ARTICLE 62
by
Stephen M. Phillips
Flat 4, Oakwood House, 117119 West Hill Road. Bournemouth.
Dorset BH2 5PH. England.
Website: http://smphillips.mysite.com
Abstract
Previous articles by the author have demonstrated that sacred
geometries, such as the five 3‑dimensional, regular polyhedrons, embody the root
structure of the E_{8}×E_{8} heterotic superstring symmetry
and various structural parameters of the UPA. This is the subatomic particle
inside atoms that Theosophists Annie Besant & C.W. Leadbeater claimed to
observe with ESP and which the author has interpreted as the subquark state of
an E_{8}×E_{8} heterotic superstring. This article will
examine the composition of the six convex regular polychorons — the
4dimensional counterparts of the Platonic solids — when they are constructed
from triangles and tetractyses. The 600cell (the polychoron with the most
faces) is found to embody the superstring structural parameter 840, which has
been discussed in many previous articles and shown to characterise various
sacred geometries, being the number is the number of turns in half a helical
whorl of the UPA. It also embodies the superstring structural parameter 8400,
which is the number of turns in an outer or inner half of the UPA. The
possession by the 600cell of these and other structural parameters of holistic
systems argues against such properties occurring by chance. The 600cell emerges
uniquely from the analysis as the polychoron counterpart of the inner form of 10
overlapping Trees of Life, just as previous articles have shown that the
disdyakis triacontahedron is their polyhedral analogue. The Gosset
4_{21} polytope, whose 240 vertices define the 240 root vectors of
the Lie group E_{8}, is a compound of two 600cells. The construction of
the 1200 faces of each 600cell from triangles requires 8400 geometrical
elements. The fact that 16800 geometrical elements are needed to construct the
faces of both 600cells making up the 4_{21} is striking evidence
that the UPA with 16800 turns in its 10 helices is an
E_{8}×E_{8} heterotic superstring. The five
halfrevolutions making up the inner half of the UPA and the five
halfrevolutions of its outer half would then correspond to the five 24cells
making up each 600‑cell. The 120+120 and 10×24 patterns of vertices in this
compound have been previously found in several sacred geometries. It is further
evidence of the archetypal nature of the 4_{21} polytope as the
polytope representation of the forces and 3dimensional structure of the UPA,
the subquark state of the E_{8}×E_{8} heterotic
superstring. A vertex of each of the 10 24cells denotes an
E_{8} gauge charge. Its counterpart in the UPA is a set of 70 turns
of the whorls.

1
Table 1. Gematria number values of the ten Sephiroth in the four Kabbalistic
Worlds.

SEPHIRAH

GODNAME

ARCHANGEL

ORDER OF
ANGELS

MUNDANE
CHAKRA

1

Kether
(Crown)
620 
EHYEH
(I am)
21 
Metatron
(Angel of the
Presence)
314 
Chaioth ha Qadesh
(Holy Living
Creatures)
833

Rashith ha Gilgalim
First Swirlings.
(Primum Mobile)
636 
2

Chokmah
(Wisdom)
73 
YAHWEH, YAH
(The Lord)
26, 15

Raziel
(Herald of the
Deity)
248 
Auphanim
(Wheels)
187 
Masloth
(The Sphere of
the Zodiac)
140 
3

Binah
(Understanding)
67 
ELOHIM
(God in multiplicity)
50

Tzaphkiel
(Contemplation
of God)
311

Aralim
(Thrones)
282

Shabathai
Rest.
(Saturn)
317 

Daath
(Knowledge)
474 




4

Chesed
(Mercy)
72 
EL
(God)
31 
Tzadkiel
(Benevolence
of God)
62 
Chasmalim
(Shining Ones)
428

Tzadekh
Righteousness.
(Jupiter)
194 
5

Geburah
(Severity)
216

ELOHA
(The Almighty)
36

Samael
(Severity of God)
131

Seraphim
(Fiery Serpents)
630

Madim
Vehement Strength.
(Mars)
95 
6

Tiphareth
(Beauty)
1081

YAHWEH ELOHIM
(God the Creator)
76 
Michael
(Like unto God)
101

Malachim
(Kings)
140

Shemesh
The Solar Light.
(Sun)
640 
7

Netzach
(Victory)
148

YAHWEH SABAOTH
(Lord of Hosts)
129

Haniel
(Grace of God)
97 
Tarshishim or
Elohim
1260

Nogah
Glittering Splendour.
(Venus)
64 
8

Hod
(Glory)
15

ELOHIM SABAOTH
(God of Hosts)
153

Raphael
(Divine Physician)
311

Beni Elohim
(Sons of God)
112

Kokab
The Stellar Light.
(Mercury)
48 
9

Yesod
(Foundation)
80

SHADDAI EL CHAI
(Almighty Living
God)
49, 363

Gabriel
(Strong Man of
God)
246

Cherubim
(The Strong)
272

Levanah
The Lunar Flame.
(Moon)
87 
10

Malkuth
(Kingdom)
496

ADONAI MELEKH
(The Lord and King)
65, 155

Sandalphon
(Manifest Messiah)
280 
Ashim
(Souls of Fire)
351

Cholem Yesodoth
The Breaker of the
Foundations.
The Elements.
(Earth)
168 
The Sephiroth exist in the four Worlds of Atziluth,
Beriah, Yetzirah and Assiyah. Corresponding to them are the Godnames,
Archangels, Orders of Angels and Mundane Chakras (their physical
manifestation). This table gives their number values obtained by the ancient
practice of gematria, wherein a number is assigned to each letter of the
alphabet, thereby giving a number value to a word that is the sum of the
numbers of its letters. Numbers from the table are written in
boldface in the article.

2
1. The 6
polychorons
A
polytope is a geometric object with straight sides that exists in any number of dimensions.
In two dimensions a polytope is a polygon and in three dimensions it is a polyhedron. If
a polytope contains all the points on a segment joining any pair of points in it, the
polytope is convex. An npolytope is an ndimensional polytope. An
npolytope is regular if any set consisting of a vertex, an edge
containing it, a 2dimensional face containing the edge, and so on up to
n−1 dimensions, can be mapped to any other such set by a symmetry of the
polytope. The regular polygons form an infinite class of 2polytopes. There are five
regular convex polyhedra, or 3polytopes, known as the Platonic solids. There exist six
types of convex, regular 4polytopes known as the polychorons, but only three types of npolytopes when
n≥5.
The elements of a polytope are
its vertices, edges, faces, cells (polyhedra) and so on. These elements can be represented
by the following Schäfli symbols:
regular pgon:
{p}.
regular polyhedron with q
regular pgon faces meeting at a vertex: {p,q}.
regular 4polytope with r {p,q}
regular polyhedral cells around each edge: {p,q,r}.
and so on. The dual of the
{p,q} polyhedron is the {q,p} polyhedron, i.e., vertices and faces are interchanged. It is
selfdual when p = q. The dual of the 4polytope {p,q,r} has the
Schäfli
symbol {r,q,p}. A 4polytope is selfdual if it is identical to its dual. This means that a
selfdual 4polytope has the Schäfli symbol {p,q,p}. The Schäfli
symbols of the five Platonic solids are:
tetrahedron:
{3,3}.
octahedron:
{3,4}.
cube: {4,3}.
icosahedron:
{3,5}.
dodecahedron:
{5,3}.
If a vertex of a polyhedron is
sliced off, the surface created is bounded by a polygon called
the vertex figure. A regular polyhedron has always a regular polygon as
its vertex figure. A {p,q} regular polyhedron has the regular vertex figure {q}. The
vertex figure of a polychoron is a polyhedron, seen by the arrangement of neighbouring
vertices around a given vertex. For regular polychora with the
Schäfli
symbol {p,q,r}, this vertex figure is a {q,r} regular polyhedron. The vertex figure for
an npolytope is an (n−1)polytope. An edge
figure for a polychoron is a polygon,
seen by the arrangement of faces around an edge. For regular {p,q,r} polychora, this edge
figure will always be a regular polygon {r}. Regular
polychora with Schäfli symbol {p,q,r} have r polyhedral cells of type {p,q} around each
edge, faces of type {p}, edge figures {r} and vertex figures
{q,r}.
Table 2 lists the geometrical
elements making up the six convex, regular polychorons:
Table 2. The geometrical
composition of the regular polychorons.
Polychoron

Schäffli symbol
{p,q,r}

Vertices
V
{q,r}

Edges
E
{r}

Faces
F
{p}

Cells
C
{p,q}

Dual

5cell

{3,3,3}

5
{3,3}

10
{3}

10
{3}

5
{3,3}

Self

8cell

{4,3,3}

16
{3,3}

32
{3}

24
{4}

8
{4,3}

16cell

16cell

{3,3,4}

8
{3,4}

24
{4}

32
{3}

16
{3,3}

8cell

24cell

{3,4,3}

24
{4,3}

96
{3}

96
{3}

24
{3,4}

Self

120cell

{5,3,3}

600
{3,3}

1200
{3}

720
{5}

120
{5,3}

600cell

600cell

{3,3,5}

120
{3,5}

720
{5}

1200
{3}

600
{3,3}

120cell

(The
Schäfli
symbols of the vertex and edge figures are given for each vertex and edge count of a
polychoron).
An ncell is a
polychoron with n polyhedral cells:
3

The 5cell (also called the
4simplex, or pentachoron) with
five vertices, 10 edges, 10 triangular faces & five tetrahedral cells is
the 4dimensional counterpart of the tetrahedron, both being
selfdual;

The 8cell (also called the
4cube, or tesseract) with 16
vertices, 32 edges, 24 square faces & eight cubic cells (four around each
of its 32 edges) is the 4dimensional counterpart of the cube;

The 16cell with eight vertices, 24 edges, 32
triangular faces & 16 tetrahedral cells (four meeting at each of 24 edges) is
the 4dimensional counterpart of the octahedron;

The 24cell, which was first discovered by
mathematician Ludwig Schäfli in
1852, has no Platonic counterpart and —
like the 5cell — is selfdual. It has 24 vertices, 96 cells, 96 triangular faces
& 24 octahedral cells, three being around each of its 96 edges.

