ARTICLE 55
by
Stephen M. Phillips
Flat 4, Oakwood
House, 117119 West Hill Road. Bournemouth. Dorset BH2 5PH. England.
Website: http://smphillips.mysite.com
Abstract
Construction of the faces of
the tetrahedron, octahedron, cube & icosahedron requires 248 points &
straight lines. Believed by the ancient Greeks to be the shapes of the particles
of the four Elements Earth, Water, Air & Fire, these Platonic solids,
therefore, embody E_{8}, the
rank8, exceptional Lie group with dimension 248 at the heart of
superstring theory that describes the symmetry of the unified force between
superstrings — the basic particles of matter. It is implausible that this is
coincidence, because the numbers of points & lines in various subsets of the
set of 248 points & lines are the numbers of roots in the four exceptional
subgroups of E_{8}. 2480
geometrical elements surround the axes of the five Platonic solids when their
faces and interiors are constructed from triangles. This is the number of
spacetime components of the 248 10dimensional vector gauge fields that
transmit superstring forces. This cannot be due to chance because numbers that
quantify the geometrical compositions of various subsets of elements are (apart
from the Pythagorean factor of 10) equal to the numbers of roots of the
exceptional subgroups of E_{8}. On
average, 496 geometrical elements other than vertices surround the axes of the
five Platonic solids, 248 elements making up each half. This is the regular polyhedral counterpart
of E_{8}×E_{8}, the symmetry group with dimension 496 which is
associated with one of the two types of heterotic superstrings. 1680
geometrical elements (including their vertices) surround the axes of the first
four Platonic solids. This number is the number of circular turns in each of the
ten helices making up the basic constituents of matter, as described over a
century ago by C.W. Leadbeater with the aid of a yogic siddhi. Many earlier articles have identified
these particles as the subquark state of the E_{8}×E_{8} heterotic
superstring. That this is not coincidental is indicated by how the
gematria number values of the Hebrew words making up the Kabbalistic name of the
Mundane Chakra of Malkuth refer naturally to the sides, corners & triangles
making up these Platonic solids. It is confirmed by the fact that the same
numbers manifest in other sacred geometries, such as the outer & inner Trees
of Life and the disdyakis triacontahedron. This is because, as sacred geometry,
the Platonic solids embody the same mathematical archetypes as those found in
these geometries. It is also confirmed by the fact that 168 yods on average in
the faces and axes of the five Platonic solids surround their centres when they
are constructed from tetractyses.

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.

(Numbers in this table referred to in the article will be written in
boldface).
2
1. The faces of the first four Platonic solids embody the dimension 248
of E_{8}
The five Platonic solids are known to
mathematicians as the five regular, convex, 3dimensional polyhedra. They are named after
Plato, who propounded in his Timaeus the
Pythagorean doctrine that particles of the four Elements Earth, Water, Air & Fire had the
shapes of the cube, icosahedron, octahedron & tetrahedron.
Figure 1. The five Platonic solids have
50 faces with 180 sectors.
The Pythagoreans associated the
fifth regular polyhedron, the dodecahedron, with the celestial sphere. It came, later, to be
identified with Aether, the fifth Element. The faces of the five Platonic solids are either
equilateral triangles (tetrahedron, octahedron & icosahedron), squares (cube) or
pentagons (dodecahedron). They have 50 vertices, 50 faces & 90 edges
(Fig. 1). The number 50 is the number value of
ELOHIM, the Godname of Binah (see Table 1), showing how this Godname prescribes their
shapes. On average, the five solids have 10 vertices, 10 faces & 18 edges, i.e., 28
vertices & edges, where
28 = 1 + 2 + 4 + 7 +
14.
1, 2, 4, 7 & 14 are
the factors of 28, so that it is the second perfect number [1]. Later, we shall
reveal how the five Platonic solids embody the third perfect number 496.
Table 2. Geometrical
composition of the faces of the five Platonic solids.
Polyhedron

