are all triangular, and
21 have either nontriangular faces or a combination of both, showing that the
Godname EHYEH with number value 21 prescribes the latter. The ten polyhedra
with triangular faces form a tetractys array:
The Platonic solids are at the corners and the seven Catalan solids are at
the corners and centre of a hexagon (shown in grey above), the disdyakis triacontahedron
occupying the latter position because this corresponds to the Sephirah Malkuth, whose
superstring manifestation the polyhedron embodies. The ten solids have 210 polyhedral vertices,
where
11
12
showing how EHYEH prescribes this set of polyhedra.
We found earlier that, in case A, where its faces are single tetractyses,
the triakis tetrahedron has 66 triangles with 24 corners and 78 sides surrounding its axis,
these numbers generating the number values of the Hebrew words ‘Cholem’ and ‘Yesodoth’ in the
name of the Mundane Chakra of Malkuth. They are symbolised in Fig. 7 by the 168 yods other than corners associated with
the first six regular polygons enfolded in the inner Tree of Life. The 24 corners are
denoted by the 24 yods of the pentagon outside the shared edge. The 78 sides are symbolised
by the 78 yods of the triangle, square & decagon and the 66 triangles correspond to
the 66 yods of the hexagon & octagon. We also found that, in case A, the
disdyakis triacontahedron has 660 triangles with 240 corners and 780 sides surrounding its
axis. As the first Catalan solid, the triakis tetrahedron is the polyhedral manifestation of
the first six polygons enfolded in the inner Tree of Life. As the last Catalan solid, the
disdyakis triacontahedron is the polyhedral version of the 60
polygons enfolded in ten overlapping Trees of Life, each yod symbolising a
corner, side or triangle. Because each Sephirah of the Tree of
Life is itself tenfold, the complete, expanded version of the latter is the set of ten
overlapping Trees of Life. This
illustrates the holistic nature of the disdyakis triacontahedron: its
remarkable geometry makes it the counterpart of the holistic system of ten Trees of Life.
Table 2 indicates that none of the Archimedean solids has the property that
the number of geometrical elements surrounding an axis passing through two vertices is ten
times that of another polyhedron of this type. The disdyakis triacontahedron is unique in
this regard for the Catalan solids as well.
When constructed from tetractyses, the dodecagon has 73
yods. Twelve of these are at its corners and one is at its centre, so that 60 extra yods are
needed to turn its sectors into tetractyses (Fig. 8). This is the number of hexagonal yods in the 16 triangles of the Tree
of Life — a sign (which will be confirmed shortly) that the last of the seven regular
polygons making up the inner Tree of Life embodies properties of its outer form. With its
sectors divided into three tetractyses, the dodecagon has 181 yods, i.e.,
168 yods other than its corners and centre (Fig. 9). As
13^{2} –1 = 168 = 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17
+ 19 + 21 + 23 + 25,
13
this is the sum of the twelve odd integers after 1 that can be assigned to
the corners of the dodecagon. Just as there are 168 yods other than corners
associated with the first six enfolded polygons, so there are 168 yods other
than corners associated with the last polygon enfolded on each side of the central Pillar of
Equilibrium. The dodecagons enfolded on one side of ten overlapping Trees of Life have 1680
such yods — exactly the same number of yods as in the first six
polygons enfolded in ten Trees of Life. What is embodied in the first six
polygons as potential degrees of freedom manifests in the seventh polygon as the yods
needed to be added to express its form. Fig. 10 indicates
14
the correspondence between the triakis tetrahedron and the dodecagon and the
correspondence between the disdyakis triacontahedron and the dodecagons enfolded in ten Trees
of Life.
We found earlier that, in case B, when its faces are constructed from three
tetractyses, 72 more geometrical elements are added to the
168 elements surrounding the axis of the triakis tetrahedron. The counterpart
of this in the inner Tree of Life is that there are 168 yods other than
corners associated with the first six polygons and 72 yods associated with the
dodecagon. Likewise, 720 more geometrical elements are added to the 1680 elements surrounding
the axis of the disdyakis triacontahedron when its faces are constructed from three
tetractyses. Its counterpart in the inner Tree of Life is that there are 720 yods associated
with the ten dodecagons enfolded in ten Trees of Life. Construction of both Catalan solids
according to Case A generates the counterpart of the first six regular polygons and
construction according to case B creates the polyhedral counterpart of all seven polygons. The
240 geometrical elements of the triakis tetrahedron are symbolised by the 240 hexagonal yods
belonging to the seven separate polygons (Fig. 11).
The extra 720 geometrical elements in the 120 faces of the disdyakis