The 120cell with 600 vertices, 1200 edges, 720
pentagonal faces & 120 dodecahedral cells is the 4dimensional counterpart of
the dodecahedron with 12 pentagonal faces. It has three dodecahedra around each of
its edges, just as the dodecahedron has three pentagonal faces meeting at each of
its 20 vertices;

The 600cell with 120 vertices, 720 edges, 1200
triangular faces & 600 tetrahedral cells is the 4‑dimensional counterpart of
the icosahedron with 20 triangular faces. It has five tetrahedra meeting at every
one of its edges, just as the icosahedron has three triangles meeting at each of
its 12 vertices.
2. Geometrical composition of the
polychorons
A polychoron has V vertices, E edges, F faces
& C cells, where
V − E + F − C = 0.
As regular mgons (m = 3, 4 or 5), the
lattermost are divided into their m sectors. Joining its vertices to its centre generates E
internal triangles. Treating them as Type A triangles generates (E+1) new corners and (V+3E)
new sides of 3E new triangles.
Faces
Number of corners of mF triangles in F
faces ≡ C = V + F.
Number of sides of mF triangles
≡ S = E + mF.
Number of triangles ≡ T = mF.
Number of corners, sides &
triangles ≡ N = C + S + T = V + E + (2m+1)F.
Interior
Number of corners of 3E triangles
≡ C′ = 1 + E
Number of sides of 3E triangles =
S′ = V +
3E.
Number of triangles = T′ = 3E.
Faces + interior
Number of corners, sides & triangles =
N′ =
C′ +
S′ +
T′ = 1 + V +
7E.
Total number of corners ≡ c = C + C′ = 1 + V + E + F.
Total number of sides ≡ s = S + S′ = V + 4E + mF.
Total number of triangles
≡ t = T + T′ = 3E + mF.
Total number of corners, sides &
triangles ≡ n = N + N′ = 1 + 2V + 8E +
(2m+1)F.
Table 3 lists the geometrical composition of
the 6 polychorons:
Table 3. Geometrical composition of faces and interiors of the 6
polychorons.
Polychoron

Faces

Interior

Total


C

S

T

N

C′

S′

T′

N′

c

s

t

n

5cell

15

40

30

85

1+10

35

30

1+75

1+25=26

75

60

1+160

8cell

40

128

96

264

1+32

112

96

1+240

1+72=73

240

192

1+504

16cell

40

120

96

256

1+24

80

72

1+176

1+64=65

200

168

1+432

24cell

120

384

288

792

1+96

312

288

1+696

1+216=217

696

576

1+1488

120cell

1320

4800

3600

9720

1+1200

4200

3600

1+9000

1+2520=2521

9000

7200

1+18720

600cell

1320

4320

3600

9240

1+720

2280

2160

1+5160

1+2040=2041

6600

5760

1+14400


(“1” denotes the centre of the polychoron).
4
Comments
5cell
1) The number value 15 of YAH (YH), the
shortened Godname of Chokmah, is the number of points needed to construct the faces of the
4d tetrahedron from triangles. The value 5 of H is the number of vertices and the value 10
of Y is the additional number of corners of the 30 triangles in its faces. The number
value 26 of YAHWEH, the full Godname of this Sephirah, is the total number of
corners of the 60 triangles making up the 5cell. The number value 21 of EHYEH, the Godname of
Kether, is the number of points in it other than vertices. The total number of their corners
& sides is (26+75=101). This is the 26th prime number. 100
(=1^{3}+2^{3}+3^{3}+4^{3}) corners & sides surround the centre of the 5cell.
2) The 30 triangles in the 10 faces comprise
(15+40=55)
corners & sides, where


1





2

3


55
=

4

5

6



7

8

9

10

is the 10th triangular number. The faces
comprise 85 geometrical elements, where
85 = 4^{0} +
4^{1} + 4^{2} +
4^{3},
of which 15 are points and 70 are
lines & triangles. Compare these properties with fact that the 2ndorder tetractys
contains 85 yods, of which 15 are corners of the 10
1storder tetractyses and 70 are hexagonal yods (Fig. 1):
The four rows of a 1storder tetractys traditionally symbolise
the sequence:
0simplex→1simplex→2simplex→3simplex.
We now see that the 2ndorder tetractys symbolises the
4simplex, the 15 black corners of the 10
1storder tetractyses symbolizing the 15 points
in 4d space that are the corners of the 30 constituent triangles in its faces, the 10 green hexagonal yods at
the centres of the 10 1storder tetractyses denoting its 10 edges and the remaining 60
hexagonal yods (30
red, 30 blue) denoting the 30 triangular sectors of its 10 faces
and their 30 remaining
Figure 2. The 141 triangular sectors of the 7 enfolded Type B polygons have
85 corners unshared with the outer Tree of Life.
sides. Just as the tetrahedron
(3simplex) is the starting point in the buildup of the five regular polyhedra in
3dimensional space, so the 4simplex as the simplest polychoron commences the buildup of
the six regular polychora in 4‑dimensional space. The Pythagorean representation of holistic
systems — the 2ndorder tetractys — represents the 4simplex. The counterpart of this in the
inner Tree of Life are the 85 corners of the 141 triangles making up the 47 sectors of the
seven enfolded Type B polygons that are intrinsic to them because none are corners of
triangles belonging to the outer Tree of Life (Fig. 2). Notice that no coloured corners are
depicted at the top, centre & bottom of the hexagon because these points are located at
Chokmah, Chesed & Netzach. It might be thought that the seven enfolded polygons with 88
corners of their 141 triangles could, alternatively, embody the number 85 as the 85 corners
outside the root edge that are
5
intrinsic to each set of polygons enfolded in
successive Trees of Life. However, this cannot be the ‘right’ embodiment because, in the
former scheme, the embodiment of the superstring structural
parameter 168 emerges naturally as the
(84+84=168)
corners of the 282 triangles composing the (47+47=94) sectors of the (7+7) enfolded
polygons that are not corners of triangles in the outer Tree of Life, whereas in the latter
scheme, the (7n+7n) polygons enfolded in n Trees of Life have 282n triangles with (170n+2)
corners, but there is no reason for removing just one corner from each set of seven polygons
so as to leave the required set of (168n+2) corners. The property
that the (70+70) polygons enfolded in the 10‑tree should have 2820 triangles with 1680
corners (the representation of the 1680 turns of each helical whorl of the 10dimensional
UPA), leads, therefore, to a unique geometrical interpretation of the number 85 as a
holistic parameter. Its presence in the 4simplex, shaping its faces, indicates that the
Pythagorean archetypal symbol of the tetractys extends beyond 3dimensional space into four
dimensions through its next higherorder version.
3) (85−5=80) geometrical elements are
needed to create the faces of the 5simplex from five equidistant points in 4dimensional
space. This is the number of Yesod, the penultimate Sephirah. Its interior
has 76 geometrical elements, where 76 is the number value of
YAHWEH ELOHIM, the Godname of Tiphareth. 155 geometrical elements
other than vertices surround its centre, with 50 points & lines in its
faces. This shows how ELOHIM, the Godname of Binah with number
value 50, and ADONAI MELEKH, the Godname of Malkuth with number
value 155, prescribes the geometrical composition of the 5simplex. It
includes 65 lines other than edges, where 65 is the number value of
ADONAI.
There are 30 vertices, edges, triangular faces
& tetrahedral cells, where
30 = 1^{2} +
2^{2} + 3^{2} +
4^{2}.
8cell
1) The 96 triangular sectors of its 24 square
faces are formed by 40 points and 128 straight lines, i.e., 168 points & lines. This is how the tesseract embodies the
superstring structural parameter 168. 264
geometrical elements make up its faces. This is the number of yods in the seven enfolded
polygons that constitute each half of the inner Tree of Life (Fig. 3).
Figure 3. The 47 sectors of the 7 enfolded polygons of the inner Tree of Life have 264 yods
when the sectors are tetractyses.
2) 240 geometrical elements inside the 8cell
surround its centre. They comprise 128 corners & triangles
and 112 sides. The 240 sides comprise 128 sides in its faces
and 112 sides in its interior.
3) The 192 triangles in the 8cell
have 73 corners, where 73 is the number value of
Chokmah.
4) There are 72 vertices, edges, square
faces & cubic cells. Including the eight cubic cells, the 8cell
has 80 vertices, edges, faces & cells,
where 80 is the number of Yesod.
16cell
1) Its 32 triangular faces contain 256
geometrical elements, where 256 = 4^{4}, showing how the Tetrad
expresses its geometrical composition. They have 65 corners
of 168 triangles, where 65 is the number of ADONAI,
the Godname of Malkuth, and 168 is the number
of Cholem Yesodoth, the Mundane Chakra of this Sephirah. 432 geometrical elements surround
its centre, demonstrating the role of the integers 1, 2, 3 & 4 symbolised by the
tetractys in expressing properties of regular polyhedra and polychorons.
2) There are 80 vertices, edges,
triangular faces & tetrahedral cells.
24cell
1) The 96 triangular faces consist of 288
triangles with 120 corners and 384 sides, where 120 is the sum of the first 10 odd integers
after 1 and 288 = 1^{1} + 2^{2} +
3^{3} + 4^{4}. It has 288 internal
triangles, so that 576 triangles are present, where 576 = 24^{2} =
1^{2}×2^{2}×3^{2}×4^{2}. Compare these properties with the following facts:

288 yods line the boundaries of the (7+7) separate, Type A
polygons;

288 yods surround the centres of the 7 separate, Type A
polygons;

120 yods line the sides of each set of seven enfolded
polygons;