V

E

F

m

C

e

T

Total = C + e + T

Tetrahedron

4

6

4

3

8

18

12

38

Octahedron

6

12

8

3

14

36

24

74

Cube

8

12

6

4

14

36

24

74

Icosahedron

12

30

20

3

32

90

60

182

Subtotal

30

60

38

–

68

180

120

368

Dodecahedron

20

30

12

5

32

90

60

182

Total
=

50

90

50

–

100

270

180

550

(V = number of vertices, E = number of edges, F
= number of faces, m = number of sectors in a face, C = V + F = number of corners of
sectors, e = E + mF = number of sides, T = mF = number of triangles in F faces).
Division of their faces into their sectors generates 180 triangles with
(50+50=100) corners and (90+180=270) sides. Table 2 lists the number of corners, sides and triangles in
the faces of each solid. The faces of the five Platonic solids are composed of 550 points,
lines & triangles, where
55
55 55
550
=
55 55 55
55 55 55
55
and
1
2 3
55 =
4 5
6
7 8 9 10 .
As 55 is the
tenth triangular number, the Pythagorean Decad (10) determines their geometrical
composition.
[1]
Perfect numbers are numbers that are the sum of their
factors. The first three perfect numbers are 6, 28 & 496.
3
This beautiful property reveals the archetypal character of the polyhedral
sacred geometry of the five Platonic solids. ELOHIM with number
value 50 prescribes them because their 50 faces have 500
(=50×10) geometrical elements other than
their 50 vertices.
Table 3 shows the numbers of
polyhedral vertices (C) and nonpolyhedral (V) corners, edges (E) and internal sides (e) of
the triangular sectors of the faces of the first four Platonic solids:
Table 3. Geometrical composition
of the faces of the first four Platonic solids.
Polyhedron

C

V

E

e

Total

Tetrahedron

4

4

6

12

26

Octahedron

6

8

12

24

50

Cube

8

6

12

24

50

Icosahedron

12

20

30

60

122

Total =

30

38

60

120

248

Notice how the Godname YAHWEH with number
value 26 prescribes the tetrahedron (the simplest Platonic solid)
with 26 corners & sides of the 12 triangles in its four faces. Notice
also that the octahedron and cube each has 26 corners & polyhedral
edges and 50 corners & sides, where 50 is the number value of
ELOHIM, the Godname of Binah. The 24 triangles in the faces of each of these two polyhedra
have 36 sides, where 36 is the number value of
ELOHA, the Godname of Geburah, which is the Sephirah next below Binah on the Pillar of
Judgement of the Tree of Life. The 120 triangular sectors of the 38 faces of the first four
Platonic solids have 68 corners and 180 sides, i.e., 248 corners & sides.
This shows how they embody the dimension 248 of
E_{8}, the
rank8, exceptional Lie group. The ancient Greeks believed that the tetrahedron, octahedron,
cube & icosahedron were the shapes of the particles of, respectively, the elements Fire,
Air, Earth & Water. We now see that these Platonic solids,
indeed, do represent the properties of physical matter because their
geometrical composition is quantified by the very number that, according to
E_{8}×E_{8} heterotic superstring theory, is the number of particles
transmitting the unified force between superstrings of ordinary matter.
The dimension of a Lie group is the
number of its socalled “roots,” the definition of which need not detain us (see
here).
This set of roots contains a subset of “simple roots” (the number of which defines the rank
of the group). The exceptional Lie group with the largest rank (8) is
E_{8}. It has eight simple roots
and 240 roots. The four vertices and the four facecentres in the tetrahedron constitute
eight points, so it seems, intuitively speaking, more natural to associate them with the
eight simple roots of E_{8}, rather than the eight
centres of the faces of the octahedron or the eight vertices of the cube. The remaining
240 points and straight lines making up the first four Platonic solids correspond to its
240 roots. The 248 corners & sides comprise 128 corners & edges and 120
internal sides of sectors.
There are (6+12=18) lines
in the tetrahedron and (6+8+8+6+12+20=60) points in the octahedron, cube & icosahedron
(see Table 3). Hence, they form a set of 78 points & lines. (30+60=90) lines are in the
faces of the icosahedron. There are (78+90=168)
points & lines other than the (12+24+12+24=72) lines in the octahedron & cube. 168 is the number value of Cholem Yesodoth, the Mundane Chakra of Malkuth, 78 being the
number of Cholem and 90 being the number of
Yesodoth (Fig. 2).
Figure 2. The gematria number value of Cholem Yesodoth, the Mundane Chakra of Malkuth, is
168.
The number 72 could be the sum of other
combinations of numbers in Table 3. However, it is readily verified that the simplest
and most natural combination is the two pairs of numbers 12 & 24. Article 53 proves that sacred geometries embody the number 240
and that it displays the factorization 10×24. This means that we
should expect the number 72 appearing in the
geometrical composition of the faces of the first four Platonic solids to display the
factorization 3×24 if it refers to the correct combinations. This, indeed, is the case for
the 72 lines in the octahedron and cube, which have 24 edges, each
polyhedron having 24 more sides of its 24 sectors. Notice also that, as the tetrahedron has
8 points in its faces, the number 80 (=8+72
) arises naturally in the set of the first four
Platonic solids. The division:
4
248 = 80 + 168
is characteristic of holistic systems. An
example is the set of 248 yods up to Chesed of the
fifth Tree (the 31st SL) when the triangles in the lowest five Trees are tetractyses, the
lowest Tree having 80 yods (see here).
The fact that the
first four Platonic solids embody the holistic
parameter 248 in their geometry illustrates the Tetrad Principle formulated in
Article 1. It states that the first four members of a class of
mathematical object or, alternatively, the fourth member of that class, always express
parameters of holistic systems.
2. Counterparts of exceptional subgroups of
E_{8}
The sceptic would find it difficult to defend
his suggestion that the number 248 arises in the first four
Platonic solids simply by chance if it were pointed out that the numbers of roots in the
four exceptional subgroups of E_{8}:

G_{2} 
F_{4} 
E_{6} 
E_{7} 
Number of roots: 
12 
48 
72 
126 
also manifest in the
points & lines making up the faces of the first four Platonic solids, as is now shown.
Ignoring the eight points in the tetrahedron (its four vertices & four
facecentres) that correspond to the eight simple roots, Table 4 indicates that the
octahedron, cube & icosahedron have (26+34=60) points and 54 edges, whilst the tetrahedron has 12
internal lines. They are indicated by the numbers in the yellow cells, which sum to 126.
They correspond to the 126 roots of
E_{7}, the largest exceptional subgroup of
E_{8}.
Table 4. The 126:114 division of the 240 points & lines in the first four Platonic
solids.
Polyhedron

C

V

E

e

Total

Tetrahedron

–

–

6

12

18

Octahedron

6

8

12

24

50

Cube

8

6

12

24

50

Icosahedron

12

20

30

60

122

Total
=

26

34

60

120

240

The numbers in the
remaining orange cells add up to 114. As
G_{2}≤F_{4}≤E_{6}≤E_{7}≤E_{8}, the
geometrical counterpart of the roots of an exceptional group must contain all the
geometrical elements corresponding to a smaller, exceptional subgroup. The choice of
numbers adding to 240 is consistent with this requirement because:
E_{7}:
the 240 points & lines contain
a subset of 126 points & lines:
126 = 26 + 34 + 12 + 12 + 30 +
12;
E_{6}:
the subset of 126 points &
lines contains a subset of 72 points & lines:
72 = 26 + 34 + 12;
F_{4}:
the subset
of 72 points & lines contains a subset
of 48 points & lines:
48 = 6 + 8 + 8 + 6 + 20
or
48 = 8 + 12 + 8 + 20;
G_{2}:
the subset
of 48 points & lines contains a subset of
12 points & lines:
5
12 = 6 +
6
or, alternatively, the 12 vertices of the
icosahedron. The ambiguity of choice is, of course, irrelevant to the point demonstrated
here, namely, that the original selection of 240 points & lines contain subsets that
correspond to the roots of the exceptional subgroups of E_{8}.
Suppose that
the 50 faces of the five Platonic solids are divided into their sectors
and that their 50 vertices and 50 facecentres are
joined to their centres (Fig. 3). Sides of sectors are sides of triangles inside the
polyhedra with vertices and their centres at their corners. For consistency with the
division of all F polygonal faces of a Platonic solid with C vertices and E edges into
their nF sectors (n = 3, 4 or 5), these internal triangles must be divided likewise,
i.e., regarded as Type A triangles. Tabulated below are the numbers
of nonvertex corners, sides & triangles in their faces and interiors that
surround their axes, each containing two vertices and two sides of internal
triangles:
Table 5. Formulae for nonvertex corners, sides
& triangles in a Platonic solid.