triacontahedron for case B are symbolised by the 720 yods in the seven separate polygons that
surround their centres when their 48 triangular sectors are each turned into
three tetractyses (Fig. 12).
It cannot be coincidence that the same number should appear in contexts
where the polygonal and polyhedral forms of the inner Tree of Life are transformed in the same
way. When constructed from tetractyses, the five Platonic solids have 820 yods in their
50 faces, of which 720 are hexagonal yods.^{3} The
very surfaces of the five regular polyhedra embody this number as well.
It cannot also be coincidence that the disdyakis triacontahedron has
62 vertices (31 vertices and their inversions), in view of
the facts that the number value of Chesed, the first Sephirah of Construction, is
72, its Godname EL has the number value 31 and its Archangel
Tzadkiel (“Benevolence of God”) has number value 62 (Table 2).
As 49^{2} – 1 = 2400 = 3 + 5 + 7 + 9 + … +
97, the number (2400) of geometrical elements
15
in the disdyakis triacontahedron in case B is the sum of the first
48 odd integers after 1. As Table 2 indicates, Kokab, the Mundane Chakra of Hod, has the number
value 48 and Haniel (“Grace of God”), the Archangel of Netzach,
has the number value 97.
The integers 1, 2, 3 & 4 symbolised by the tetractys express the number
2400 as:
2400 = 1×2×3×4(1^{3} + 2^{3} + 3^{3} + 4^{3})
= 1×2×3×4(1 + 2 + 3 + 4)^{2}.
As 24 = 5^{2} – 1 = 3 + 5 + 7 + 9,
illustrating how the Tetrad Principle^{4} determines this number.
The disdyakis triacontahedron has 30 polyhedral vertices and their 30
inversions surrounding its axis, where
This again shows how the Tetrad Principle determines the properties of
this solid.
We found earlier that, as the first Catalan solid, the triakis tetrahedron
has 168 geometrical elements surrounding its axis, where
This further illustrates the Tetrad Principle, for the odd integers 3, 5, 7,
etc are arranged along the sides of a foursided square, four to a side.
Since 168 = 13^{2} – 1,
16800 = 100×168 = 10^{2}×(13^{2} – 1) =
130^{2} –100.
But
1300 = 1^{5} + 2^{5} + 3^{5} + 4^{5}
and
100 = 1^{3} + 2^{3} + 3^{3} + 4^{3} = (1 + 2
+ 3 + 4)^{2}.
Therefore,
and
This shows how the integers 1, 2, 3 & 4 express the superstring
parameters 16800 and 1680. The former number has also the remarkable representation:
16
The number 2400 has the representation:
There are 240 gauge fields corresponding to the 240 nonzero roots of the
superstring gauge symmetry group E_{8}. Each of these fields has ten components
because superstring spacetime is 10dimensional. This means that the 240 fields have 2400
components. Their counterparts in the disdyakis triacontahedron are the 2400 geometrical
elements surrounding its axis that are needed to define its shape when it is constructed
from the building block of the tetractys.
Table 1 shows that the truncated tetrahedron — the dual of the triakis
tetrahedron — has 244 geometrical elements surrounding its axis, which consists of five
elements (three vertex points and two edges). The total number of elements is therefore 249.
Surrounding the centre of the solid are 248 elements. Starting from the
mathematical point, 248 more geometrical elements are needed to build the
simplest Archimedean solid. Remarkably, this is the dimension of the superstring gauge
symmetry group E_{8}! The tetractys therefore reveals the following beautiful
properties of four solids that speak eloquently of their archetypal nature:
It was because of its power to describe the nature of the universe that the
followers of Pythagoras swore the following oath:
“I swear by the discoverer of the tetractys
Which is the spring of all our wisdom,
The perennial fount and root of Nature.”
They would have been delighted by the connections between superstring
physics and solid geometry that the tetractys uncovers. This universal template has
application both on a metaphysical level as the symbol of the tenfold nature of God and on
a physical level as the representation of holistic systems such as the superstring and — in
its higherorder version — the tetrahedron, the basic unit of solid geometry.
Let us now calculate how many yods make up the Archimedean and Catalan
solids
17
when they are constructed from tetractyses. Formulae are simplified by
making use of Euler’s equation connecting the number of vertices (C), edges (E) and faces
(F) in a convex polyhedron:
C – E + F = 2.
Surface
Number of corners surrounding axis ≡ V_{s} = C + F – 2 = E;
Number of hexagonal yods surrounding axis ≡ H_{s} = 2E + 3L;
Number of yods surrounding axis ≡ N_{s} = V_{s} + H_{s} = 3E + 3L;
Interior
Number of corners surrounding axis ≡ V_{i} = E;
Number of hexagonal yods surrounding axis ≡ H_{i} = 2C + 9E – 4 = 11E – 2F;
Number of yods surrounding axis ≡ N_{i} = V_{i} + H_{i} = 12E – 2F;
Total number of corners surrounding axis ≡ V = V_{s} + V_{i} = 2E;
Total number of hexagonal yods surrounding axis ≡ H = H_{s} + H_{i} = 13E – 2F
+ 3L;
Total number of yods surrounding axis ≡ N = N_{s} + N_{i} =
15E – 2F + 3L.
Table 3. Populations of nonaxial yods in the Archimedean and Catalan
solids.
Archimedean solid 
L

