ARTICLE 30
by
Stephen M. Phillips
Flat 4, Oakwood House, 117119 West Hill Road. Bournemouth.
Dorset BH2 5PH. England.
Website: http://smphillips.mysite.com
Abstract
Earlier analysis of the Archimedean and Catalan solids
constructed from the tetractys — the template of straightline, sacred geometry
— revealed that the disdyakis triacontahedron (the Catalan solid with the most
vertices) embodies the structural parameter 1680 of the
E_{8}×E_{8} heterotic superstring. This article completes the
analysis by proving that this polyhedron also embodies its structural parameter
840. The triakis tetrahedron (the simplest Catalan solid) and the disdyakis
triacontahedron are the polyhedral version of Plato’s Lambda Tetractys, whose
numbers generate the tone ratios of the notes of the Pythagorean musical scale.
The 120 faces of the disdyakis triacontahedron contain 480 hexagonal yods. They
symbolize the 480 roots of E_{8}×E_{8} and are the counterparts
of the 480 hexagonal yods of the two similar sets of seven regular polygons that
constitute the inner form of the Tree of Life. The direct product nature of this
gauge symmetry group reflects the fact that each face and vertex of the
disdyakis triacontahedron has a mirror image.

In previous articles, it was proved that, when each of the 120 triangular
faces of the disdyakis triacontahedron (Fig. 1) is constructed from three tetractyses and the triangles in its
interior are turned into single tetractyses (case C), there are 1320 geometrical elements
surrounding its axis. When its faces are single tetractyses and the interior triangles are
divided into three tetractyses (case A), there are 1680 elements surrounding its axis. When
its faces are each divided into three tetractyses and its internal triangles are divided
into three triangles (case B), there are 2400 elements surrounding its axis. In this
article, we shall consider the final possibility (case D), wherein internal triangles and
faces are single tetractyses. Of the four possible cases, only cases B and C have
counterpart transformations in the Archimedean and Catalan solids because consistency of
construction requires a face that is a regular polygon with n sides to be divided into n
triangular sectors.
1
Cases A and D are exceptions to the general rule of construction of
polyhedra from tetractyses. This does not mean, however, that they lack legitimacy or
significance. On the contrary, the disdyakis triacontahedron in case A embodies the superstring structural parameter 1680 and (as we shall see shortly) provides
another such parameter in case D. What it does mean is that polyhedra like the disdyakis
triacontahedron with only triangular faces are a special case because their faces can be
constructed from tetractyses in two ways, not one, i.e., from one or three tetractyses. As
none of the Archimedean solids has only triangular faces, only the Catalan solids need to
be considered in case D. As in earlier articles, we shall consider an axis passing through
two diametrically opposite vertices of a polyhedron and then calculate the number of
geometrical elements and the number of yods that surround this axis, simplifying
expressions by using Euler’s equation for a polyhedron:
C – E + F = 2.
The central axis passing through two such vertices consists of three corners
of tetractyses (two polyhedral vertices and its centre) and two edges of tetractyses
(Fig. 2). Seven yods lie on the axis (three corners and four hexagonal
yods).
For case D:
Number of vertices ≡ V = C + 1. Number of vertices surrounding axis ≡ V' = V – 3 = C – 2 = E –
F.
Number of edges ≡ e = C + E. Number of edges surrounding axis ≡ e' = 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 1. Numbers of geometrical elements in the Catalan solids (case D).
Catalan solid

C

E

F

V

V'

e

e'

T

N

N'