384 corners, sides &
triangles and (288+288=576) yods surround the centres of the
(7+7) separate
polygons.
6
As the 600cell is the convex hull of five
disjoint 24cells, this display of holistic parameters in the 24cell suggests that the
former polychoron, too, is holistic in nature. Later discussion of the 600cell will confirm
this.
2) 216 points
(=6^{3})
surround the centre of the 24cell with Type A faces and internal triangles. This is the
number of Geburah, the sixth Sephirah.
3) 1488 geometrical elements surround the
centre. They include 24 vertices and 96 edges, i.e., 120 points & lines, so that
(1488−120=1368) new geometrical elements are
needed to construct the 24cell. As the (7+7) enfolded Type B polygons contain 1370 yods (Fig.
4),
Figure 4. The (7+7) enfolded Type B polygons have 1370 yods.
of which two yods (topmost corners of the two
hexagons) are shared with the two similar sets of polygons enfolded in the next higher Tree,
1368 yods are intrinsic to each set of polygons.
4) There are 240 vertices, edges, triangular
faces & octahedral cells.
120cell
1) The 18720 geometrical elements surrounding
its centre include 720 centres of its pentagonal faces and 1200 centres of the internal
triangles formed by its 1200 edges, i.e. 1920 points. The (18720−1920=16800) geometrical elements
other than these centres include (9720−720=9000=90×10×10) elements in faces
and (4200+3600=7800=78×10×10) internal sides & triangles. This is remarkable, because
the number value 168 of Cholem
Yesodoth, the Mundane Chakra of Malkuth, is the sum
of the number 78 of Cholem and the number 90 of Yesodoth. The 120cell is the
only polychoron that is built from a number of geometrical elements that exceeds 16800, so
that it is the only one that could embody this superstring structural parameter, namely, the
number of turns in the 10 helices of the UPA/superstring. It is implausible that
both these
properties could arise by chance. Moreover, the algebraic expression for the population of
geometrical elements other than centres of faces or internal triangles formed by edges
is
2V + 7E + 2mF = (2V+2mF) + 7E
= (1200+7200=8400) + 7×1200 = 8400 + 8400 =
16800
for the 120cell. As 2V = E for this
polychoron, the expression may be rearranged in this case as (E+2mF) + (6E+2V). It enables
ready interpretation, for “(E+2mF)” is the number (8400) of geometrical elements in the
faces other than points and “(6E+2V)” is the number (8400) of vertices & internal
geometrical elements other than corners of triangles. The 120cell therefore embodies the
16800 turns of the 10 whorls of the UPA as the 8400 turns in their 2½ outer revolutions and the 8400 turns in their 2½ inner
revolutions.
2) The general algebraic expression discussed
above may be rewritten:
6E + (2V+E) + 2mF
7
Figure 5. Correspondences between
the geometrical composition of the first four Platonic solids, the disdyakis triacontahedron
and the outer & inner Trees of Life.
For the 120cell, “6E” is the number (7200) of internal lines
& triangles formed by the 1200 edges, “(2V+E)” is the number (2400) of vertices, edges
& internal sides joining vertices to the centre of the polychoron and “2mF” is the number
of lines & triangles generated by dividing the faces into their sectors. Therefore, 16800 =
7200 + 2400 + 7200, where the first and third numbers refer to elements created by turning,
respectively, internal triangles and faces into Type A triangles and where the second term
refers to the number of external & internal points & lines before this transformation.
Apart from a Pythagorean factor of 10, this sum has its counterparts in the outer and inner
forms of the 1tree (the lowest of any set of Trees), the first four Platonic solids and the
disdyakis triacontahedron (Fig. 5).
Outer & inner forms of the
1tree
When its 19 triangles are Type A, the 1tree
contains 240 yods other than their 11 corners. Each set of the seven separate Type B
polygons making up its inner form has 720 yods surrounding their centres.
1st 4 Platonic
solids When the F faces of a
Platonic solid with E edges are divided into their n sectors and their vertices & face
centres are joined to the centre of each polyhedron, creating internal triangles that are
treated as Type A, the number of corners, sides & triangles that surround an axis
passing through two opposite vertices = 10E + 9mF. The tetrahedron (E = 6; mF = 12) has
168 such geometrical elements, both the
octahedron (E = 12; mF = 24) & cube (E = 12; mF = 24) have 336 elements and the
icosahedron (E =
8
30; mF = 60) has 840 elements. The axes of the
first four Platonic solids, which the ancient Greeks believed were the shapes of the
particles of the elements of Fire, Water, Earth & Air, are surrounded by 1680
geometrical elements (840 in the tetrahedron, octahedron & cube and 840 in the
icosahedron). They comprise 240 corners and 720 sides & triangles in each set of their
halves.
Disdyakis triacontahedron
When the
62 vertices of the disdyakis triacontahedron
with 180 edges are joined to its centre and the resulting 180 internal triangles turned into Type A
triangles, 1680 corners, sides & triangles in its faces and interior surround an axis
passing through any two diametrically
opposite vertices. They
Figure 6. 3dimensional projection of the 600cell. (Credit: uploaded to Wikipedia by Jgmoxness).
comprise (60+180=240)
vertices & edges and 1440 corners (720 in each half of the polyhedron).
3) There are 2640 (=264×10) vertices, edges,
pentagonal faces & dodecahedral cells, where 264 is the number of yods in the seven
enfolded Type A polygons (see Figure 3).
4)
600cell
1) Figure 6 shows a threedimensional
projection of the 600cell. Surrounding the centre of the 600cell are 14400
(=120^{2})
geometrical elements, where


3





5 
7 

120 = 
7

9

11



13

15 
17 
19 
Is the sum of the first 10 odd integers after
1. This beautiful property shows how the Decad determines the geometrical composition of the
600cell. As 120 = 5!, it is also determined by the Tetrad because 14400 is the square of
the factorial of the fourth integer after 1.
The number of points surrounding its centre = 2040 = 24×85 =
1×2×3×4(4^{0}+4^{1}+4^{2}+4^{3}), showing that it, too, is expressed by the Tetrad. Of these, 120 (=
2^{3} + 2^{4} +
2^{5} + 2^{6}) are vertices, leaving
1920 points, where 1920 = 64×30 =
2^{6}(2^{1}+2^{2}+2^{3}+2^{4}) = 2^{7} +
2^{8} + 2^{9} +
2^{10}.
The number of triangles = 5760 = 10×24^{2} =
(1+2+3+4)×1^{2}×2^{2}×3^{2}×4^{2}. The number of triangles in its 600 faces = 3600 = 4×900 =
4×30^{2} = 4×(1^{2}+2^{2}+3^{2}+4^{2})^{2}. The 9240 points, lines
& triangles in its faces include (120+720=840) vertices & edges and 8400
new geometrical elements created by its construction from triangles.
This is easily seen from the fact that inside each of the 1200 faces as Type A triangles
are a corner & three sides of three triangles, i.e., seven geometrical elements, so
that (1200×7=8400) new elements are generated when the faces become Type A triangles. 840
= 84×10 = (1^{2}+3^{2}+5^{2}+7^{2})×(1+2+3+4) and 8400 = (1^{2}+3^{2}+5^{2}+7^{2})×(1^{3}
+2^{3}+3^{3}+4 ^{3}). As 120 =5! and 720 = 6!,
840 = 5! + 6!. As each edge is the side of an internal Type A triangle, the 5160 internal
geometrical elements surrounding the centre comprise 5! sides terminating on vertices and
(7×720=7!) geometrical elements generated by the edges, i.e., 5160 = 5! + 7!.
The possession by the 600cell of 840 vertices
& edges and 10 times as many geometrical elements in its faces when they are Type A
polygons is an unambiguous indication of the relevance of this polychoron to superstring
theory. This is because both the number 840 and the number 8400 are superstring structural
parameters, being, respectively, the number of turns in an outer or inner half of,
respectively, a single helical whorl and all 10 whorls of the UPA examined with micropsi by
Besant & Leadbeater. The 120cell embodies the superstring 16800, but only as a subset
of its geometrical elements, namely, the 16800 elements surrounding its centre other than
centres of Type A faces and Type A internal triangles. Is, however, the display of
superstring structural parameters like 168, 840, 1680, etc unique to the
600‑cell? The number of geometrical elements other than vertices and edges needed to divide
each of the F faces of a polychoron into m sectors is (2m+1)F. Table 4 shows these values
for the six polychorons. Also displayed is the number (C+S) of the corners & sides of
the triangles in their faces.
9
Table 4. Number of geometrical elements other than
vertices & edges generating Type A faces.
Polychoron

Vertices
V

Edges
E

Faces
F

V+E

m

(2m+1)F

C+S

5cell

5

10

10

15

3

70

70

8cell

16

32

24

48

4

216

168

16cell

8

24

32

32

3

224

160

24cell

24

96

96

120

3

672

504

120cell

600

1200

720

1800

5

7920

6120

600cell

120

720

1200

840

3

8400

5640

We see that the numbers 840 and
8400 appear only in the 600cell. However, the 24 faces of the 8cube have
168 points & lines. Notice from
Table 3 that it also appears in the 16cell (the dual of the 8cell) as the
number of triangles in its 32 faces and interior. It suggests that both polytopes play a
role in the structure of the UPA/superstring that is yet to be discovered.
Let us next consider two 600cells sharing the
same centre, one a smaller version of the other. Surrounding their centre are
(1320+1320=2640=264×10) points in their 2400 faces. They consist of 240 (=24×10) vertices
and 2400 (=240×10) new points generated by dividing their faces into their sectors. Compare
this with the fact that the seven enfolded polygons of the inner Tree of Life contain 264
yods, of which 24 lie outside the root edge as corners of the first six enfolded polygons (a
subset of the set of seven polygons that is holistic in itself). It is evidence that
the two concentric 600cells conform to the inner form of
the Tree of Life. The significance of this for
superstring physics will be revealed later. Surrounding their shared centre, the two
600cells have 240 vertices and 3840 other points in their faces and interiors, where 240 =
2^{4} + 2^{5} +
2^{6} + 2^{7} and 3840 =
2^{8} + 2^{9} +
2^{10} + 2^{11}.
2) There are 2640 (=264×10) vertices, edges, triangular
faces & tetrahedral cells. The 600cell embodies the Tree of Life parameter 264 (see
Fig. 3). It is yet more evidence for this polychoron being the polytope counterpart of
either half of the inner Tree of Life.
3. Yod composition of the
polychorons
Faces
Number of yods at corners of mF tetractyses ≡ C = V + F.
Number of hexagonal yods ≡ H = 2E + 3mF.
Total number of yods ≡ N = C + H = V + 2E + (3m+1)F.
Number of yods on sides of tetractyses ≡ B = V + 2E + (2m+1)F.
Interior
Number of yods at corners of 3E tetractyses ≡ C′ = 1 + E.
Number of hexagonal yods ≡ H′ = 2V + 9E.
Total number of yods ≡ N′ = C′ +
H′ = 1 + 2V + 10E.
Number of yods on sides of tetractyses ≡ B′ = 2V +
C′ + 6E = 1 + 2V + 7E.
Faces + interior
Number of yods at corners of (mF+3E) tetractyses ≡ c = C + C′ = 1 + V + E +
F.
Number of hexagonal yods ≡ h = H + H′ = 2V + 11E +
3mF.
Total number of yods ≡ n = N + N′ = 1 + 3V + 12E +
(3m+1)F.
Total number of boundary yods ≡ b = B + B′ = 1 + 3V + 9E +
(2m+1)F.
Table 5 lists the yod composition of the 6
polychorons:
Table 5. Yod composition of
the faces and interiors of the 6 polychorons.
Polychoron

Faces

Interior

Total


C

H

B

N

C′

H′

B′

N′

c

h

b

n

5cell

15

110

95

125

1+10

100

1+80

1+110

1+25=26

210

1+175

1+235

8cell

40

352

296

392

1+32

320

1+256

1+352

1+72=73

672

1+552

1+744

16cell

40

336

280

376

1+24

232

1+184

1+256

1+64=65

568

1+464

1+632

24cell

120

1056

888

1176

1+96

912

1+720

1+1008

1+216=217

1968

1+1608

1+2184

120cell

1320

13200

10920

14520

1+1200

12000

1+9600

1+13200

1+2520=2521

25200

1+20520

1+27720

600cell

1320

12240

9960

13560

1+720

6720

1+5280

1+7440

1+2040=2041

18960

1+15240

1+21000

(“1” denotes the centre of the
polychoron).
10
Comments
5cell
1) In remarkable confirmation of the evidence
presented earlier for the holistic nature of the 5cell, the table indicates that it contains
236 yods. This is the number of yods on the boundary of the (7+7) enfolded polygons (Fig. 7).
70 yods inside the triangular faces line
tetractyses. This is the number
of




The 7 enfolded polygons are
composed of 176 corners, sides & triangles.

