ARTICLE 66
by
Stephen M. Phillips Flat 4, Oakwood House, 117119
West Hill Road. Bournemouth. BH2 5PH. England.
Website: http://smphillips.mysite.com
Abstract
The 16cell is the 4dimensional counterpart of the
octahedron. This article presents evidence for its holistic character by
demonstrating that all the gematria number values of the 10 Sephiroth, their
Godnames, Archangels, Orders of Angels & Mundane Chakras quantify the
geometrical and yod composition of the 16cell when it is constructed from
Type A triangles. These 50 parameters emerge so naturally in the analysis as
to make it highly improbable that their appearances are due to chance. The
holistic parameter 384 manifests in the 16cell as its 384 rotations &
rotationsreflections. Each half of the 16cell consists of 84 triangles (48
in its faces, 36 in its interior). This 48:36 division of the holistic
parameter 84 is characteristic of various sacred geometries, as well as
quantifying Pythagorean intervals between the notes in the four Authentic
Modes and the three distinct Plagal Modes. The 84:84 division of its 168
triangles has a counterpart in the 84 intrinsic corners of the 141 triangles
associated with each half of the inner Tree of Life when composed of Type B
polygons, as well as in each half composed of Type A polygons as the 84
geometrical elements outside the root edge in either the triangle, square,
pentagon & dodecagon or the hexagon, octagon & decagon that surround
the five centres that are not corners of polygons. The 16cell embodies the
number 137 determining the finestructure constant and the dimension 496 of
the two symmetry groups SO(32) &
E_{8}×E_{8}^{′} governing superstring forces
that are free of quantum anomalies. The yod population of the 16cell is the
sum of the two holistic parameters 384 and 248, where the latter is the
dimension of E_{8}. It indicates that the 16cell embodies the root
composition of E_{8} and
E_{8}×E_{8}^{′}.^{.} This confirms its
holistic status because previous articles have proved that other sacred
geometries embody these symmetry groups.

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
16cell Absent from the seven types of regular
polygons making up the inner form of the Tree of Life are the heptagon, the nonagon and the
undecagon. Article
58explored the analogy between these three
absent polygons and the three Sephiroth of the Supernal Triad, as implied by the analogy
between the seven polygons and the seven Sephiroth of Construction. Article
59 revealed that the three absent polygons embody the root
composition of E_{8} and
E_{8}×E_{8}.
This remarkable property has been encountered in other sacred geometries and analysed in
many earlier articles. has revealed the 16cell to be — in that context —the most fundamental of the six polychorons because
the 24cell that they encode is a compound of three 16cells, it is worthwhile to analyse
its properties here in order to uncover evidence of its holistic character. This will
manifest by the appearance of the gematria number values of the 10 Sephiroth in the four
Kabbalistic Worlds quantifying its properties and by its displaying structural parameters
of sacred geometries established elsewhere in this website. These numbers are listed in
Table 1 shown above.
The 16cell is the 4dimensional version of
the octahedron. It is composed of:

eight points (vertices, or 0polytopes) with
coordinates ±1 along each of the four perpendicular Cartesian axes; each is met by
six edges, 12 faces & eight cells;

24 lines (edges, or 1polytopes), joining all pairs
of vertices except diametrically opposite ones; each one is met by four faces &
four cells;

32 triangles (faces, or 2polytopes); each is met by
two cells;

16 tetrahedra (cells, or 3polytopes); four meet at
each edge.
It has 80 0, 1, 2 & 3polytopes, where 80 is the number value of Yesod ("Foundation"). This is the
number of yods in the 1tree constructed from tetractyses and the number of corners of the
(47+47=94) sectors of the (7+7) enfolded polygons that are its inner form
(see here). As evidence that
the appearance of this number in the 16cell is not coincidental but indicative of its
holistic nature because it is described by the system of gematria number values of the 10
Sephiroth, observe that the distinction between the seven Sephiroth of Construction and the
rest of the 1tree generates the division:
80
= 32 + 48
in yods (48 yods up to Chesed, the first Sephirah of Construction, and
32 blue yods above it). This appears in the inner Tree of Life as the 48 sector corners that are either black centres of the seven
enfolded polygons in one set (see picture below) or the 41 red corners of the other set of
seven enfolded polygons and as the 32 blue corners of the former set of seven polygons. It
appears in the 16cell as its 32 vertices & edges and its 48 faces & cells (Fig. 1):
Figure 1
It would be incorrect to regard the 32 faces
of the 16cell as corresponding to the 32 yods above Chesed because one would, intuitively,
expect the degree of complexity of polytopes in the 16cell to increase with the downward
descent from Kether towards Malkuth of the yods that symbolise them, but this correspondence
disrupts the order by placing 2polytopes before 1polytopes, instead of the more natural
progression: 0polytopes→1polytopes→2polytopes→3polytopes, which has been followed in
Figure 1.
3
Suppose that the 16cell is constructed both externally and internally
from Type A triangles. The geometrical & yod compositions of the six polychorons are
discussed here. The two
compositions of the 16cell are reproduced below in Table 2:
C/C′/c = number of
corners; S/S′/s = number of sides; T/T′/t = number of triangles; N/N′/n = total number of
geometrical elements.
H/H′/h = number of hexagonal yods; B/B′/b = number of
boundary yods; Y/Y′/y = total number of yods.
Table 2.
Geometrical composition of
the 16cell