Corners

Sides

Triangles

Total

Faces:

F

E + nF

nF

E + F + 2nF

Interior:

E + nF

C − 2 + F + 3E + 3nF = 4E +
3nF

3E + 3nF

8E + 7nF

Total:

E + F + nF

5E + 4nF

3E + 4nF

9E + F + 9nF

(Euler's formula for a polyhedron: C − E + F =
2 has been used to simplify expressions). Tabulated below are the numbers of:

nonvertex corners (V), sides (S) & red triangles
(T) in the faces of the five Platonic solids that surround their axes, where V = F,
S = E + nF and T = nF;

corners (c), sides (s) & triangular sectors (t)
in blue interior triangles formed by their edges that surround their axes, where c
= E, s = C − 2 + F + 3E and t = 3E;

corners (c′), sides (s′) & triangular sectors
(t′) in green interior triangles formed by sides of sectors that surround their
axes, where c′ = nF, s′ = 3nF and t′ = 3nF.
Table 6. Geometrical composition of the faces
& interiors of the five Platonic solids surrounding their axes.
Platonic solid

Faces

Interior


V

S

T

Subtotal

c

s

t

Subtotal

c′

s′

t′

Subtotal

Total

Tetrahedron (n = 3)

4

18

12

34

6

24

18

48

12

36

36

84

166

Octahedron
(n = 3)

8

36

24

68

12

48

36

96

24

72

72

168

332

Cube (n =
4)

6

36

24

66

12

48

36

96

24

72

72

168

330

Icosahedron (n = 3)

20

90

60

170

30

120

90

240

60

180

180

420

830

Subtotal
=

38

180

120

338

60

240

180

480

120

360

360

840

1658

Dodecahedron (n = 5)

12

90

60

162

30

120

90

240

60

180

180

420

822

Total
=

50

270

180

500

90

360

270

720

180

540

540

1260

2480

We find that 2480 points, lines &
triangles are needed to construct the five Platonic solids around their axes, given their
50 vertices. This is the number of geometrical
elements that surround their axes, each of which contains three points and two lines.
Amazingly, as the unified superstring force with E_{8} symmetry is transmitted by 248 vector gauge fields, each with 10 spacetime components,
these fields have 2480 vector components. To every such component, there is a corresponding
geometrical element in the five Platonic solids. 248 is the number value
of Raziel, the Archangel of Chokmah. YAHWEH, the Godname of this Sephirah,
prescribes the dimension of E_{8} because it is the
mean of the squares of all the integers after 1 up to 26, which is the number value
of YAHWEH:
248
= (2^{2} +
3^{2} + 4^{2} + ... + 26^{2})/25.
The following evidence indicates that this
result is not due to coincidence. The five exceptional groups are G_{2} (dimension = 12+2), F_{4} (dimension = 48+4), E_{6}
(dimension = 72+6), E_{7} (dimension = 126+7) & E_{8} (dimension = 240+8), where "+n" means that the group has n
simple roots. As G_{2}≤F_{4}≤E_{6}≤E_{7}≤E_{8}, the
geometrical counterpart of an exceptional group must contain all the geometrical elements
corresponding to any smaller, exceptional subgroup of this group. Table 6 shows that
inside the five Platonic solids and surrounding their axes are 1260 geometrical elements other than vertices that belong to
triangles with sides of face sectors (green triangles of the type indicated in Fig. 3).
1260 is the number value of Tarshishim, the Order
6
of Angels assigned to Netzach. They include
(180+540=720) points & lines. Of these, the table indicates that (120+360=480) belong to
the first four Platonic solids, which contain 120 points. Compare the correspondence between
the roots of the five exceptional groups and these sets of elements (in the case of
E_{8}, its eight simple roots are included):
G_{2}: 12 roots ↔
120 internal points in first four Platonic
solids;
F_{4}:
48 roots ↔ 480
internal points & lines in first four Platonic
solids;
E_{6}:
72
roots ↔ 720 internal points & lines in five Platonic
solids;
E_{7}: 126 roots ↔ 1260
internal points, lines & triangles in five Platonic
solids;
E_{8}:
248 roots ↔ 2480 points,
lines & triangles in five Platonic solids.
We see that not only do the five Platonic
solids have 2480 geometrical elements other than vertices surrounding their axes but they
also are composed of sets of elements that are ten times the numbers of roots (excluding
simple roots) in the four exceptional subgroups of E_{8}. This cannot be as well the
result of chance! Rather, the more sensible conclusion to be drawn is that an isomorphism
exists between the geometrical composition of the five Platonic solids and the root
composition of E_{8} and its exceptional subgroups. Given that these famous polyhedra
have been shown in previous articles to constitute a holistic system with features analogous
to other sacred geometries, it should not seem surprising that the number 2480 manifests in
their geometrical composition. This remarkable analogy demonstrates that superstring theory
is part of the holistic pattern embodied in these geometries.
The icosahedron
is the fourth Platonic solid. According to Table 2 & Table 6, it has
(20+60=80)
points & triangles in its faces. They also have 80 points in its faces and in
internal triangles formed by sides of their sectors. Intuitively speaking, the latter seems the
more likely counterpart of the eight simple roots of E_{8} because they are all points,
whereas the former mixes points with triangles. It is readily verified by combining the various
numbers in the table that no other Platonic solid has a combination of
80 geometrical
elements of the same kind. The 60 corners of the internal triangles of the icosahedron
formed by the sides of the sectors in its faces are the counterparts of the six simple
roots of E_{6}, whilst the centres of its 20 faces are the counterparts of the two
extra simple roots of E_{8}. The remaining 2400
geometrical elements consist of the following:
Faces