176 hexagonal yods line the 88
sides of the 47 tetractyses in the 7 enfolded polygons.

Figure 7. 236 yods line the boundaries of the (7+7)
enfolded polygons.

Figure 8. The number 176 is a structural parameter of the inner Tree of
Life.

corners of the (7+7)
enfolded polygons that belong to the set of 236 boundary yods. The yod population of the
5cell is the number of yods shaping the inner
Tree of Life. Moreover, 176 yods line its 60 tetractyses. This is both the number of
geometrical elements in the 47 sectors of the seven enfolded polygons and the number of
hexagonal yods on the 88 sides of the 47 tetractyses (Fig. 8). In the former, the sum “1 +
175” appears as the topmost corner of the hexagon that coincides with the lowest corner of
the hexagon enfolded in the next higher Tree and as the 175 geometrical elements intrinsic
to the seven enfolded polygons. In the latter, the sum appears as one of the two hexagonal
yods on the root edge that can be associated with one set of seven polygons and as the 175
hexagonal yods lining their 47 tetractyses. The possession of Tree of Life parameters by the
5cell is strong evidence of its holistic nature as the simplest polychoron.
2) The 5cell contains 210
(=21×10)
hexagonal yods, where 21 is the number value of EHYEH, the Godname of Kether. It has
15 corners in its 10 faces and 11 corners in its interior,
totalling 26, where 15 is the number of YAH
and 26 is the number of YAHWEH.
8cell
1) It has 73 corners of 192
tetractyses, where 73 is the number value of Chokmah.
2) 296 yods line the 96 tetractyses in its 24
faces. 16 of these are vertices, leaving 280 yods needed to create
these faces, where 280 is the number value of Sandalphon, the Archangel of
Malkuth.
16cell
1) 280 yods line the
tetractyses. This number reappears because the 16cell has eight fewer vertices than the
8cell but eight more faces, whilst it has eight fewer edges, so that it has (8×2=16) fewer
yods in its faces, i.e., the same number of yods as in the 8cube other than its 16
vertices.
2) Its 168 tetractyses have
65 corners.
This is a remarkable conjunction of the numbers of the Godname and Mundane Chakra of
the same Sephirah (Malkuth). 64 (=4^{3}) corners surround its
centre.
3) Its 32 faces have 336 hexagonal yods. This
number has been discussed in many previous articles as a superstring structural parameter,
being the number of turns in one revolution of each helical whorl of a UPA around its axis
of spin.
24cell
1) 216 corners of 576
(=24^{2}=1^{2}×2^{2}×3^{2}×4^{2}) tetractyses surround its centre, where 216 is the number value of
Geburah. 120 of them (24 vertices) are in its 96 triangular faces, where 120 is the sum of
the first 10 odd integers after 1. The 24cell consists of 24 octahedral cells.
120cell
1) It has 1320 (=132×10) yods at corners of
3600 tetractyses in its 720 pentagonal faces. 132 is the 66th even integer, where 66 is
the 65th
integer after 1 and 65 is the number value of ADONAI, the Godname of Malkuth.
600cell
1) The number of yods in its 1200 faces is
13560, that is, the number of yods in 1356 tetractyses. The
11
number 1356 is the number of yods surrounding
the centres of the 14 enfolded Type B polygons, which contain 1370 yods (see Fig. 4). 13560
yods surround the (70+70=140) centres of the (70+70)
polygons enfolded in 10 separate Trees of Life.
This is remarkable evidence for the holistic character of the 600cell, for in each case
this number determines the form of the two equivalent
geometries. The inner Tree of Life shares with its outer form seven yods on the vertical
diameter of each hexagon, one of which is the centre of this polygon, and the centre of each
triangle. Each set of seven polygons shares six yods that are not centres of polygons. The
number of these shared yods in the (7+7)n polygons enfolded in the ntree = 5n + 1 + 5n +1 =
10n + 2. The number of yods in these polygons = 1368n + 2. The number of yods surrounding
their centres = 1368n + 2 – 7n – 7n = 1354n + 2. The number of such yods that are intrinsic
to the polygons in the sense that they are unshared with the ntree = 1354n + 2 – (10n+2) =
1344n. The (70+70) polygons enfolded in the 10tree have 13440 intrinsic yods surrounding
their 140 centres. This is the number of yods other than vertices in the faces
of the 600cell: 13560 – 120 = 13440. Given the 10tree and the 140 centres of the polygons
making up its inner form, 13440 more yods are needed to construct the latter from Type B
triangles. In an analogous way, given the 120 vertices of the 600cell, 13440 more yods are
needed to construct it in 4dimensional space. Discounting the possibility of coincidence as
implausible, this suggests that the 600cell is the 4dimensional polychoron counterpart of
the inner form of 10 overlapping Trees of Life. The archetypal nature of the latter has been
analysed in depth in many previous articles, as has its equivalence to other sacred
geometries, such as the disdyakis dodecahedron. Moreover, the analogy is unique, for the
faces of the 120cell (the only other polychoron whose yod population is an integer multiple
of 10) have 14520 yods — more than the 13682 yods in the (70+70) polygons enfolded in the
10tree.
2) The number of yods lining the 720 edges
connecting its 120 vertices = 120 + 2×720 = 1560 = 156×10, where 156 is the
155th integer
after 1. ADONAI MELEKH, the Godname of Malkuth with number value 155, appropriately prescribes
the form of the 600cell.
3) 21000 (=21×1000) yods surround its
centre. This is a remarkable example of how Godnames prescribe objects with holistic
character (as we shall see shortly), for 21 is the number value of
Kether, the first Sephirah.
4) The 600cell is composed of 600 tetrahedra,
joined five to an edge.
5) 9960 yods line the 3600 tetractyses in its
faces. Of these, 1560 yods line the 720 edges. This
leaves (9960−1560=8400) more
boundary yods needed to construct its 1200 faces
from Type A triangles. This is the same as the number of new geometrical elements needed for
their construction, as found in the discussion of the 600cell in Section 2. Once again, the
superstring structural parameter 8400 appears as the number of yods lining the 3600
tetractyses other than those on its edges.
4. The
4_{21} polytope Firstly, we need to consider a few geometrical
concepts. A skew polygon, is a polygon whose
vertices do not lie in a plane. Skew polygons must have at least four vertices. A
regular skew polygon has equal edge lengths. A
facet of a polyhedron is any polygon whose
corners are vertices of the polyhedron, and need not be a face. More generally, a facet of
an npolytope is an (n–1)polytope formed by some of its vertices. ndimensional
facets
are also called nfaces. For example, a 0dimensional face
is a vertex of a polyhedron, a 1face is a 1dimensional edge and a 2face is a
2dimensional face of a polyhedron. In higherdimensional geometry, the facets of a
npolytope are the (n1)faces of dimension one less than the polytope itself. A
Petrie polygon for a regular polytope of n
dimensions is a skew polygon such that every (n−1) consecutive sides (but no n) belong to one of the
facets. For every regular polytope, there exists an orthogonal projection onto a plane such
that one Petrie polygon becomes a regular polygon
12
with the
remainder of the projection interior to it. The plane in question is the Coxeter plane of the symmetry group of the polygon. Figure 9
shows the Petrie polygons for the five Platonic solids.
Shown in Figure 10 are the Petrie polygons of
the six polychorons.
Figure 10. The Petrie polygons of the six
polychorons.
It is amusing that the pentagram, which the
ancient followers of Pythagoras displayed on their arms as a public sign of their membership
in his brotherhood, should appear in the Petrie polygon of the 5cell, which is the
simplest, 4dimensional, regular polyhedron. They would certainly have approved, given their
emphasis of the central role of the number 4 in the mathematics that describes nature, as
enshrined in the author’s Tetrad Principle formulated in Article 1 and as demonstrated in nearly every one of his previous articles!
The Petrie polygon of this polyhedron is a simple pentagon because the 5cell is dual to
itself.
In 1900, Thorold Gossett (18691962), an
English lawyer and amateur mathematician, discovered [2] an 8‑dimensional, semiregular
polytope that has proved to be relevant to the theory of superstrings (or at least to the
mathematics of one of the two symmetry groups describing superstring forces). The nface
composition of the socalled "4_{21} polytope" is shown below in Table 6:
Table 6. Numbers of nfaces in the 4_{21} polytope.
Vertices
(0faces)

Edges
(1faces)