triakis tetrahedron 
8

18

12

9

6

26

24

30

65

60

rhombic dodecahedron 




















triakis octahedron 
14

36

24

15

12

50

48

60

125

120

tetrakis hexahedron 
14

36

24

15

12

50

48

60

125

120

deltoidal icositetrahedron 




















pentagonal icositetrahedron 




















rhombic triacontahedron 




















disdyakis dodecahedron 
26

72

48

27

24

98

96

120

245

240

triakis icosahedron 
32

90

60

33

30

122

120

150

305

300

pentakis dodecahedron 
32

90

60

33

30

122

120

150

305

300

deltoidal hexacontahedron 




















pentagonal hexacontahedron 




















disdyakis triacontahedron 
62

180

120

63

60

242

240

300

605

600

According to Table 1, the triakis tetrahedron — the Catalan solid with fewest vertices —
consists of 60 vertices, edges and triangles surrounding its axis. The disdyakis
triacontahedron — the Catalan solid with most vertices — consists of 600 geometrical
2
elements surrounding its axis. Indeed, for the latter, the numbers of
vertices (V'), edges (e') and triangles (T) surrounding its axis are ten times the
corresponding numbers for the former. The Godname ADONAI with number value 65
is assigned to Malkuth.^{1} It prescribes the triakis tetrahedron because the simplest
Catalan solid comprises 65 vertices, edges and triangles. It has
36 vertices and triangles, where 36 is the number value of
ELOHA, Godname of Geburah. The Godname EHYEH with number value 21
assigned to the Sephirah Kether prescribes this polyhedron because, counting
firstly the five Platonic solids, then the 15 Archimedean solids and, finally,
the Catalan solids, the triakis tetrahedron is the 21st polyhedron. The
Godname YAHWEH with number value 26 assigned to Chokmah prescribes the triakis
tetrahedron because its 18 tetractyses have 26 edges. Two edges form its axis,
so that 24 edges surround it. The 60 geometrical elements surrounding the axis comprise 24
edges and 36 vertices and triangles.
The counterpart of this in the Tree of Life is the fact that, constructed
from 16 tetractyses, it has 60 yods other than the ten Sephiroth at their corners (Fig. 3). Each yod symbolizes one of the geometrical elements in the triakis
tetrahedron. The 24 yods below the apex of the Tree of Life down to the path connecting
Chesed and Geburah denote the 24 edges around the axis and the 36 yods
below the path denote the 36 vertices and triangles that surround it. This
is striking evidence that the triakis tetrahedron is the polyhedral counterpart of the outer
form of the Tree of Life.
The numbers of yods in the faces and in the interiors of a polyhedron with
triangular faces are calculated below:
Faces
Number of corners of tetractyses = C. Number of corners surrounding axis = C – 2 = E – F.
Number of hexagonal yods on edges of tetractyses = 2E.
Number of hexagonal yods at centres of tetractyses = F.
Total number of hexagonal yods ≡ H_{s} = 2E + F.
Number of yods surrounding axis ≡ N_{s} = C – 2 + H_{s} = C – 2 + 2E + F =
3E.
Interior
Number of corners of tetractyses = 1 (centre of polyhedron).
Number of hexagonal yods on edges of tetractyses ≡ 2C.
Number of hexagonal yods at centres of tetractyses = E.
Total number of hexagonal yods surrounding axis ≡ H_{i} = E + 2C – 4 = 3E – 2F.
Total number of yods surrounding axis ≡ N_{i} = H_{i} = 3E – 2F.
3
4
Total number of corner yods surrounding axis ≡ C' = C – 2 = E – F.
Total number of hexagonal yods surrounding axis ≡ H = H_{s} + H_{i} = 5E –
F.
Total number of yods surrounding axis ≡ N = C' + H = 6E – 2F.
Table 2. Numbers of yods in the Catalan solids (case D).
Catalan solid 
C

E

F

C'

H_{s}'