Interior

Interior


Corners

Sides

Triangles

Subtotal

Corners

Sides

Triangles

Subtotal

Corners

Sides

Triangles

Subtotal

Total

30

270

180

480

90

360

270

720

120

540

540

1200

2400

As pointed out above, the 720 geometrical elements in the
blue interior triangles generated by the edges correspond to
the 72 roots of E_{6}. The (480+1200=1680)
geometrical elements in the faces and in the interior green triangles generated by the
sides of sectors of faces correspond to the remaining 168 roots of
E_{8}.
Previous articles have given many examples of the 72:168 division of the
parameter 240 embodied in sacred geometries. It reappears naturally in the geometrical
composition of the five Platonic solids as the 720 internal geometrical elements
generated by edges that surround the axes of the five Platonic solids and as the
remaining 1680 elements other than the 80 corners in either the
faces of the icosahedron or the internal triangles formed by sides of their
sectors.
Referring to Table 6, surrounding
the axis of a Platonic solid are, on average, (2480/5=496) geometrical elements other
than vertices. 248 geometrical elements other than vertices on average surround its
axis in each half. This is the regular polyhedral basis of the E_{8}×E_{8} heterotic superstring, the counterpart of the direct
product being the fact that each geometrical element in onehalf of the “average” Platonic
solid has a corresponding mirrorimage element in its other half. 320 points other
than vertices surround the axes of the five Platonic solids, the average number of points
being 320/5 = 64, which is the number of Nogah, the Mundane Chakra
of Netzach. Including their centres, the average number of points other than vertices is
65. This is the number value of ADONAI, the Godname of Malkuth. Including
the 40 vertices surrounding their axes, the average number of points surrounding the
axis of a Platonic solid = 65 + 40/5 = 73. This is the
number value of Chokmah, the second Sephirah of the Tree of Life. According to Table 6, the
number of sides surrounding the axes of the five Platonic solids = 270 + 360 + 540 = 1170.
The number of triangles = 180 + 270 + 540 = 990. Therefore,
(1170+990=2160=216×10) sides & triangles surround their axes, where
216 is both the number of Geburah, the fifth Sephirah, and the average
number of sides & triangles in each half of a Platonic solid that surround its
axis.
3. First four Platonic solids embody the
superstring structural parameter 1680
Including their 30 vertices, there are (30−8=22) vertices surrounding the
axes of the first four Platonic solids. The table above shows that 1658 other geometrical
elements surround their axes. The total number of such elements = 1658 + 22 = 1680 =
168×10. This is
another fundamental parameter of superstring physics that string theorists have yet to
discover, being the number of circular turns that C.W. Leadbeater counted in each helical
whorl of the UPA/subquark superstring (Fig. 4). Confirmation that this number does not
appear by chance in the context of the first four Platonic solids is provided by their
composition when their vertices are included:
Corners