Faces
(2faces)
{3}

Cells
(3faces)
{3,3}

4faces
{3,3,3}

5faces
{3,3,3,3}

6faces
{3,3,3,3,3}

7faces
{3,3,3,3,3,3} + 4_{11}

Total

240

6720

60480

241920

483840

483840

207360

17280 + 2160 =
19440

1503840

It has 6720 edges, where 6720 = 672×10.
Compare this with the fact that the first four Platonic solids:
tetrahedron, octahedron, cube &
icosahedron
13
Figure 11. The inner form of 10 Trees of Life
has as many yods as that which line the edges of the 4_{21} polytope.
have 672 yods when their faces and interiors are
constructed from tetractyses [1]. It is an amazing example of how the same
parameters quantifying global properties of holistic systems appear in different sacred
geometries. Another example is the following: suppose that we turn each triangular
face into a tetractys. The number of yods lining its edges = 240 + 6720×2 =
13680. As was pointed out in the earlier discussion of the properties of the 120cell (see
comment 3), the (7+7) enfolded Type B polygons of the inner Tree of Life have 1368 intrinsic
yods, so that the (70+70) Type B polygons enfolded in 10 overlapping Trees of Life have
13680 yods intrinsic to them (Fig. 11). We see that the number measuring the shape
of the 4_{21} polytope as the number of yods
aligning its edges is exactly the same as the yod population of the inner form of 10
overlapping Trees! Moreover, the (7+7) enfolded
polygons have 26 yods that are either centres or yods shared with the outer Tree.
Of these, two yods are corners of hexagons that coincide with corners of hexagons
enfolded in the next higher Tree, so that 24 of these yods belong to each set of (7+7)
polygons. It means that 240 yods in the 140 polygons enfolded in 10 overlapping Trees of Life are either
centres or shared. They correspond to the 240 vertices of the
4_{21} polytope. (13680–240=13440) yods intrinsic to the (70+70) polygons surround their centres. They correspond
to the 13440 hexagonal yods lining the 6720 edges of the 4_{21}. Such correspondences make the conclusion seem inescapable that the
4_{21} polytope central to superstring theory has archetypal character
because it is the polytope analogue of the inner form of 10 overlapping Trees of Life,
each representing a Sephirah. Figure 12 shows its Petrie projection onto the Coxeter
plane. Its Petrie polygon is a 30gon.
Figure 12. Petrie projection
of the 4_{21} polytope. (Place cursor over image to enlarge it).
As found earlier, the 1200
faces of the 600cell contains 9240 points, lines & triangles when constructed from Type
A triangles. They comprise the original 840 vertices & edges and 8400 new points, lines & triangles generated from the seven
points, lines &
triangles inside each face.
14
The 2400 faces
of two such polytopes comprise (840+840=1680) vertices & edges and
(8400+8400=16800) new points, lines & triangles (4800 points or lines and 3600
triangles, or 4800 points or triangles and 3600 sides). What this means is that 8400
geometrical elements are needed to construct the faces of each 600cell from simple
triangles in a way that is consist with the construction of other polytopes from triangles,
a requirement which demands that all polygonal 2faces be divided into their sectors. They
comprise 1×1200 points, 3×1200 lines and 3×1200 triangles, that is, 7×1200 geometrical
elements. Many previous articles have shown that this inner form embodies the numbers 840
and 1680, and now we encounter these very numbers in the numbers of vertices & edges
making up the two concentric 600cells whose 240 vertices are root vectors of E_{8}.
As these numbers are the numbers of circular turns in, respectively, an outer or inner half
of a helical whorl and a whole whorl, the inevitable conclusion to be drawn is that the UPA
is, indeed, an E_{8}×E_{8} heterotic superstring the symmetry of whose
forces is described by E_{8}. In other words, the 4_{21} polytope
embodies not only the 240 roots of E_{8} but also the number of circular turns in
the 10 helical whorls along which the 240 E_{8} gauge charges are spread. Each
of the 16800 circular turns would correspond to one of the 16800 geometrical elements within
the faces of the two 600cells generated by their
construction from triangles. One of the 600cells
with 120 vertices and 840 vertices &
edges would correspond to the outer half of the UPA as its 10 whorls make 25 revolutions,
the 2½ revolutions in each helical whorl comprising 840
turns, and the other 600cell with 120 vertices would correspond to its inner half as its 10
whorls make another 25 revolutions. The appearance of the numbers 840 and 1680 in the two
600cells making up the 4_{21} polytope is sufficient
to establish the identity of the UPA as an E_{8}×E_{8} heterotic superstring.
The appearance of the numbers 8400 and 16800 in their construction from Type A triangles is,
so, to speak, the “icing on the cake.” They refer to both the geometrical composition and
boundary yods of the two 600cells, for — as we found earlier — 8400 yods lining tetractyses
in addition to those on edges are needed to construct the faces of each 600cell, so that
(8400+8400=16800) extra boundary yods are needed to create the faces of both 600cells
making up the 4_{21} polytope.
It is known to mathematicians that the 120
vertices of a 600cell can be portioned into those of five


Figure 15. (120+120=240) points,
lines & triangles surround the centres of the two separate Type B
dodecagons.


Figure 14. The (10+10) Type A
dodecagons enfolded in 10 Trees of
Life have 240 sectors with 240
corners. 120 corners are associated
with each set of 10 dodecagons.

Figure
16. (120+120=240) yods line the sides of the (7+7) polygons enfolded in
the separate halves of the inner Tree of Life. 
15
disjoint 24cells, each with
24 vertices (Fig. 13). This compares with the fact that the dodecahedron with 20
vertices is formed from five distinct tetrahedra. The two 600cells comprise 10 disjoint
24cells. This 10fold character was shown in Article 53 to be a natural property of sacred geometries,
indicating that it is the geometrical basis of either the 10 whorls of the UPA or its 10
halfrevolutions. A similar property now emerges in the geometry of the
4_{21} polytope whose 4dimensional projection is the compound of
two 600cells. As we saw earlier, it
has its striking counterpart in the inner form of 10 overlapping Trees of Life
as the 10 sets of 24 yods that are either centres of the (70+70)
enfolded polygons or shared with these Trees. This 10fold
pattern in the E_{8} lattice
of 240 roots suggests that the whorl “carries” 24 E_{8} gauge charges corresponding to 24 roots that are represented by the 24
vertices of a 24cell. The UPA would have 10 whorls because the two 600cells consist of 10
disjoint 24cells. Alternatively, a 24cell could represent a halfrevolution of
all 10 whorls, the two 600cells determining the outer and inner halves of the UPA,
each of which comprises 2½ revolutions of 10 whorls around the axis of the UPA, the five
halfrevolutions being represented by the five 24cells.

Figure 17. The 120:120 pattern in the faces of the five Platonic
solids. 
That the physics of matter
might be determined by the (120+120=240) vertices of two concentric 600cells that are the
projection of the 240 root vectors of the lattice of E_{8} is indicated by the two dodecagons in the inner form of the
Tree of Life. When they are Type A, their 24 sectors have 24 corners, so that the (10+10)
dodecagons enfolded in 10 Trees of Life have 240 sectors with 240 corners (Fig. 14). 120
corners can be associated with either set. Moreover, the centre of a Type B dodecagon is
surround by 120 points, lines & triangles, 10 geometrical per sector, so that the two
dodecagons belonging to the inner Tree of Life comprise (120+120=240) geometrical elements
arranged in 10 sets of 24 elements (Fig. 15). In fact, 120 yods line the sides of each set
of seven enfolded polygons (Fig. 16), whilst the first four Platonic solids have 480
hexagonal yods in their faces, the tetrahedron, octahedron & cube having 240 hexagonal
yods and the icosahedron having 240 hexagonal yods (Fig. 17). Both the icosahedron and the
dodecahedron have 120 hexagonal yods in the faces of each half of the polyhedron, as do the
first three solids. It is easily verified that each set of 120 hexagonal yods comprise 10
sets of 12, so that the 240 hexagonal yods in the faces of the first three Platonic solids
comprise 10 sets of (12+12=24) hexagonal yods, as do the last two solids (for more
examples, see Article 53).
Many previous articles
have shown that the Catalan solid called the “disdyakis triacontahedron” is the polyhedral
counterpart of 10 overlapping Trees of Life. It has 62 vertices (30 A vertices, 12 B vertices & 20 C vertices) and 180 edges that are of three types.
Sixty edges join A & B vertices, 60 edges join B & C vertices and 60 edges join C
& A vertices. The B vertices are vertices of an icosahedron and the C vertices form a
dodecahedron. Surrounding an axis that passes through two diametrically opposite
B vertices are 20 C vertices, 30 A vertices &
10 B vertices, i.e., 10 sets of six vertices (3A, B &
2C).
Figure 18. Surrounding an axis of the
disdyakis triacontahedron are 10 sets of 24 vertices & edges.
Surrounding the axis are 10 sets of six AB edges, 10
sets of six BC edges & 10 sets of six CA edges.
16
Therefore, the axis is surrounded by 240 vertices & edges
comprising 10 sets of 24 vertices & edges, each set comprising six vertices (3A, B &
2C) and 18 edges (6AB, 6BC & 6CA) (Fig. 18). The disdyakis triacontahedron therefore
manifests the same 10×24 pattern as the Platonic solids, the inner Tree of Life and the
4_{21} polytope. In
fact, the pair of separate dodecagons not only generates
the 10×24 and
Figure 20. The connection between the
UPA/superstring, the 4_{21} polytope and 4dimensional polytopes.
17
(120+120) patterns characteristic of sacred geometries
through their (12+12) sectors but also the 13 Archimedean
solids and their duals (the Catalan solids) can be assigned to the 26 corners of these
sectors, the disdyakis triacontahedron and its dual (the truncated icosidodecahedron)
being assigned to the centres of the two dodecagons (Fig.
19).
The
compound of two 600cells that is the 4d projection of a 4_{21} polytope is the 4dimensional manifestation of this archetypal
pattern exhibited by sacred geometries. Constructed from Type A triangles, it contains 4080 points,
where
4080 =
(2^{4} +
2^{5} +
2^{6} +
2^{7}) +
(2^{8} +
2^{9} +
2^{10} +
2^{11}),
240 =
2^{4} +
2^{5} +
2^{6} +
2^{7}
is the number of vertices of
the two 600cells and
3840 =
2^{8} +
2^{9} +
2^{10} +
2^{11}
is the number of other corners
of their triangles. The two 600cells have 13200 sides of 11520 triangles, that is, 11760
sides other than their 1440 edges, where 11760 =
1680×7. This is the
number of turns in the seven helical minor whorls of the UPA. Each cell has 5760 triangles with 6600 sides, where
6600 = 55×120. The number 55 is the sum of the first 10 integers and 120 is the sum of
the first 10 odd integers after 1, showing how the Decad determines the number of
straight lines constructing the 600cell.
This pattern manifests in the subatomic world
in the E_{8}×E_{8} heterotic superstring as the 120 E_{8} gauge charges that are spread along the inner and the
outer halves of the 10 whorls of the UPA, 12 charges per halfwhorl (Fig. 20). The
composition of each 600cell representing the inner or outer half of the UPA in terms of
five 24cells determines the five halfrevolutions of each half. As 16800 = 240×70, it
means that each E_{8} gauge charge manifests as 70 turns of a helical whorl. This is
both the number of yods in the Tree of Life created from tetractyses and the number of
corners of the (7+7) enfolded polygons of its inner form. As might be expected, each
charge is a whole in itself and so must be represented by a Tree of Life whose 16
tetractyses are composed of 70 yods. The boundary of the 24cell is composed of 24
octahedral cells with six meeting at each vertex, and three at each edge. Together they
have 96 triangular faces, 96 edges, and 24 vertices. The vertexfirst parallel projection
of the 24cell into 3dimensional space has a rhombic dodecahedral envelope. Twelve of
the 24 octahedral cells project in pairs onto six square dipyramids that meet at the
centre of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12
rhombic faces of the rhombic dodecahedron.
The superstring structural
parameter 1680 and its connection to the 240 roots/gauge charges of
E_{8} manifests in another context. The sphere in 3dimensional
space R^{3} has a 2dimensional surface S^{2}. In (n+1)dimensional space R^{n+1}, the nsphere S^{n} with radius r is the set of points {x_{1},x_{2},…x_{n+1}}
equidistant from a point {c_{1},c_{2},…c_{n+1}}
such that