H_{i}

H

N_{s}

N_{i}

N

triakis tetrahedron 
8

18

12

6

48

30

78

54

30

84

rhombic dodecahedron 




















triakis octahedron 
14

36

24

12

96

60

156

108

60

168

tetrakis hexahedron 
14

36

24

12

96

60

156

108

60

168

deltoidal icositetrahedron 




















pentagonal icositetrahedron 




















rhombic triacontahedron 




















disdyakis dodecahedron 
26

72

48

24

192

120

312

216

120

336

triakis icosahedron 
32

90

60

30

240

150

390

270

150

420

pentakis dodecahedron 
32

90

60

30

240

150

390

270

150

420

deltoidal hexacontahedron 




















pentagonal hexacontahedron 




















disdyakis triacontahedron 
62

180

120

60

480

300

780

540

300

840

Table 2 lists the yod populations of those Catalan solids that have only
triangular faces (other solids have cells with dashes). The triakis tetrahedron has 84 yods
surrounding the seven yods along its axis, where
84 = 1^{2} + 3^{2} + 5^{2} + 7^{2}.
This can be expressed as:
This provides a 4×4 array of odd integers adding to 84 (Fig. 4, bottom). The ten integers enclosed in the grey triangle form the
upper half of the rhombic array shown in Fig. 4. The 2ndorder tetractys (Fig. 5) — the next higher order of differentiation of the
(1storder) tetractys — has 85 yods, 84 of them surrounding its centre. The
1, 2, 3 & 4 points in the four rows of the tetractys symbolise the sequence of generation
of the tetrahedron from a mathematical point:
5
point line triangle tetrahedron.
Its next higherorder version denotes the triakis tetrahedron, which is
formed from the tetrahedron by raising the centres of its four faces (mathematicians call this
process ‘cumulation’). The 2ndorder tetractys also symbolises the tetrahedron in case B
because the latter consists of 15 vertices (symbolized by the
15 corners of the ten tetractyses), 40 edges and 30 triangles (symbolized by
their 70 hexagonal yods), that is, 85 geometrical elements.^{2}
The polyhedron has 91 yods, where
91 = 1^{2} + 2^{2} + 3^{2} + 4^{2} +
5^{2} + 6^{2},
and
1^{2} =
1
2^{2} =
1 + 3
3^{2} =
1 + 3 + 5
4^{2} =
1 + 3 + 5 +
7
5^{2} =
1 + 3 + 5 + 7 +
9
6^{2} =
1 + 3 + 5 + 7 +
9 + 11.
The 21 odd integers adding to 91 demonstrate how the
Godname EHYEH with number value 21 prescribes the yod population of the
triakis tetrahedron. As 15 integers lie on the boundary of this triangular
array, it also shows how the Godname YAH with number value 15 prescribes this
number.
Starting from a point, 90 more yods are needed to construct the triakis
tetrahedron from its 30 tetractyses. As discussed in Article 27,^{3} the Lambda array of powers of 2 and 3 made famous by Plato in
Timaeus, his treatise on cosmology, is but seven of ten integers arranged in a
tetractys (Fig. 6). Including the three interpolated integers (shown in red in Fig. 6), the ten integers add up to 90, which is the number of yod
surrounding the centre of the triakis tetrahedron. The integer 6 at the centre of the
tetractys
corresponds to the number of yods on the axis of the polyhedron that
surround its centre. The sum (36) of the integers at the corners of the
tetractys corresponds to the 30 internal yods and the six vertices surrounding the axis. The
sum (48) of the six integers at the corners of a hexagon in the Lambda
Tetractys corresponds to the 48 hexagonal yods in the faces of the
polyhedron.
This correlation demonstrates that the Lambda Tetractys and the triakis
tetrahedron are, respectively, the arithmetic and geometrical expressions of the same
mathematical archetype. It manifests spiritually as the 90 subplanes of the seven cosmic
planes
6
above the lowest subplane of the physical plane described by
Theosophy.^{4} The seven yods on the axis symbolize the seven subplanes of the
physical plane, the 42 yods on the 18 edges symbolize the 42 subplanes of the six planes of
the cosmic physical plane above the physical plane and the 42 yods either inside the