V_{s}

H_{s}

N_{s}

V_{i}

H_{i}

N_{i}

V

H

N

truncated tetrahedron 
36

18

144

162

18

182

200

36

326

362

cuboctahedron 
48

24

192

216

24

236

260

48

428

476

truncated cube 
72

36

288

324

36

368

404

72

656

728

truncated octahedron 
72

36

288

324

36

368

404

72

656

728

rhombicuboctahedron 
96

48

384

432

48

476

524

96

860

956

snub cube 
120

60

480

540

60

584

644

120

1064

1284

icosidodecahedron 
120

60

480

540

60

596

656

120

1076

1196

truncated cuboctahedron 
144

72

576

648

72

740

812

144

1316

1460

truncated icosahedron 
180

90

720

810

90

926

1016

180

1646

1826

truncated dodecahedron 
180

90

720

810

90

926

1016

180

1646

1826

rhombicosidodecahedron 
240

120

960

1080

120

1196

1316

240

2156

2396

snub dodecahedron 
300

150

1200

1350

150

1466

1616

300

2666

2966

truncated icosidodecahedron 
360

180

1440

1620

180

1856

2036

360

3296

3656

Catalan solid 
L

V_{s}

H_{s}

N_{s}

V_{i}

H_{i}

N_{i}

V

H

N

triakis tetrahedron 
36

18

144

162

18

174

192

36

318

354

rhombic dodecahedron 
48

24

192

216

24

240

264

48

432

480

triakis octahedron 
72

36

288

324

36

348

384

72

636

708

tetrakis hexahedron 
72

36

288

324

36

348

384

72

636

708

deltoidal icositetrahedron 
96

48

384

432

48

480

528

96

864

960

pentagonal icositetrahedron 
120

60

480

540

60

612

672

120

1092

1212

rhombic triacontahedron 
120

60

480

540

60

600

660

120

1080

1200

disdyakis dodecahedron 
144

72

576

648

72

696

768

144

1272

1416

triakis icosahedron 
180

90

720

810

90

870

960

180

1590

1770

pentakis dodecahedron 
180

90

720

810

90

870

960

180

1590

1770

deltoidal hexacontahedron 
240

120

960

1080

120

1220

1340

240

2180

2420

pentagonal hexacontahedron 
300

150

1200

1350

150

1530

1680

300

2730

3030

disdyakis triacontahedron 
360

180

1440

1620

180

1740

1920

360

3180

3540

Table 3 lists the various nonaxial yod populations of the Archimedean and
Catalan solids (enantiomorphic partners are excluded because their numbers are the same).
The disdyakis triacontahedron has V = 360 (=36×10) corners surrounding its
axis, showing how the Godname ELOHA of Geburah with number value 36
prescribes this solid. It also prescribes the triakis tetrahedron because it has
36 corners surrounding its axis
18
and 354 yods surrounding its axis made up of seven yods, i.e., it has 361
yods in total, 360 (=36×10) of them surrounding its centre. As there are three
vertices on the axis, the total number of corners in the disdyakis triacontahedron is
363. This is the number value of SHADDAI EL CHAI (“Almighty Living God”), the
Godname of Yesod, the penultimate Sephirah (see Table 1). If its faces are single tetractyses, it has 2460
(=246×10) yods surrounding its axis. 246 is the number
value of Gabriel (“Strong Man of God”), the Archangel of Yesod.
We found earlier that the numbers of vertices, edges and triangles
surrounding the axis of the disdyakis triacontahedron are ten times the corresponding numbers
for the triakis tetrahedron. Table 3 indicates that the numbers of corners or hexagonal yods in its faces
and interior that surround the axis are ten times the corresponding numbers for the triakis
tetrahedron. This is true only for surrounding numbers, not for the total numbers.
It is also true only for the first and last members of the Catalan solids, not for the first
and last members of the Archimedean solids, although some of their corresponding numbers do
have this property.
The number of boundary yods in the faces of a polyhedron surrounding its
axis = C + 2E + 2L + F – 2 = 3E + 2L. The triakis tetrahedron with 12 triangular faces has 126
such yods on the boundaries of its (12×3=36) tetractyses in its faces, and the
disdyakis triacontahedron has 1260 boundary yods in the (120×3=360)
tetractyses of its 120 faces, i.e., ten times as many yods. The number of boundary yods inside
a polyhedron surrounding its axis = 2C + 7E – 4 = 9E – 2F. The triakis tetrahedron has 138 such
yods and the disdyakis triacontahedron has 1380 such yods.
The former has (126+138=264) boundary yods and the latter has
(1260+1380=2640) boundary yods. We see that the numbers of yods on the
boundaries of tetractyses inside the disdyakis triacontahedron and on its faces are ten times
the corresponding numbers for the triakis tetrahedron. The division: 264 = 126 + 138, created
by the surface and interior of the triakis tetrahedron, appears in the inner form of the Tree
of Life as the 126 yods of the last two enfolded polygons and as the 138 yods of the first five
enfolded polygons! The fact that the number of boundary yods in the disdyakis triacontahedron
is ten times that for the triakis tetrahedron and ten times the yod population of the inner
Tree of Life is further evidence that this solid is the polyhedral counterpart of the inner
Tree of Life.
1260 is the number of yods in 126 tetractyses. It is
remarkable that the number 126 is the sum of the number values of the four types of
combinations of the letters A, H and I in EHYEH (AHIH), the Godname of Kether:
A = 1, H = 5, I = 10
1. 
A + H + I 
= 
16 
2. 
AH + HI + AI + HH 
= 
42 
19
3. 
AHI + HIH + AHH 

= 
47 
4. 
AHIH 

= 
21 


Total 
= 
126 
This shows how EHYEH prescribes the population of yods that shape the 360
tetractyses in the faces of the disdyakis triacontahedron. It is further evidence of the way
the Godnames of the ten Sephiroth prescribe the disdyakis triacontahedron as the polyhedral
form of the inner Tree of Life.
The number 126 has the remarkable property that it is the arithmetic mean of
the first 26 triangular numbers, where 26 is the number value
of YAHWEH:
This shows how YAHWEH prescribes the shape of the disdyakis triacontahedron
by mathematically determining the number of yods on boundaries of tetractyses in its
surface.
According to Table 3, the number of yods in its faces that surround its axis
is 1620. Of these, 60 are vertices of the polyhedron, leaving 1560 (=156×10) extra yods needed
to create its faces. 156 is the sum of the number values of the four types of combinations of
the letters Y, H and V in YHVH, the Godname of Chokmah:
Y = 10, H = 5, V = 6
1. 
Y + H + V 