n+1 
r^{2} =

∑ (x_{i} −
c_{i})^{2} 

^{i=1} 
A question that has occupied
mathematicians for many years is: how many nspheres can be packed in
R^{n+1} so as to touch a central nsphere? This is called the “kissing number
problem,” the number K(n) being called the “kissing number.” In 1dimensional space, the
0sphere is a point and the answer is 2 because a point at the middle of a finite straight
line is equidistant from the two end points of this line. In 2dimensional space, the
1sphere is a circle and six similar circles can be arranged with their centres at the
corners of a hexagon so as to touch a circle of the same radius at its centre. In
3dimensional space, K(3) = 12, although it took until 1953 to prove this rigorously. In
4dimensional space, K(4) = 24 (first proved in 2003 by Oleg Musin [3]). The densest lattice
packing of spheres in this space is for 24 3spheres centred on the vertices of a 24cell.
It explains why the 600cell with 120 vertices can be regarded as five disjoint 24cells.
The problem is still unsolved for general n>4, although various upper and lower limits
for K(n) have been derived for certain value of n. However, the kissing number problem is
exactly solvable for spheres in eight and 24 dimensions. It was first proved in 1979 that
K(8) = 240. Then a Ukrainian mathematician, Maryna Viazovska, proved in March,
2016 that the packing of 240 spheres in the 8dimensional E_{8} lattice space
is the densest possible [4]. Amazingly, the number of 7spheres that can touch a
central sphere in 8dimensional space is the same as the number of vectors of the
8dimensional root lattice space of E_{8}! Not
only that, their centres coincide with its 240 lattice points! A point in 8dimensional
space has the spherical coordinates (r, φ_{1}, φ_{2}, φ_{3}, φ_{4}, φ_{5}, φ_{6}, φ_{7}). The
positions of points on 240 7spheres require (240×7=1680) angular coordinates to specify
them. This is the number of 1storder spirillae in a whorl of the UPA that Leadbeater
painstakingly counted 135 times [5]. It is another way in which this superstring structural
parameter appears naturally in the group mathematics of
E_{8}×E_{8} heterotic superstrings.
Of course, the author does not
claim that this property is the complete explanation of the string winding number 1680. This because the number
refers to a property of spheres packed in a special,
18
8dimensional lattice, whereas in the context of the UPA it
refers to the number of times a closed curve winds around the axis of a 2torus. The crucial
point is that it is highly unlikely that it could be just by chance that this number
with an alleged paranormal provenance appears naturally twice in a narrow
context connected to the group mathematics of superstrings, namely, the
4_{21} polytope, whose vertices determine the 240 roots
of E_{8} whose 4d
projection is the compound of two 600cells with 1680
vertices & edges, and the specification of the 240 root vectors of
E_{8} by 1680 angular coordinates. The argument against coincidence has even
more force visàvis
the UPA's structural parameters 840, 8400 & 16800, which we have seen manifest
uniquely in this polytope. The conclusion is inescapable that the UPA is a state of
the E_{8}×E_{8} heterotic superstring and that both its spacetime structure and the
symmetry of its forces are laid out in the 4dimensional sacred geometries discussed in
this article, as well as in the 3dimensional sacred geometries discussed in previous
articles.
5. The Tetrad determines the superstring structural parameter
16800
According to Table 3, 14400
geometrical elements surround the centre of a 600cell constructed from Type A triangles.
Two 600cells contain 28800 geometrical elements surrounding their centres, where 28800 =
1!×2!×3!×4!(1^{3}+2^{3}+3^{3}+4^{3}).
According to Table 5, the centre of a 600cell is surrounded by 21000 yods, so that the
centres of two 600cells are surrounded by 42000 yods, where 42000 = 1680×25. Noting
that 168=
13^{2}− 1 = 3
+ 5 + 7 +... + 25, i.e., it is the sum of the first 12 odd integers after 1, the superstring
structural parameter 1680 is the sum of the 12 integers 30, 50, 70, ... 250 that can be
assigned to the 12 yods on the boundary of a Type A square, which has 25 yods, whilst 4200
is the sum of the same integer 1680 assigned to all its yods:
If the sceptic thinks that it
is just coincidence that the number of vertices & edges in two 600cells should be the
structural parameter of the basic particle that C.W. Leadbeater claimed to remoteview,
despite their being connected to the 4_{21} polytope representation of the roots of the symmetry group
E_{8} in
E_{8}×E_{8} heterotic superstring theory, then — to remain consistent — he has to
accept that it is also just a matter of chance that the square should simultaneously
generate this number and the yod population of the two 600cells. How many such miraculous
coincidences is he willing to accept before he grudgingly admits that a profound,
mathematical design embodied in the UPA is being revealed here? Any argument based upon
chance that he uses to justify his scepticism is untenable, given the degree of correlation
established in this article between the features of the UPA, the geometry of the two
600cells as the 4dimensional projection of the 4_{21} polytope and the fivefold
distribution of their vertices in 24cells.
The Pythagorean integers 1, 2,
3 & 4 determine the 2400 faces of the two 600cells because

1^{1}


1^{2}


1^{3}


1^{4}


4^{3}


4^{4}


2^{1} 2^{1}


2^{2} 2^{2}


2^{3} 2^{3}


2^{4} 2^{4}


3^{3} 3^{3}


3^{4} 3^{4}

2400 =

3^{1} 3^{1}
3^{1}

+

3^{2} 3^{2}
3^{2}

+

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

+

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

+

2^{3} 2^{3}
2^{3}

+

2^{4} 2^{4}
2^{4}


4^{1} 4^{1}
4^{1} 4^{1}


4^{2} 4^{2}
4^{2} 4^{2}


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


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


1^{3} 1^{3}
1^{3} 1^{3}


1^{4} 1^{4}
1^{4} 1^{4}

They determine the number
1680 because
They determine the number
16800 because
19
6. The holistic 72:168 division in the two
600cells making up the 4_{21}
polytope As additional evidence of the holistic
nature of the compound of two 600cells, it will be next proved that the yod composition of
their faces displays the 72:168 division characteristic
of holistic systems.
Table 5 indicates that the 1200 faces of the
600cell contains 13560 yods. Its 720 edges comprise (120 + 2×720 = 1560) yods.
(13560−1560=12000) new yods are needed to
turn the faces into Type A triangles. Of these, 3600 hexagonal yods are at the centres of
the (3×1200=3600) tetractyses, leaving 8400 new yods on their sides. Therefore, 12000 =
3600 + 8400. (2×12000=24000) new yods are needed to construct the 2400 faces of the two
600cells. They contain (2×3600=7200) hexagonal yods at the centres of the 7200
tetractyses and (2×8400=16800) new yods on their sides. Hence:
24000 = 7200 +
16800.
Ignoring the Pythagorean double factor of 100
originating from the fact that the two 600cells have 2400 (=24×100) faces, we see that the
compound of two 600cells exhibits the 72:168 division that is found
in all other sacred geometries. Moreover, the 36:84 division differentiating
between central hexagonal yods and boundary yods is exactly the same as the division of the
120 yods on the boundaries of each separate set of seven enfolded polygons
into 36 corners and 84 hexagonal yods (Fig. 21).
Article 62 discusses how other sacred geometries manifest this pattern in the
holistic parameter 240. The fact that it is revealed in the very compound of two
600cells that generates the 4_{21} polytope when they are
constructed from tetractyses is truly remarkable evidence of the archetypal nature of this
polytope as sacred geometry. In terms of the 240 roots of E_{8}, this division is between
the 72 roots of its exceptional subgroup E_{6} and
its 168 remaining roots. Here is the sacred geometrical counterpart of this
property.

Figure 21. The 120
yods lining the sides of each set of 7 enfolded polygons consist of
36 brown yods at
their corners and 84 turquoise hexagonal yods.

7. The holistic 192:192 division in the two
600cells
We found in comment (2) on
the 600cell that the centre of a 600cell is surrounded by 1920 corners of sectors in its 1200 faces and 720 interior triangles
that are not vertices. Therefore, (192+192=384)×10 points other than vertices surround the
common centre of two 600cells. As


1^{2}





2^{2}

3^{2}


385 =

4^{2}

5^{2}

6^{2}



7^{2}

8^{2}

9^{2}

10^{2},

384 is the sum of the squares of 210, so that
1920 = 5×(2^{2} + 3^{2} + ... +
10^{2}).
This means that the sum of the 90 squares of 210 arranged in a 10fold array surrounding
the central square of 1 (Fig. 22) is 3840 — the number of corners other than vertices of the
7200 sectors of the 2400 faces of two 600cells. This demonstrates the remarkable way in
which the Decad determines the number of points other than vertices that surround the common
centre of the two 600cells.
Alternatively, if the 120
vertices of the 600cell are joined to its centre, creating the sides of triangles, and the
internal triangles and the faces treated as simple triangles, then there are (1200+720=1920)
triangles and 1920 lines & triangles in its faces. It has (120+720=840) internal sides
& triangles, as well
20

Figure 22. Each pentagram array of
the squares 2^{2}10^{2} sums to 1920. This is the number of
corners of triangles in each 600cell that are not its vertices or,
alternatively, the number of its external and internal triangles when both
types are simple triangles.

as (720+120=840) external & internal lines
and (120+720=840) points & lines in its faces. Therefore, there are three possible sets
of 840 geometrical elements:
Assuming that their sets of 840 elements are
compounded in the same way, the two 600cells sharing the same centre are composed of four
possible sets of 1680 geometrical elements:

(840+840) points & lines in their 2400
faces;

(840+840) external & internal lines;

(840+840) internal lines &
triangles;