Sides

Triangles

Total

22 + 38 + 60 + 120 = 240

180 + 240 + 360 = 780

120 + 180 + 360 = 660

1680

780 (=78×10) sides and (240+660=900=90×10) corners
& triangles surround their axes. The number 168 is the gematria number
value of Cholem Yesodoth
7
the Mundane Chakra of Malkuth (see Table 1). As
indicated in Figure 2, the number value of Cholem is 78 and the number
value of Yesodoth is 90.
We find that the first
four Platonic solids, thought by the ancient Greeks to be the shapes of the particles of the
Elements Earth, Water, Air & Fire, not only embody the superstring structural parameter
1680 but also reproduce in their geometrical makeup the number values of the two Hebrew
words in the Kabbalistic name of the Mundane Chakra of Malkuth! It is improbable in the
extreme that both numbers, as well as their
sum, should appear by chance in the geometry of these Platonic
solids in such a natural way. Instead, we are witnessing here the manifestation of the
mathematical archetypes embodied in the Kabbalistic system of Godnames, Archangelic Names,
Orders of Angels and Mundane Chakras (see Table 1). They determine the properties
of all sacred geometries because the latter are isomorphic, so that they
possess properties that are quantified by the same numbers — namely, the gematria
number values of the Godnames, etc.
Figure 4. The first four Platonic solids
embody superstring structural parameters.
As further confirmation that this amazing property of the first four
Platonic solids does not arise by chance, consider the geometrical composition of each
Platonic solid when its vertices are included:
Table 7. Geometrical composition (including vertices) of the first four
Platonic solids.
Platonic
solid

Faces

Interior 
Interior

Total

Tetrahedron 
36 
48 
84 
168 
Octahedron 
72 
96 
168 
336 
Cube 
72 
96 
168 
336 
Icosahedron 
180 
240 
420 
840 
Total = 
360 
480 
840 
1680 
We find that surrounding its axis:

the
tetrahedron has 168 geometrical
elements ((36+48=84) in faces & interior
generated by edges, 84 internal generated by sides of
sectors);

the
octahedron has 336 geometrical elements
((72+96=168) in faces & interior generated by
edges, 168 internal generated by sides of
sectors);

the cube
has 336 geometrical elements (168 in faces & interior
generated by edges, 168 internal generated by sides of
sectors);

the
icosahedron has 840 geometrical elements (420 in faces & interior generated by
edges, 420 internal generated by sides of sectors);