= 
21

2. 
YH + HV + YV + HH 

= 
52 
3. 
YHV + HVH + HYH 

= 
57 
4. 
YHVH 

= 
26 


Total 
= 
156 
It illustrates the creative quality embodied in this wellknown Godname, for
the yods in 156 tetractyses are needed to form the faces of the disdyakis triacontahedron
marked out by its 62 vertices. 156 is the 155th integer after
1, showing how ADONAI MELEKH, the Godname of Malkuth with number value 155
(see Table 1), prescribes this polyhedron.
156 is the sum of the first 12 even integers that can be arranged in a
square:
This illustrates how the Tetrad Principle determines this number as a square
array of even integers, starting with 2, the first even integer.
Let us now consider a polyhedron whose faces and internal triangles are
single tetractyses (case C). The number of internal yods = 2C + E + 1. The number of yods in
its faces = C + 2E + F = 3E + 2. The total number of yods = 2C + 4E + 3. The triakis
tetrahedron has 91 yods, that is, 84 yods surround its axis made up of seven yods. The
disdyakis triacontahedron has 847 yods, that is, 840 yods surround its axis. Once again, we see
that the yod population of the last Catalan solid is ten times that of the
20
first one. The number 840 was encountered in Article 26 as the number of geometrical elements inside the faces
of the disdyakis triacontahedron in case B.^{5} It is also the number of circularly polarised oscillations in an
outer or an inner half of each whorl of the E_{8}×E_{8} heterotic
superstring (Fig. 6). The triakis tetrahedron with single tetractyses as its faces has as
many yods surrounding its axis as the tetrahedron has geometrical elements surrounding its
centre (case B).
For case C:
Number of corners ≡ V = C + 1. Number of corners surrounding axis ≡ V' = V – 3 = C –
2.
Number of sides ≡ e = C + E. Number of sides surrounding axis ≡ e' = e – 2 = C + E – 2.
Number of triangles ≡ T = E + F. Number of triangles surrounding axis = E + F.
Total number of geometrical elements ≡ N = 2C + 2E + F + 1 = C + 3E + 3.
Number of geometrical elements surrounding axis ≡ N' = C + 3E – 2.
Table 4. Numbers of geometrical elements in the Catalan solids.
Catalan solid 
V

V'

e

e'

T

N

N'