(840+840) points & triangles.
There can, therefore, be no doubt whatsoever
that the pair of 600cells embodies the paranormally obtained superstring
structural parameter 1680. It does not need its construction from Type A triangles to
demonstrate that. But the latter is required to show that the
compound embodies the numbers 8400 and 16800 as well. Notice that even
a single 600cell can embody the number 1680, for its faces are composed of
840 points & lines, whilst its interior has 840 lines & triangles. Alternatively, it
has 840 lines and 840 points & internal triangles. The question now arises: which of all
these possibilities is the correct geometrical analogue of the 1680 turns in a helical whorl
of the UPA? To answer this question, it is necessary to point out that it is not sufficient
that a parameter of holistic systems be embodied in a geometry for the latter to count as
sacred. True parameters of holistic systems are not isolated; they are also interconnected,
forming patterns that are always the same from one holistic system to another. Study of
these systems has shown that one such pattern is the sum:
192 = 24 +
168
21
(or a pattern with 10 times these numbers). A
holistic parameter may be present in a sacred geometry in more than one way. Its "correct"
way will be that which is consistent with the holistic patterns that the geometry displays.
In the case of the 600cell constructed from simple triangles, there are (720+1200=1920)
lines & triangles in its faces and (720+1200=1920) triangles in its faces and interior.
In neither case, however, does this number include the number 240. This appears only as the
sum of the 120 vertices and the 120 internal lines, but in each case the number 120 is also
part of the number 840 instead of being separate from it, which it needs to be in order to
create the sum: 1920 = 240 + 1680. This holistic pattern only manifests in
the pair of 600cells, which contains 240 points (vertices) and 1680 lines
(edges & internal lines), i.e., 1920 points & lines. As we have found that sacred
geometries embody the numbers 384 and 192 (or 10 times these numbers) as:
384 = 192 + 192,
it indicates that the truly complete, holistic system exhibiting
the division: 3840 = 1920 + 1920 is one in which the compound is replicated, i.e., there should
be two 4_{21} polytopes, each with
240 vertices determining the 240 roots of E_{8}. But this is, precisely,
the E_{8}×E_{8} heterotic superstring theory, which this website demonstrates is
encoded in the inner Tree of Life, the five Platonic solids and the disdyakis
triacontahedron! Hence, this particular superstring theory is the simple consequence of
holistic systems having two suitably defined "halves", each embodying the number 192 (or
1920). The correct way of seeing how the superstring structural parameter 1680 is
embodied in either the 600cell or two 600cells composed of simple triangles is the one
that is consistent with both the sums: 1920 = 240
+ 1680 and 1680 = 840 + 840. The only one is a compound of two 600cells
with 240 points, 1680 lines and 1920 points & lines, i.e., each 600cell with 840
sides of external and internal simple triangles represents half a helical whorl with
840 turns. This is consistent with the conclusion reached in Article 62 that, in terms of its vertices, the two 600cells composed of Type A
triangles represent the outer and inner halves of a UPA/E_{8}×E_{8} heterotic superstring,
the 120 vertices in each denoting 120 E_{8} roots/gauge charges that
are spread along either half, whilst each of the five halfrevolutions of the 10 whorls in
each half is represented by one of the five disjoint 24cells that make up a 600cell. The
division: 840 = 120 +720 observed for the sides of exterior and interior triangles in each
600cell is the exact counterpart of what is found in the inner & outer Trees of Life,
where 120 yods making up half of the 1tree constructed from Type A triangles can be
associated with the 720 yods that surround the centres of each set of seven separate Type B
polygons (see Fig. 5). That the superstring divides into an inner and an outer half due to
this 120:120 division of vertices/E_{8} gauge charges is
confirmed by the fact that, when the faces and interior triangles of each 600cell are Type
A triangles instead of simple triangles, (7×1200=8400) geometrical elements are added in the
1200 faces of each 600cell. In other words, these elements are the counterpart of the 8400
helical turns in the outer or inner halves of the UPA.
8. The Decad determines 120, 1200, 720 &
840
The ancient Pythagoreans
regarded 2 as the first true integer because they taught that the number 1 (the Monad) was
not a number but the source of all numbers,
including integers. They would have treated the number 11, the tenth integer after 1, as the
tenth true integer and 3 as the first true odd integer. As
11^{2} − 1 = 121 − 1 = 120 = 3 + 5 + 7 +... + 21,
the Decad determines the
120 vertices of the 600cell arithmetically as the sum of the first 10 odd integers. It also
determines this number in a geometrical way because the centre of a Type B dodecagon is
surrounded by 120 points, lines & triangles and the dodecagon is the tenth type of regular polygon. The Decad determines its 1200
faces

Figure 23. 720 yods surround the
centre of a decagon whose sectors are 2ndorder tetractyses.

22
because
1200 = 10×120 = 10(3+5+7+...
+21) = 30 +
50 + 70 +...+210.
The tenth prime number is 29. As
29^{2} − 1 = 840 = 3 + 5 + 7 +... + 57,
the number 840 is the sum of
the first 28 odd integers after 1. Therefore,
840 − 120 = 720 = 23 + 25 + 27
+... + 57,
i.e., 720 is the sum of 18 odd
integers. The Decad determines arithmetically both the number of vertices in the 600cell
and its 840 vertices & edges. It determines the 720 edges as well because there are 720
yods surrounding the centre of a decagon whose sectors are 2ndorder tetractyses (Fig.
23).
9. The holistic 168:168 division in the two
600cells
When the 600cell is composed
of simple tetractyses, (720×2=1440) hexagonal yods line edges of the 1200 tetractyses in the
faces and (120×2=240) hexagonal yods line sides of the 720 internal tetractyses. Therefore,
(1440+240=1680) hexagonal yods line the 840 sides of the (1200+720=1920) tetractyses. The
superstring structural parameter 1680 recorded by C.W. Leadbeater when he remoteviewed the
UPA is the number of hexagonal yods in a single 600cell that correspond to the six
Sephiroth of Construction above Malkuth. Here is yet more confirmation of the superstring
nature of the UPA. The pair of 600cells has (1680+1680=3360) hexagonal yods on the 1680
sides of their (1920+1920=3840) tetractyses. This is the number of turns in one revolution
of the 10 whorls. Once again, we see that each 600cell is representing all 10 whorls, but
this time a halfrevolution of them instead of five halfrevolutions. Previous articles have
discussed numerous examples of how sacred geometries display
the 168:168 or
1680:1680 divisions of their components. The remarkable ways in which polygonal numbers and
sacred geometries represent the superstring structural parameter 3360 (number of turns in
one revolution of the 10 whorls of the UPA) are shown here.
10. The holistic 120:720 division in two
600cells and in sacred geometries
Figure 5 displays how the
number 1680 is embodied in 1. the disdyakis triacontahedron, 2. the 1tree & the two
sets of 7 separate Type B polygons, and 3. the first four Platonic solids. Their geometrical
or yod compositions conform to the pattern:
1680 = 720 + 240 +
720.
This pattern manifests in a
single 600cell composed of simple triangles as the 720 edges, (120+120=240) vertices &
internal sides and 720 internal triangles. It appears in two 600cells as their 240 vertices
and their pair of 720 edges. Let us examine these sacred geometries in more detail in order
to demonstrate how the holistic 120:720 division exhibited by the 600cell in both its faces
and its interior and by two 600cells in their vertices & edges manifests in each half
of these sacred geometries.
Disdyakis
triacontahedron
The polyhedron
has 62 vertices, 180 edges & 120 triangular faces. Surrounding an axis
passing through two diametrically opposite vertices are 360
(=36×10)
geometrical elements comprising 60 vertices, 180 edges, i.e., 240 vertices & edges,
and 120 triangles. Joining its vertices to its centre creates internal triangles which,
when Type A, are composed of 180 points, (60 + 3×180 = 600) lines & (3×180=540)
triangles surrounding the axis. Hence, (60+180=240) points, (180+600=780) lines &
(120+540=660) triangles surround the axis. Each half of the disdyakis triacontahedron is
composed of 120 points, 390 lines & 330 triangles. i.e., 120 points and 720 lines
& triangles. Therefore, the 1680 geometrical elements surrounding the axis consist of
two sets of elements, each comprising 840 elements made up of 120 points and 720 lines
& triangles. Each half of the disdyakis triacontahedron is the 3dimensional analogue
of a 600cell.
Outer & inner form of
1tree Constructed
from 19 Type A triangles, the 1tree contains 251 yods. Of these, 11 are corners of
triangles that coincide with Sephiroth, leaving 240 yods generated by this transformation.
The seven regular polygons making up each half of its inner form contain 727 yods when they
are Type B. 720 yods surround their centres. It was shown here that the 240 yods in the 1tree consist of 10 sets of 24
yods. Yods in triangles that are on the left or the right of the central Pillar of
Equilibrium can be associated with the yods in the seven regular polygons on that side. The
yods that line this pillar as the axis of symmetry of the 1tree can be divided into two
sets containing the same number. Of course, unlike the yods in each set of polygons, these
central yods are not mirror images of each other. However, one set can be associated with
the lefthand set of polygons and the other set associated with
23
the righthand set. This means that 120 yods in the 1tree can be
associated with each set of polygons. The 1680 yods can be divided into two sets of
(120+720) yods.
First four Platonic
solids
How the tetrahedron,
octahedron, cube & icosahedron embody the number 1680 is
calculated here. When their
faces and interiors are constructed from Type A triangles, the numbers of corners, sides
& triangles that surround their axes passing through two opposite vertices are tabulated
below in Table 7.
Table 7. Geometrical
elements surrounding the axes of the Platonic solids.
Platonic solid