polyhedron or at the centres of the 12 faces symbolize the 42 subplanes of the six
superphysical, cosmic planes. The three yods on the axis that are corners of tetractyses
denote what Theosophy calls the three ‘dense physical subplanes’ and the four hexagonal yods
on the axis denote the four etheric subplanes. The correlation between the Lambda Tetractys
and the triakis tetrahedron manifests physically as the (10×9=90) coordinate variables
needed to track the motion of the ten closed curves in 10dimensional spacetime that
constitute the heterotic superstring constituent of quarks. Notice in both cases that the
total number of degrees of freedom — subplanes or spacetime coordinates of the curves — is
91, which is the number of yods in the triakis tetrahedron. The reason for this is that the
seven planes and the heterotic superstring are, respectively, the macrocosmic and
microscopic manifestations of the Tree of Life blueprint.^{5}
The Lambda Tetractys generates the tone ratios of the notes of the
Pythagorean musical scale as ratios of pairs of integers.^{6} The sums 36 and 54 of, respectively, the
integers at its three corners and the integers at the centre and corners of a hexagon also
define the perfect fifth with tone ratio 3/2 because 54/36 = 3/2, whilst
the sums 36 and 48 define the perfect fourth with tone
ratio 4/3 because 48/36 = 4/3. The triakis tetrahedron also defines the
perfect fifth because its faces and interior have, respectively, 36 and 24
geometrical elements surrounding its axis, and 36/24 = 3/2.
For case D, according to Tables 1 and 2, every number of yods or geometrical elements surrounding the axis of the
disdyakis triacontahedron is ten times its counterpart in the triakis tetrahedron.
Surrounding its axis are 600 geometrical elements and 840 yods. Among the latter are 60
vertices and (180×2=360) hexagonal yods on its edges, that is, (360+60=420) yods form its
edges, leaving 420 yods either at centres of faces or inside the polyhedron. As
29^{2} – 1 = 840 = 3 + 5 + 7 + … + 57,
840 is the sum of the first 28 odd integers after 1. The number 29 is the
largest of the first ten prime numbers:
2
3 5
7 11 13
17 19 23 29 .
The number (120) of faces of the disdyakis triacontahedron is the sum of the
first ten odd integers after 1:
3
5
7
120 = 9
11 13
15 17 19 21 .
As the largest of these integers, the number value 21 of
EHYEH, Godname of Kether, determines the number of faces of this Catalan solid. These faces
contain 480 hexagonal yods (see the value of H_{s} for the disdyakis triacontahedron in
Table 2). This is also the number of hexagonal yods in the 32 faces of the
icosahedron and the dodecahedron when they are constructed from tetractyses, as well as the
number of hexagonal yods in the 38 faces of the tetrahedron, octahedron, cube and
icosahedron
7
that the ancient Greeks believed were the shapes of the particles of the
elements Fire, Air, Earth and Water.^{7} The superstring gauge symmetry group E_{8}×E_{8} has 480 nonzero
roots (240 for each E_{8}). This grouptheoretical
parameter of superstrings is therefore embodied in the disdyakis triacontahedron as the
number of hexagonal yods creating its outer form. In other words, the polyhedral form of the
Tree of Life is associated with the 480 gauge bosons corresponding to these roots. The direct product
structure of the E_{8}×E_{8} heterotic
superstring gauge symmetry group arises from the inversion symmetry (Fig. 7) of the polyhedral Tree of Life because 60 faces contain 240 hexagonal
yods and their 60 mirror images contain 240 hexagonal yods as well.
The counterpart of this mirror symmetry in the polygonal, inner form of
the Tree of Life is its two sets of seven regular polygons, one being the mirror image
of the other (Fig. 8).
An nsided, regular polygon has (6n+1) yods, of which 5n are hexagonal.
Each set of seven separate polygons has 48 corners.
The total number of hexagonal yods = Σ 5n =
5×48 =
240. Therefore, the two sets of polygons have
(240+240=480) hexagonal