triakis tetrahedron 
9

6

26

24

30

65

60

rhombic dodecahedron 














triakis octahedron 
15

12

50

48

60

125

120

tetrakis hexahedron 
15

12

50

48

60

125

120

deltoidal icositetrahedron 














pentagonal icositetrahedron 














rhombic triacontahedron 














disdyakis dodecahedron 
27

24

98

96

120

245

240

triakis icosahedron 
33

30

122

120

150

305

300

pentakis dodecahedron 
33

30

122

120

150

305

300

deltoidal hexacontahedron 














pentagonal hexacontahedron 














disdyakis triacontahedron 
63

60

242

240

300

605

600

Table 4 lists the various numbers of elements in the Catalan solids with
triangular faces. The triakis tetrahedron has 26 sides prescribed by YAHWEH
and 65 geometrical elements prescribed by ADONAI, Godname of Malkuth,
because its number value is 65 (see Table 1). The latter property is remarkable because, together with the
168 elements surrounding its axis in case A, the solid embodies number
values of the same Sephirah (Malkuth) in two Kabbalistic worlds — Atziluth
and Assiyah. This is yet another indication that the simplest Catalan solid, which is
prescribed by the Godname EHYEH of Kether with number value 21 because it
is the 21st member of the set of Platonic, Archimedean and Catalan solids,
is one that possesses a Tree of Life pattern.
21
This is confirmed by the fact that the disdyakis triacontahedron has numbers
of geometrical elements that — once again — are ten times their counterparts in the triakis
tetrahedron. As was discussed in Article 26,^{6} ADONAI prescribes the lowest ten Trees of Life of an extended
sequence of trees because they have 65 Sephirothic emanations or levels
(what in previous articles were called ‘SLs’). The 65 bits of geometrical
information needed to define the triakis tetrahedron are signalling these ten trees, the
properties of their enfolded polygons being embodied in the disdyakis
triacontahedron.^{7}
The 60 vertices surrounding the axis of the disdyakis triacontahedron are
degrees of freedom symbolised by the 60 extra yods needed to construct the Tree of Life from
tetractyses (Fig. 14). As the last of the seven regular polygons constituting the inner
Tree of Life, the dodecagon — the tenth regular polygon — has 60 hexagonal yods symbolising
these degrees of freedom.
3. The triakis tetrahedron as the
Cosmic Tree of Life
Because the superphysical and physical cosmos constitute a whole, they are
together represented by the tetractys. Or, rather, following the ancient hermetic axiom “as
above, so below,” a tetractys that is modified to express this statement. This is accomplished
by replacing the tetractys at the centre of the next higherorder tetractys by another such
tetractys (Fig. 15). This is the ‘Cosmic Tetractys.' It maps what the author has called
the ‘Cosmic Tree of Life,’ namely, 91 overlapping Trees of Life, each tree representing a
possible jump in evolutionary consciousness.^{8} This map of all levels of consciousness is encoded in the inner
Tree of Life. The seven hexagonal yods at the centre of its central tetractys (shown
coloured in Fig. 15)
symbolise the seven physical subplanes of consciousness discussed in
Theosophical literature. The (6×7=42) hexagonal yods in the central higherorder tetractys that
surround the central, 1storder tetractys denote the 42 subplanes of the six planes of
consciousness above the physical plane: astral, mental, buddhic, atmic,
22
anupadaka and adi planes. The 42 hexagonal yods surrounding those in the
central 2ndorder tetractys denote the 42 subplanes of the six cosmic superphysical planes. The
91 hexagonal yods in the Cosmic Tetractys symbolise the 91 subplanes. 91 is the sum of the
squares of the first six integers:
Let us now compare this map of consciousness with the pattern of yods in the
triakis tetrahedron. It has 91 yods, of which seven yods lie along its axis. They are
surrounded by 84 yods distributed as 42 yods on the edges of the solid (6 vertices,
36 hexagonal) and 42 hexagonal yods either at the centre of its faces or
inside the solid. We see that the axis of the solid with seven yods corresponds to the central
tetractys with seven hexagonal yods symbolising the seven physical subplanes. The 42 yods lying
on the boundary of the solid correspond to the 42 subplanes above the physical plane and
belonging to the cosmic physical plane (hence their function in defining the shape of the
solid). The 42 hexagonal yods not delineating its faces (and therefore playing an invisible
role) correspond to the 42 subplanes of the cosmic superphysical planes (notice the
aptness of the correspondence). The triakis tetrahedron may be regarded as the polyhedral