Corners

Sides

Triangles

Total

Tetrahedron

24

78

66

168

Octahedron

48

156

132

336

Cube

48

156

132

336

Icosahedron

120

390

330

840

Subtotal =

240

780

660

1680

Dodecahedron

120

390

330

840

Total =

360

1170

990

2520

The 1680 geometrical elements
surrounding the axes of the first four Platonic solids comprise 240 corners and 1440 sides
& triangles. Surrounding the axes of the first three Platonic solids are 120 corners and
720 sides & triangles, i.e., 840 geometrical elements. Similarly, 120 corners and 720
sides & triangles, i.e., 840 geometrical elements, surround the axis of the icosahedron.
Amazingly, the first four regular 3polytopes display the same 120:720
division as the 600cell, which is the 4dimensional counterpart of the icosahedron,
the fourth Platonic solid! The compound of two 600cells is the
fourdimensional counterpart of the first four Platonic solids. Together with the disdyakis
triacontahedron, they are the 3dimensional and 4dimensional blueprints for the
E_{8}×E_{8} heterotic superstring, which was paranormally described by Besant
& Leadbeater over a century ago. The same pattern is displayed by the icosahedron and
the dodecahedron because the latter is the dual of the former and has the same geometrical
composition. This is not repeated in their 4dimensional counterparts because the 120cell
(the 4dimensional counterpart of the dodecahedron) has 600 vertices, not 120 vertices. It
is, however, repeated for its 120 dodecahedra and 720 pentagonal faces. Just as the
disdyakis triacontahedron is based upon the rhombic triacontahedron as a compound of the
icosahedron and dodecahedron, so the 4_{21} polytope is based upon
the compound of two 600cells. In this sense, it can be regarded as the 8polytope analogue
of the disdyakis triacontahedron — a 3polytope. What is crucial to recognize is that the
highly mathematical, hyperdimensional objects that are being discovered to underpin
E_{8}×E_{8} heterotic superstring
physics have their exact parallels in the
sacred geometries of mystical traditions and ancient philosophies.
Why? Because they represent the same thing.
11. How the Godnames prescribe the 600cell
and the pair of two 600cells
Kether: EHIEH
= 21.
The first six rows in
Pascal's triangle that contain binary coefficients other than 1
comprise 21 such numbers that add up to
240. This is the number of vertices in the two 600cells:
The centre of the 600cell constructed from Type A triangles is surrounded
by 21000 (=21×10×10×10) yods.
Chokmah: YAH
= 15.
The sum of the
first 15 positive integers is 120, i.e., it is
the 15th
triangular number. This is the number of vertices in the 600cell. The sum of the
first 15 even integers is 240, which is the number of vertices in the two
600cells. The two 600cells have 1680 vertices & edges. The sum of
the 15 combinations of the four
integers 21, 42,
63 & 84 (21 units apart) = 1680. The 15th prime number is 41, where 41^{2} − 1 = 1680.
24
Binah: ELOHIM
= 50.
2040 corners of 5760 triangles surround the
centre of the 600cell. Each half has 2880 triangles with 1020 (=102×10) corners. 102 is
the 50th even integer after 2.
Chesed: EL = 31.
The two 600cells have 1680 vertices &
edges. 1680 is the sum of the 31 combinations of the five
integers 7, 14, 21, 28 & 35, which are 7 units apart. Construction of the faces of the
600cell from Type A triangles adds (7×1200=8400) geometrical elements. The number 8400 is
the arithmetic mean of the 31 square numbers
5^{2},
10^{2}, 15^{2} … 155.^{2}
Geburah: ELOHA
= 36.
The 1200 faces of the 600cell have 3600
(=36×10×10)
sectors. It has 720 edges, where 720 = 72×10
and 72 is the 36th even integer.
Tiphareth: YAHWEH ELOHIM
= 76.
The pair of 600cells composed of simple
triangles has (1200+1200=2400) triangles in its faces and (720+720=1440) internal triangles,
making a total of 3840 (=384×10) triangles. 384 is the 383rd integer after 1, where 383 is
the 76th prime number.
Netzach: YAHWEH SABAOTH
= 129. Construction of the 2400
faces of two 600cells from Type B triangles adds 16800 geometrical elements.
As 168 = 13^{2} − 1,
16800 = 168×10^{2} =
(13^{2} − 1)10^{2} =
130^{2} −10^{2} =
(2×65)^{2} − (2×5)^{2}
= 4(65^{2} −
5^{2}).
65^{2} = 1 + 3 + 5 + 7 + 9...
+ 129
and
5^{2} = 1 + 3 + 5 + 7 +
9.
Therefore,
16800 = 4(11 + 13
+ 15 +... + 129).
The number 16800 is the sum of (4×60=240) odd
integers, the smallest of which is the 10th integer after 1 and the largest of which
is 129, the 65th odd integer,
where 65 is the sum of the first 10 integers after 1.
Hod: ELOHIM SABAOTH
= 153.
1560 (=156×10) yods line the 720 edges of the
600cell when constructed from tetractyses or Type A triangles. 156 is
the 153rd integer after 3, the first odd integer.
Yesod: EL ChAI
= 49.
Inside the 600cell are 2160 triangles with
2280 (=228×10) sides, where 228 is the 227th integer after 1 and 227 is
the 49th prime number. Two 600cells have 2400 faces.
As 49^{2} = 2401, 2400
= 49^{2} − 1. This is the
dimension of SU(49).
Malkuth: ADONAI
= 65.
The 600cell has (120+720=840) vertices &
edges. As 120 = 5! and 720 = 6!, 840 = 6! + 5!. These integers make up the gematria number
value of ADONAI. The 3600 triangles in the 1200 faces of the 600cell have 1320 (=132×10)
corners, where 132 is the 65th even integer after 2. The
5760 triangles in its faces and interior have 6600 (=66×10×10) sides, where 66 is
the 65th integer after 1.
ADONAI MALEKH
= 155.
The number of yods lining the 720 edges
connecting the 120 vertices of the 600cell = 120 + 720×2 = 1560 (=156×10). 156 is
the 155th integer after 1.
The sum of the Godname numbers of the first
six Sephiroth of Construction is 240, which is the number of vertices of the
4_{21} polytope denoting roots of E_{8}, as well as the number of
vertices of the two 600cells:
240 = 21 + 26 + 50 + 31 + 36 + 76.
The 240 vertices of
the 4_{21} polytope can be
constructed in two sets: 112 (2^{2}×^{8}C_{2}) with coordinates obtained
from (±2, ±2, 0, 0, 0, 0, 0,
0) by taking an arbitrary combination of signs
and an arbitrary permutation of coordinates, and 128 roots (2^{7}) with coordinates obtained
from (±1,
±1, ±1, ±1, ±1, ±1, ±1, ±1) by taking an even
number of minus signs (or, equivalently, requiring that the sum of all the eight
coordinates be even). Remarkably, the division 240 = 128 + 112 is
reproduced in the sums of the
25
Godname numbers of the first four Sephiroth in the set of
six:
128
= 21 + 26 + 50 + 31
and the last two Sephiroth:
112
= 36 + 76.
112 is the number value
of Beni Elohim (the "sons of God"
referred to in biblical Genesis). Table
3 indicates that the same division manifests in
the sides of the triangles making up the 8cell. Its 24 square faces have 40
corners (16 vertices & 24 corners) and 128 sides (32 edges & 96 sides) of 96
triangular sectors, i.e., (40+128=168) corners & sides
(48 vertices & edges; 120 corners & sides). This is how the
8cell embodies the superstring structural parameter 168. Its 96 interior
triangles have 112 sides. The
(96+96=192) triangles have (128+112=240)
sides.
Table 7 above shows that, when
constructed from Type A triangles, there are 840 geometrical elements surrounding the axis of
the icosahedron. They comprise 120 corners and 390 sides of 330 triangles, i.e., 720
sides & triangles. Compare this with its 4dimensional counterpart: the 1200 faces of the
600cell have 120 vertices & 720 edges, i.e., 840 vertices & edges. It is an example of
how parameters of holistic systems are related by the same patterns, in this case: 840 = 120 +
720. In fact, the same pattern appears in the tetrahedron, octahedron & cube because,
according to Table 7, 120 corners of 720 sides & triangles surround their axes, so that the
axes of the first four Platonic solids are surrounded by (120+120=240) corners of
(720+720=1440) sides & triangles. This is the 3polytope analogue of the compound of two
600cells, which have (120+120=240) vertices and (720+720=1440) edges. The 840 = 120 +720
division does exist in the 120cell, but as its 120 dodecahedral cells and 720 pentagonal
faces. Whilst a pair of
120cells has 240 dodecahedra and 1440 faces, i.e., 1680 faces & dodecahedra, this
combination (unlike the pair of 600cells) has no geometrical connection to the
4_{21} polytope. Instead, a single 120cell embodies the number 16800 as the
number of geometrical elements other than centres of Type A faces and internal triangles (see
comment 1 about the 120cell). As a
600cell has 8400 geometrical elements within its Type A faces and 8400 yods line the sectors
of its faces other than those lining its edges, it requires two such polytopes to generate the
superstring structural parameter 16800. Only a pair of 600cells can display simultaneously the
holistic parameters 240 and 16800. A single 120cell displays the number 120 and the number
16800, whilst a pair of them displays the numbers 240 and 33600. But the latter is not a
structural parameter of a single UPA. Instead, it is the number of turns in the whorls of two UPAs. So,
even though it displays the divisions: 240 = 120 + 120 and 840 = 120 + 720, a pair of 120cells
cannot be regarded as a polytope model for the UPA because it does not embody the number 240 in
a way that clearly refers to the 240 roots of E_{8} whose gauge charges are present in the
E_{8}×E_{8} heterotic superstring.
The fact that the
same patterns and superstring structural parameters 840 & 1680 appear in both the
first four Platonic solids and the compound of 600cells that is the 4dimensional projection
of the 240 vertices of the 4_{21} polytope is yet
another argument against the possibility of these paranormallybased numbers appearing by
chance in the two 600cells. It is also strong evidence that they conform to the
universal patterns exhibited by sacred geometries discussed in many previous research articles.
It implies that the pair of 600cells does, indeed, constitute sacred geometry. Given their
mathematical connection to the 4_{21} polytope, whose 240 vertices define
the 240 root vectors of the Lie group E_{8} describing one of the two
types of heterotic superstrings, the conclusion is inescapable that this type of superstring
exists. How can it not exist and yet its group mathematics be embodied in
analogous ways in the sacred geometries of some of the world's religions? Just a coincidence?
How can it not exist and yet purported paranormal description of the
constituents of matter match exactly this geometrical embodiment in both qualitative and
quantitative ways? Just another coincidence? Whilst a sceptic might speculate that this
happened by chance or make accusations of cherrypicking if he were presented with only one
example as alleged evidence that superstrings exist, even he would have to admit that it is
improbable in the extreme that his facile explanation could apply as well to all the other
sacred geometries discussed on this website that display the same patterns and parameters.
Unless he can provide a more plausible alternative, his scepticism is indefensible.
References
1. See Article 3, Table 1, p. 6.
2. Gosset, Thorold (1900). "On the regular and semiregular figures in space
of n dimensions". Messenger of Mathematics
29: 43–48.
3.
http://www.jstor.org/discover/10.2307/40345407?uid=3738032&uid=2&uid=4&sid=21104360280127.
4. Viazovska, Maryna, http://arxiv.org/abs/1603.04246. See also:
https://www.quantamagazine.org/20160330spherepackingsolvedinhigherdimensions/
5. Occult Chemistry, by Annie Besant & C.W. Leadbeater, 3rd edition (1951), Theosophical
Publishing House, Adyar, Chennai, India, p. 23.
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