^{n}

8
yods. A yod denotes a gauge boson and gauge charge associated with those
generators of the Lie algebra of E_{8} that are defined by its 240 roots. As with the
disdyakis triacontahedron, the fact that one set of polygons is the mirror image of the other
explains why the gauge symmetry group of this superstring is the direct product of two
identical groups E_{8}.
As
48 = 7^{2} – 1 = 3 + 5 + 7 + 9 + 11 + 13
and
10 = 1 + 2 + 3 + 4,
480 = 48×10 =
(3+5+7+9+11+13)×(1+2+3+4).
As
24 = 5^{2} – 1
= 3 + 5 + 7 + 9,
240 = 24×10 = (3+5+7+9)×(1+2+3+4).
Therefore,
480 – 240 = 240 = (11+13)×(1+2+3+4)
= (11+13+22+26) + (33+39+44+52)
= 72 +
168,
where 72 is the sum of the four smallest
integers and 168 is the sum of the four largest ntegers. This division is
therefore determined by the Tetrad (4).
The same 72:168 division of 240 was found
in Article 27^{8} to appear in the triakis tetrahedron. In case A, 168
vertices, edges & triangles surround its axis and in case B, 240 such geometrical
elements surround its axis, that is, 72 elements are added by the
construction of each of the 12 faces into three tetractyses. In terms of a tetractys array
of the integer 24:
9
the transformation amounts to considering a whole tetractys array instead of
just a hexagon. In terms of the equivalence between the Tree of Life and tetractys, this means
that all ten Sephiroth are considered instead of just the seven Sephiroth of Construction. In
terms of the root structure of E_{8}, the
72:168 division denotes the 168 roots of E_{8} that are not also roots of its exceptional subgroup E_{6},
which has 72 roots.^{9}
Fig. 9 depicts the UPA (‘ultimate physical atom’), the fundamental unit of
matter described by Annie Besant & C.W. Leadbeater. Identified by the author as the
E_{8}×E_{8}
heterotic superstring constituent of quarks, this object consists of ten closed curves, or
‘whorls.’ In the ‘positive’ type of UPA, the whorls wind clockwise around the axis when
viewed from the top. In the ‘negative’ variety, they wind anticlockwise. Fig. 9 shows a positive type of UPA. The uppermost three whorls, called
‘major whorls,’ are different from the seven others (called ‘minor whorls’) in ways that
need not be discussed here. The 3:7 division between these two types of whorls is the
manifestation of the ontological distinction in the Tree of Life map of all beings between
the Supernal Triad of Kether, Chokmah and Binah and the seven Sephiroth of Construction. In
the context of the tetractys, wherein each yod symbolises a whorl, the major whorls
correspond to the yods at the corners of the tetractys and the seven minor whorls correspond
to the seven yods at the corners and centre of a hexagon. According to the author’s
research, 24 gauge charges of E_{8} are spread along
each curve. This means that the three major whorls carry 72 charges of
E_{8} and the seven minor whorls carry
168 charges. This is the superstring counterpart of the
168 geometrical elements surrounding the axis of the triakis tetrahedron in
case A and the extra 72 elements added to its faces in case B. The 240
geometrical elements that surround the axis of this polyhedron in the latter case represent
degrees of freedom whose counterparts in the E_{8}×E_{8} heterotic superstring
are its 240 E_{8} gauge charges.
Each whorl is a helix with 1680 turns. It is a standing wave with 1680
circularly polarised oscillations. This is the string manifestation of the 1680 vertices, edges
and triangles surrounding the axis of the disdyakis triacontahedron in case A. Figure 8 shows that a whorl makes five revolutions around its axis. It winds
2½ times in an outer spiral, making 840 oscillations, and 2½ times in a narrower spiral,
making another 840 oscillations. Alternatively, there are 840 turns in a halfrevolution of
five whorls. This is the string manifestation of the 840 yods surrounding the axis of the
disdyakis triacontahedron in case D (see Table 2). There are 168 turns in a halfrevolution of a
whorl and 84 turns in a quarterrevolution. This is the string manifestation of the 84 yods
surrounding the axis of the triakis tetrahedron. The tenfold difference between the
populations of yods or geometrical elements in the first and last Catalan solids has its
counterpart in the heterotic superstring as the ten halfrevolutions of each whorl and as
the ten quarterrevolutions in the inner or outer half of a whorl. The disdyakis
triacontahedron embodies the number of oscillations in each whorl of the superstring made
during 2½ revolutions (case D) and five revolutions (case A).
According to Table 2, the 840 yods surrounding the axis of the disdyakis triacontahedron
consist of the 480 hexagonal yods in its 120 faces and 360 other yods (60 vertices and 300
hexagonal yods in its interior). Multiplying the Lambda Tetractys of integers by 10, the sum
of the integers at its corners is 360 and the sum of the integers at the corners of a
hexagon is 480:
10
Magnified by the Pythagorean factor of 10, the Lambda Tetractys is an
arithmetic version of the disdyakis triacontahedron. The central integer 6 corresponds to the
central axis. In the triakis tetrahedron, it denotes the six yods on the central axis on either
side of the centre of the polyhedron. The sum of the seven integers at the centre and corners
of a hexagon is 540. This is the number of yods on the sides of the 300 tetractyses needed to
construct the polyhedron. The number defining the edges of these building blocks is determined
aptly by the very numbers whose positions in a tetractys are occupied by yods that symbolize
the seven Sephiroth of Construction.
The sum of the seven integers forming the Lambda modified by multiplying by
10 is 540. This is the number of yods on the 120 faces of the disdyakis triacontahedron:
The two interpolated integers on the base of the Lambda Tetractys are 120
and 180. They denote, respectively, the number of hexagonal yods on the 60 edges of the 180
internal triangles (60×2=120) and the number of hexagonal yods at the centres of the latter.
Combinations of the nine integers surrounding the centre of the Lambda Tetractys when
multiplied by 10 therefore determine the numbers of various types of yods in the 300
tetractyses making up the disdyakis triacontahedron. As the latter numbers are ten times their
counterparts in the triakis tetrahedron (see Table 2), this demonstrates the equivalence between the Lambda Tetractys and
this Catalan solid:
The sum of the four integers 8, 12, 18 & 27 forming the base of the
Lambda Tetractys is 65, which is the number value of ADONAI, the Godname of
Malkuth. It is the number of vertices (9), edges (26) and triangles (30) in
the triakis tetrahedron (see Table 1), showing how this Godname prescribes the simplest Catalan solid. The
two integers 8 and 18 add up to 26, the number of edges, and the two other
integers 12 and 27 add up to 39, which is the number of vertices and triangles. The
geometrical properties of the triakis tetrahedron are therefore represented by groups of
integers belonging to the Lambda Tetractys. The reason for this is simple: they are the
arithmetic and polyhedral expressions of the archetypal blueprint of the Tree of Life, whose
inner, polygonal form can be shown^{10} to encode its own 91fold replication to generate the Tree of
Life mapping of the 91 subplanes of the seven cosmic planes of consciousness. Article
11^{11} discussed the way in which the number 540 — the sum of the nine
integers surrounding the centre of the modified Lambda Tetractys — expresses the number of
emanations of the 90 trees down to the top of the lowest Tree of Life that map the seven
planes. It is summarised here briefly: 91 overlapping Trees of Life have 550 emanations (see
ref. 11). The lowest of these has 11 emanations. There are (550–11=539) emanations
11
12
above the top of the lowest tree, that is, 540 emanations down to Kether of
the first tree. Just as the seven yods on the axis of the triakis tetrahedron surrounded by 84
yods symbolize the seven subplanes of the physical plane interpenetrated by 84 higher
subplanes, so the 540 yods in the faces of the disdyakis triacontahedron that surround its axis
symbolise the 540 emanations of the 90 trees extending down to the top of the lowest one. The
latter represents the lowest subplane, which, being the seventh, bears a correspondence to the
physical plane (the seventh plane). The numbers 84 and 540 express the same kind of information
— the former as trees and the latter as their emanations. 539 is the sum of the number values
of the Godnames assigned to the seven Sephiroth of Construction:
539 = 31 + 36 + 76 + 129 + 153 + 49 + 65
(see table in Ref. 1). This is not coincidental but, instead, an example of the remarkable,
designing potency of the Godnames.
The sum of the four integers forming the base of the Lambda Tetractys
multiplied by 10 is 650. This is the number of yods in the 144 faces of the 144 Polyhedron when
they are constructed from tetractyses, showing how ADONAI, the Godname of
Malkuth
with number value 65, prescribes the polyhedron that — together with the disdyakis triacontahedron — constitute
what was called in Article 23^{12} the ‘polyhedral Tree of Life.’ The number 650 is the sum of the
Godname numbers of the first four Sephiroth:
650 = 543 + 26 + 50 +
31.
It is also the number of yods other than corners associated with each set of
seven enfolded polygons when each of their 94 sectors is divided into three tetractyses
(Fig. 10). In other words, it is the number of extra yods needed to
construct each half of the inner Tree of Life — its actual form, in keeping with the meaning
of Malkuth.
The sum of the six integers above the base of the Lambda tetractys is 250.
It is the sum of the central integer 60 and the five integers on its sides that add to 190:
13
The significance of this is that the lowest of the 91 overlapping Trees of
Life mapping the spiritual cosmos has 250 yods below its apex when each of its 19 triangles is
divided into three tetractyses (Fig. 11). Below this point, there are 60 yods on its 25 paths and 190 yods
inside its triangles. As already pointed out, the apex is the 540th emanation, where 540 is
the sum of the six integers at the corners of a hexagon that surround the central integer
60. The sum 250 is therefore the number of yods in the lowest Tree of Life below the 540th
emanation. The lowest tree corresponds to Malkuth, so it is apt that the Godname ADONAI of
this Sephirah of Construction has the very number value 65 that specifies
the base of four integers and hence the six remaining integers of the Lambda Tetractys whose
sum is the number of yods in this tree.
The significance of the number 250 for superstring physics is that ten
independent, nonintersecting curves in 26dimensional spacetime need
(10×25+1=251) rectangular spacetime coordinate variables. They are symbolised by the 251 yods
of the lowest Tree of Life. One variable is the time coordinate, symbolized by the apex of the
tree. The 250 spatial coordinates of the ten curves are determined by the sum of the six
integers in the Lambda Tetractys above its base specified by the Godname of Malkuth. This is
truly a Malkuth level of description of the superstring as ten separate, closed curves in
26dimensional spacetime. Each curve is a whorl of the UPA described by
Besant and Leadbeater (see Fig. 9). In terms of the disdyakis triacontahedron, the
Table 3. Number values of the Sephiroth in the four 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