version of CTOL. The three vertices and four hexagonal yods making up the central axis
symbolise what in Theosophy are called, respectively, the three “dense physical” subplanes and
the four “etheric” subplanes. Just as the physical universe (including the world of shadow
matter or mirror matter) is the final product of Divine Thought issuing down through all higher
realms of consciousness, so this axis is the fulcrum for the polyhedron.
The ‘case C’ type of transformation of the triakis tetrahedron creates a
polyhedral map of all levels of consciousness, and the similar transformation of the disdyakis
triacontahedron generates the superstring structural parameter 840. The transformation
according to case A generates for these polyhedra the superstring structural parameters
168 and 1680. Transformation according to case B generates the
grouptheoretical parameters 240 and 2400 of the superstring gauge symmetry group
E_{8}.
4.
Equivalence of Plato’s Lambda and the triakis tetrahedron
Article 26 discussed the remarkable correspondence between the
tetractys of numbers extrapolated from Plato’s socalled ‘Lambda’ and the various numbers of
tetractyses in the disdyakis triacontahedron (case B).^{9} As the latter are ten times the corresponding numbers in the triakis
tetrahedron for the same kind of transformation, it might be expected that a correspondence
also exists for case C, where the faces and internal
23
triangles are single tetractyses. This, indeed, is true. 90 yods surround
its centre (Fig. 16). This is the sum of the ten integers in the ‘Lambda Tetractys’
formed from the Lambda by extrapolating the three numbers shown in red. The six yods on the
axis on either side of its centre correspond to the number 6 at the centre of the tetractys.
The 18 edges of the solid have 36 hexagonal yods, and there are 12
hexagonal yods at centres of its faces. The 48 hexagonal yods in its faces
correspond to the sum of the six integers in the Lambda Tetractys at the corners of a
hexagon: 2 + 3 + 9 + 18 + 12 + 4 = 48. The remaining 36
yods (6 vertices and 30 internal yods) correspond to the sum of the integers at the corners
of the tetractys: 1 + 8 + 27 = 36.
An alternative correspondence would have the central integer corresponding
to the six vertices surrounding the axis and the sum 36 of the integers at the
three corners corresponding to the 36 internal yods. As the number 6 is the
musical proportion of the tonic, the lowest note of the musical scale, and as its position in
the Lambda tetractys corresponds to Malkuth, the physical universe of seven subplanes
symbolised (as we found earlier) by the seven yods on the axis, the correspondence that is more consistent with
the analogy between the Cosmic Tetractys and the triakis tetrahedron is the one discussed
first in which the central
integer 6 denotes the six yods on the axis on either side of its centre.
Another alternative correspondence scheme would have, firstly, the sum (54)
of the seven integers making up the Lambda denoting the number of yods in the faces of the
solid that surround its axis (6 vertices, 48 hexagonal yods) and, secondly,
the sum (36) of the three extrapolated integers (shown in red in Fig. 17) denoting the number of yods in its interior. As before, the central
integer 6 would signify the number of yods on the axis on either side of its centre, but now
the extrapolated number 12 would denote the 12 internal hexagonal yods on the lines joining
the centre to vertices surrounding the axis and the number 18 would denote the 18 hexagonal
yods at the centres of the triangles formed by the centre and 18 edges of the triakis
tetrahedron. This is intuitively appealing because the three extrapolated (i.e.,
hidden) numbers would then determine the number of interior yods, although the
correspondence is not perfect because the number 6 would also determine the two vertices at
the ends of the axis and these are external, not internal, yods.
Whatever scheme is favoured, this correspondence exists because the Lambda
Tetractys and the triakis tetrahedron are, respectively, the arithmetic and geometric
24
expressions of mathematical archetypes working through a law of
correspondence to link the psychological qualities of the notes of the musical scale to the
pattern of varieties of consciousness in the spiritual cosmos.
It was pointed out in Article 26^{10} that the sums of the integers in the four diagonal rows of the
Lambda Tetractys are the numbers of yods in the first four polygons of the inner Tree of
Life that are outside their shared edge (Fig. 18). As if this 1:1 correlation between the numbers and the geometry
were not remarkable enough, it is further extraordinary that:

the sum (36) of the integers at the corners of the tetractys is the
number of yods outside the root edge in the triangle and square. This is also the
number of vertices and internal yods of the triakis tetrahedron;

the sum (54) of the integers arranged at the corners and centre of a
hexagon is the number of external yods in the pentagon and hexagon. This is the number
of hexagonal yods on the faces of the triakis tetrahedron and yods on its axis on
either side of its centre.
25
side of its centre.
5. Root structures of
E_{8},
E_{7} & E_{6} and their polyhedral
representation
The superstring gauge symmetry group E_{8} has 248
roots made up of eight zero roots and 240 nonzero roots. Its largest exceptional subgroup
E_{7} has 133 simple roots made up of 7 zero roots and 126 nonzero roots.
E_{7} has the rank6, exceptional subgroup E_{6} with 78 simple roots that
comprise six zero roots and 72 nonzero roots. Let us compare this root
structure with the geometrical composition of the triakis tetrahedron. Fig. 17 shows that, when we start with case C, in which
the faces and internal triangles are single tetractyses, the solid has 60 geometrical
elements surrounding its axis.
Construction of the triangular faces from three tetractyses adds
72 new elements. Construction of the internal triangles from three tetractyses
requires adding three sides within each triangle, i.e., 54 for the interior of the solid,
therefore adding (72+54=126) elements for its interior and faces. As well as
adding 54 more sides, division of each internal triangle into three tetractyses adds one corner
and two triangles, creating 54 more elements. The total number of elements = 60 +
72 + 54 + 54 = 240. Constructed from tetractyses, the triakis tetrahedron has
as many geometrical elements surrounding its axis as E_{8} has nonzero roots. The
72 new elements needed to construct its faces from three tetractyses
correspond to the 72 nonzero roots of E_{6}, whilst the minimal
number of 126 new elements needed to construct both faces and internal triangles from
tetractyses correspond to the 126 nonzero roots of E_{7}. The remaining (60+54=114)
elements correspond to the (240–126=114) nonzero roots of E_{8} that do not belong to
E_{7}. The simplest Catalan solid embodies the nonzero root structure. As we saw
earlier, the simplest Archimedean solid — the truncated tetrahedron — has, when constructed
from tetractyses, 248 geometrical elements surrounding its centre (eight
corners at the centres of its faces, 240 other elements). This solid represents the eight zero
roots and 240 nonzero roots of E_{8}; its dual expresses the nonzero root structure
of E_{8}.
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Fig. 18 displays the same type of calculation for the disdyakis
triacontahedron as that given in Fig. 17 for the triakis tetrahedron. It shows a representative face and an
internal triangle divided into three triangles. Amazingly, all corresponding numbers differ
by a factor of 10. The nonzero root structures of the superstring gauge symmetry group
E_{8} and its exceptional subgroups E_{7} and E_{6} are embodied in
the geometrical composition of the disdyakis triacontahedron. E_{6} is determined by
its faces and E_{7} encompasses both faces and its interior.
The superstring parameter 168 is the number of geometrical
elements in the triakis tetrahedron in case A. In terms of the corresponding nonzero
roots:
168 = 54 + 114,
where the first number is the number of nonroots of E_{7} that do
not belong to E_{6} and the second term is the number of nonzero roots of
E_{8} that do not belong to E_{7}. In terms of corners, sides and triangles,
there are in the triakis tetrahedron 24 corners (six external, 18 internal), 78 sides (18
external, 60 internal) and 66 triangles (12 external, 54 internal). The total number of
168 elements is the number value 168 of Cholem
Yesodoth. The number value 78 of the word ‘Cholem’ measures the 78 sides and the number
value 90 of the word ‘Yesodoth’ measures the number of corners & triangles. The triakis