all four
Platonic solids have 1680 geometrical elements (840 in faces & interior
generated by edges, 840 in interior generated by sides of
sectors).
8
The superstring structural parameter
168 (the number of turns in a halfrevolution of a whorl of the
UPA/subquark superstring) appears explicitly in the tetrahedron, octahedron & cube,
whilst its structural parameter 840 (the number of turns in the 2½ revolutions of the
outer or inner half of a whorl) appears explicitly in the icosahedron. Such
repeated appearances of the number value 168 of
Cholem Yesodoth, as well as the numbers 840 & 1680, cannot be
due to chance! Rather, it demonstrates
the truly amazing fact that the sacred geometry of the first four Platonic solids — both
individually and collectively — embed a number (in fact, several
numbers) characterizing the structure of superstrings that was paranormally obtained
over 100 years ago by the remote viewing of atoms. This is because
both superstrings and these Platonic solids are holistic systems that conform to the
archetypal pattern described in The holistic
pattern.
Figure 5. Correspondence
between the geometrical composition of the first four Platonic solids, the disdyakis
triacontahedron and the outer & inner Trees of Life.
9
When the 180 internal
triangles formed by joining adjacent vertices of the disdyakis triacontahedron to its centre
are divided into their sectors, there are 1680 corners, sides & triangles surrounding
any axis that passes through two diametrically opposite vertices (Fig. 5). They consist of
240 corners, 780 sides & 660 triangles. This is remarkable, for the table above
indicates that the same numbers of corners, sides & triangles surround the axes of the first
four Platonic solids. The reason for this is simple: both holistic systems are isomorphic
representations of the same archetypal
pattern.
The 1680 geometrical
elements surrounding the axes of the first four Platonic solids consist (referring to the
last table) of 240 points, (180+240+120+180=720) sides & triangles either in the
faces or in their interior formed by edges and (360+360=720) internal sides &
triangles formed by sides of sectors. This 240:720:720 division has its parallel in the
1tree, which has 240 yods other than Sephiroth, and the two sets of seven
separate Type B polygons making up the inner
Tree of Life, each set containing 720 yods that surround their centres (Fig.
5).
4. Yod
composition of the Platonic solids
According to Table 5, a regular polyhedron with F faces, E edges & n
sectors in each face has F corners other than vertices, (E+nF) sides & nF triangles in its
faces that surround its axis passing through two opposite vertices. If each triangle becomes
a tetractys, the number of yods other than vertices in its faces that surround its axis = F
+ 2(E+nF) + nF = F
+ 2E + 3nF. Including the (C–2) vertices surrounding its axis, the total number of yods = C
+ F – 2 + 2E + 3nF = 3E + 3nF, where Euler’s formula for a
polyhedron:
C
– E + F = 2
has been used to replace C by E and F. The number of yods surrounding the
axes of the five Platonic solids = 3∑ (E+nF) = 3(90+180) = 810. Their axes
each consist of three yods on either side of its centre. Therefore, (810 + 5×6 = 840) yods
in their faces and axes surround the centres of the five Platonic solids,
where
840 = 29^{2} – 1 = 3 + 5 + 7 + … +
57
is the sum of the first 28 odd integers after 1. This demonstrates how the
second perfect number 28, which we found in Section 1 to be the average number of vertices
& edges in the five Platonic solids, determines how many yods surround their centres.
The number 840 is a superstring structural parameter, being the number of circular turns in
an outer or inner half of a whorl of the UPA/subquark superstring (see Fig. 4). It is
expressed by the Tetrad Principle as
840 = 10×84 = (1+2+3+4)(1^{2} +
3^{2} + 5^{2} +
7^{2}).
Figure 6. When nested, the five
Platonic solids have 841 yods in their faces & axes. This is the number of yods in the
1tree and the two sets of seven separate polygons of the inner Tree of Life.
10
The centre of a Platonic solid on average is surrounded by (840/5 =
168) yods in its
faces & axes. This demonstrates how the superstring structural parameter
168 is embodied in
the five Platonic solids.
Suppose that the five Platonic solids are nested in one another, occupying
the same centre but with their axes pointing in different directions, each axis having three
yods on either side of its centre, and with no vertices coinciding. The number of yods that
surround the centre of a Platonic solid in its faces and on its axis = 3E + 3nF + 6. The
five nested Platonic solids contain 841 yods (Fig. 6). The 1tree with Type A triangles
contains 251 yods and the seven separate Type A polygons contain 295 yods, so that the
number of yods in the 1tree and both sets of seven separate Type A polygons = 251 + 295 +