central integer 60 in the Lambda Tetractys corresponds in this particular
context to the 60 vertices surrounding its axis (or, alternatively, to its 60 faces and their
mirror images. They are symbolized in the lowest Tree of Life by the 60 yods below its apex
that lie along its paths and thus create its shape. In this sense, Plato’s Lambda marks out
uniquely the disdyakis triacontahedron, for it is the only Catalan solid that has 60 vertices
and 840 yods surrounding its axis. As Table 1 shows, these 840 yods are associated with 600 vertices, edges and
triangles — a further illustration of how the root number 6 at the centre of the
original Lambda Tetractys determines the nature of the sacred geometry of the disdyakis
triacontahedron. In music, it fixes the frequency, or pitch, of the starting note (tonic) of
the musical scale as unity, so that the ratios of the other integers to 6 are the tone
ratios of the notes. The integer 6 is the starting point for the Pythagorean musical scale,
its diversification in the seven octave species, the Tree of Life counterpart of the Lambda
Tetractys and its polyhedral equivalent — the triakis tetrahedron and the disdyakis
triacontahedron.
14
References
^{1}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. Table 3 above gives their number values calculated by the practice of
gematria, wherein each letter of a word has a number assigned to it, giving a number
value to the word that is the sum of the number values of its letters. Numbers from
this table that are referred to in the article are written in
boldface.
^{2} Phillips, Stephen M. Article 27: “How the Disdyakis Triacontahedron
Embodies the Structural Parameter 1680 of the E _{8}×E _{8} Heterotic
Superstring,” ( WEB, PDF), pp. 2–3.
^{3 }Ibid, pp. 22–23.
^{5 }Phillips, Stephen M. Article 5: “The Superstring As Microcosm of the
Spiritual Macrocosm,” ( WEB, PDF).
^{6} Phillips, Stephen M. Article 11: “Plato’s Lambda — Its Meaning,
Generalisation and Connection to the Tree of Life,” ( WEB, PDF), pp. 2–3.
^{7} Phillips, Stephen M. Article 3: “The Sacred Geometry of the Platonic
Solids,” ( WEB, PDF), p. 20.
^{9} Phillips, Stephen M. Article 22: “The Disdyakis Triacontahedron As the
3dimensional Counterpart of the Inner Tree of Life,” ( WEB, PDF), pp. 3–4.
^{10} Phillips, Stephen M. “The Mathematical Connection between Religion and
Science” (Antony Rowe Publishers, England, 2009).
^{12} Phillips, Stephen M. Article 23: ““The Polyhedral Tree of Life,”
( WEB, PDF), pp. 8, 14.
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