tetrahedron is composed of 36 geometrical elements in its faces (six vertices,
18 edges & 12 triangles) that surround its axis and 132 internal elements (18 corners, 60
sides & 54 triangles). This shows how ELOHA, the Godname assigned to Geburah with number
value 36 (see Table 2), prescribes this solid and thus the superstring parameter
168 as the number of geometrical elements surrounding its axis. Surrounding
the axis of the disdyakis triacontahedron are 360 elements in its faces (60 vertices, 180
edges & 120 triangles) and 1320 elements in its interior (180 corners, 600 sides &
540 triangles). Both the oscillatory structure and the E_{8}symmetric forces of the
E_{8}×E_{8} heterotic superstring are represented in the geometry of this
remarkable polyhedron.
The 1680 circularly polarised oscillations of each whorl of the
E_{8}×E_{8} heterotic superstring are the string manifestation of the 240 gauge
charges of E_{8} associated with the 240 generators defined by its nonzero roots.
These gauge charges are spread along the ten whorls, 24 to a whorl. The seven minor whorls of
the subquark superstring carry 168 gauge charges of E_{8} and the
three major whorls carry (3×24=72) gauge charges that are also the gauge
charges of its subgroup E_{6}. The distinction between the two types of whorl amounts
to a difference in how each turn in their helices is compounded from helices formed by the
winding of each stringy whorl around six successively smaller circular dimensions of a 6torus
and does not need to be discussed here. Their difference indicates symmetry breaking of
E_{8} into E_{6}. Its counterpart in the disdyakis triacontahedron is the
topological distinction between the interior of the polyhedron and its exterior, which requires
720 extra geometrical elements to construct it in order to make its outer fashioned in the same
way from tetractyses as its inner. Analogous to a hypothetical universe where only superstring
physics governed by E_{6} operates (this symmetry is further reduced for the real
world) would be one allowing only the faces of polyhedra to be visible, so that their interiors
never played any noticeable role, despite always being present. It would be one in which, like
an ant crawling over the surfaces of things, one could be conscious only of surfaces, never of
depth. The form of superstrings is determined, however, not by E_{6} or any of its
subgroups (this determines only the difference between the major and minor whorls) but by that
part of E_{8} that is not also its subgroup E_{6}. In this sense, the analogy
between the disdyakis triacontahedron and the E_{8}×E_{8} heterotic superstring
indicates that what is now visible as matter is determined by what is now unobservable
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because the physics operated in a former time when the universe was,
momentarily, one of perfect E_{8} symmetry and is locked out by the lesser symmetry of
E_{6} and its subgroups that survived. It is not that the universe today is less
perfect in a mathematical sense than before. It is just that the perfection of the complete
symmetry is hidden from view — as the interior of a disdyakis triacontahedron would be to an
ant crawling over it. Only when consciousness has the attribute of recognising spatial depth
can the mathematical beauty of its solid structure be appreciated and its archetypal meaning
discerned.
References
1 All numbers from the table appearing in the text will be written in
boldface.
2 Phillips, Stephen M. Article 1: “The Pythagorean Nature of Superstring and
Bosonic String Theories,” (WEB, PDF), p. 5.
3 Phillips, Stephen M. Article 3: “The Sacred Geometry of the Platonic Solids,”
(WEB, PDF), Table 1, p. 20.
4 Ref. 2.
5 Phillips, Stephen M. Article 26: “How the Seven Musical Scales Relate to the
Disdyakis Triacontahedron," (WEB, PDF), p. 29.
6 Ibid., p. 21.
7 Ibid., pp. 21, 34–36.
8 Phillips, Stephen M. Article 5: “The Superstring as Microcosm of the Spiritual
Macrocosm,” (WEB, PDF).
9 Ref. 5.
10 Ibid., p. 28.
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