295 = 841. This demonstrates the Tree of Life nature of the Platonic solids. As two
hexagonal yods lie on each edge, there are (90×2 + 50 = 230) yods on
the 90 edges of the five Platonic solids. There are (5×4 + 1 = 21) yods
other than vertices on their axes when they are nested. Therefore,
(230+21=251) yods line their axes and edges, which leave 590 yods in
their faces, 295 such yods being in their upper halves and 295 yods being in their lower
halves. This is the precise counterpart of the 251 yods in the 1tree and the 295
yods in each set of seven separate polygons. It is convincing evidence that it is not by
chance that the five nested Platonic solids contain the same number of yods as the 1tree
and the two sets of seven separate polygons! The inner form of the 1tree manifests in the
five nested Platonic solids as their faces, whilst its outer form manifests as their edges
and axes — those geometrical components that determine their
forms.
Surrounding their axes
are 40 vertices, 50 centres of faces, 90 edges lined by 180 hexagonal yods and 180 hexagonal
yods at centres of the 180 tetractyses in their faces, i.e., (40+50+180+180=450) yods. There
are also 360 hexagonal yods lining the 180 sides of these tetractyses that meet at
facecentres, whilst lining the axes are 30 yods surrounding centres of Platonic solids. The 840 yods
surrounding the latter are made up of (30+360=390) yods lining either axes or interior sides
and 450 yods either lining edges or at centres of tetractys. On average, (390/5=78) yods are
of the former type and (450/5=90) yods are of the latter type. As Figure 2 indicates, the
average yod population of the axes and faces of the five Platonic solids reproduce the
number value 78 of Cholem and the value 90 of Yesodoth. Once again, this is
evidence that the number 168 does not appear by
chance in this context.
Figure 7. The five Platonic
solids embody the number value 73 of Chokmah
("Wisdom").
Referring again to Table 5, surrounding the axis of a
regular polyhedron are (E+F+nF) corners other than vertices, (5E+4nF) sides & (3E+4nF)
triangles. Therefore, the total number of yods other than vertices surrounding the axis = E
+ F + nF + 2(5E+4nF) + 3E + 4nF = 14E + F + 13nF. For the five Platonic solids,
the number of
such yods = 14×90 + 50 + 13×180 = 3650 = 365×10. On average, a Platonic solid has
(3650/5=730=73×10) yods other than vertices surrounding its axis (Fig. 7).
73 is the
gematria number value
of Chokmah ("Wisdom"), the
second Sephirah of the Tree of Life. Remarkably, it is the number of yods up to Chokmah
in the lowest of any set of overlapping Trees of Life when their triangles are
tetractyses (see here). According to Table 5, the interior of a Platonic
solid has (E+nF)
corners, (4E+3nF) sides & (3E+3nF) triangles surrounding its axis. The number of
internal yods surrounding its axis = E + nF + 2(4E+3nF) + 3E + 3nF = 12E + 10nF. The number
of internal yods in the five Platonic solids = 12×90 + 10×180 = 2880 =
(1+2+3+4)(1^{1}+2^{2}+3^{3}+4^{4}). On average, the axis of a Platonic solid is surrounded by (2880/5=576)
internal yods. This, too, is expressed by the integers 1, 2, 3 & 4 because 576 =
24^{2} = 1^{2}×2^{2}×3^{2}×4^{2}. It is a beautiful
illustration of how these integers symbolized by the tetractys express properties
of sacred geometries.
5. Conclusion
The five Platonic solids are an example of sacred geometry
not because a famous philosopher wrote about them over 2000 years ago or because their
symmetries and shapes are ubiquitous in nature — as though regularly used by a designing God
— but because they are the regular polyhedral realisation of the divine mathematical
archetypes, embodying the same patterns as those
found in the sacred geometries of various religions. Their geometrical and yod compositions
are quantified by the respective dimensions 248 & 496
of E_{8} and E_{8}×E_{8} and by the structural parameters 168, 840 & 1680 of the UPA
described by C.W. Leadbeater over a century ago. This is exemplified
by:

their faces consist of 248 points
& lines, where 248 is the
dimension of E_{8}, the rank8,
exceptional Lie group at the heart
of superstring theory;

they have 2480 geometrical elements other than vertices
surrounding their axes. On average, 496 geometrical
elements other than vertices surround the axis of a Platonic solid, each half of it
comprising 248 elements;
11

1680 geometrical elements surround the axes of the first four
Platonic solids, so that they embody the superstring structural parameter recorded
by C.W. Leadbeater — a number that many previous articles have shown is present in
other sacred geometries;

840 yods in the faces & axes of the five Platonic solids
surround their centres. On average, 168 such yods
surround the centre of a Platonic
solid.
Suggested reading
"The mathematical connection between
religion and science," by Stephen M. Phillips,
Antony Rowe, Publishing. England (2009).
12