Yod composition of the
16cell

Faces

Interior

Total

C

S

T

N

C′

S′

T′

N′

c

s

t

n

40

120

96

256

1+24

80

72

1+176

1+64=65

200

168

1+432


Faces

Interior

Total

C

H

B

Y

C′

H′

B′

Y′

c

h

b

y

40

336

280

376

1+24

232

1+184

1+256

1+64=
65

568

1+464

1+632


(“1” denotes the centre of the
16cell). 
Comments
1) Its 32 triangular faces contain 256 geometrical elements, where 256 = 4^{4}.
They comprise:

8 vertices, where 8 = 4th even integer;

24 edges, where 24 = 1×2×3×4 = 5^{2} − 1 = 3 + 5 + 7 + 9 = sum of
1st 4 odd integers after 1;

64 vertices, edges & faces, where
64 = 4^{3};

16 tetrahedral cells, where 16 = 4^{2};

40 corners (points), where 40 = (1+2+3+4)×4 = 4 + 8 + 12 +
16;

120 sides (lines), where 120 = 2^{2} + 4^{2} +
6^{2} + 8^{2} = sum of squares of 1st 4 even
integers;

96 triangles, where 96 = 4×24 = 4×1×2×3×4 = 4(5^{2}−1) =
4(3+5+7+9) = 12 + 20 + 28 + 36;

160 corners & sides, where 160 =
(1+2+3+4)×4^{2};

136 corners & triangles, where 136 = 16th triangular number and 16 =
4th square number;

216 sides & triangles, where 216 =
6^{3} and 6 is the fourth integer after 2, the first prime
number.
This demonstrates very clearly how the Tetrad (4) expresses the
geometrical composition of the faces of the 16cell. It also expresses its interior,
for there are 176 (=4×44) points, lines & triangles and 256 (=4^{4}) yods
inside the 16cell surrounding its centre. There are 248 geometrical
elements other than the eight vertices in its 32 faces, where 248 is the
number value of Raziel, the Archangel of Chokmah. The
dimension 248 of the rank8 exceptional Lie group E_{8} is
embodied in the facial geometry of a single 16cell. The 248
geometrical elements comprise 128 corners & triangles and 120 sides. Its faces have
(120+96=216) triangles & their sides, where 216 is the
number value of Geburah. As 432 geometrical elements surround its centre, each half of the
16cell contains 216 geometrical elements. The 16cell has
168 triangles with 65 corners,
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. This conjunction of two gematria numbers associated with the same Sephirah
is both remarkable and highly significant. 432 geometrical elements surround its centre ("1"),
demonstrating once again the role of the integers 1, 2, 3 & 4 symbolised by the tetractys
in expressing properties of the 16cell. 264 corners & sides (64 corners,
200 sides) surround its centre, where 264 is the number of yods in the 7 enfolded Type A
polygons that make up the inner Tree of Life (see Article 64, Table 5, and here). This embodiment of a primary
parameter of the seven enfolded Type A polygons of the inner Tree of Life is
unmistakable evidence of the Tree of Life/holistic character of the 16cell. It is
confirmed by the fact that 24 of the 200 sides are edges of the faces of the 16cell,
leaving 176 sides of the 168 triangles that are not edges. Compare this
with the fact that the 7 enfolded polygons have 176 corners, sides & triangles (see
here). In both cases, the holistic structural parameter 264
divides up in the same way:
7 enfolded polygons: 
264 yods = 176 hexagonal yods on 88 sides + 88 corners &
centres of 47 tetractyses; 
16cell: 
264 corners & sides = 176 sides + (64+24=88) corners
& edges.

The four yods in the root edge of the seven enfolded polygons correspond to any two
opposite vertices and the two sides of internal triangles that form an axis of the 16cell
(see (6) below). The 260 (=26×10)
4
yods outside the root edge correspond to the 260 corners & sides outside
this axis: 260 = 62 corners + 198 sides = 6 vertices + 56 corners + 24 edges +
174 sides. This is how YAHWEH with number value 26 prescribes the 16cell. Just
as the 264 yods in the 7 enfolded polygons show the factorisation: 264 = 3×88 as 2×88
(hexagonal yods) + 88 (corners & centres of tetractyses), so, too, the 264 corners &
sides of the 16cell comprise the 88 sides in each half that are not edges and the 88
corners & edges of the whole polychoron. Isomorphism exists between the inner Tree of
Life and the 16cell because the latter is the 4dimensional manifestation of the
former.
2) The 16cell has 64 vertices, edges & faces, where
64 is the number value of Nogah, the Mundane Chakra of Netzach. It also
has 64 corners of 168 triangles surrounding its centre. When its
faces and interior triangles are simple triangles rather than Type A triangles, there are in
its interior 24 internal triangles with one corner and eight sides, totalling nine corners,
32 sides & 56 triangles, i.e., 97 internal geometrical elements, where
97 = 21 + 26 +
50
is the sum of the Godname numbers of Kether, Chokmah & Binah; it is also
the number value of Haniel, the Archangel of Netzach. When these triangles are
tetractyses, there are (1 + 8×2 + 24 = 41) internal yods and (8 + 24×2 + 32 = 88) yods in
its faces, a total of 129 yods, where 129 is the number value of
YAHWEH SABAOTH, the Godname of Netzach. Nine yods are corners of 56 tetractyses, leaving 120
hexagonal yods, where
120 = 11^{2} − 1 = 3 + 5 + 7 +... + 21
is the sum of the first ten odd integers after 1, showing how
the Decad and EHYEH with number value 21 determine the number of hexagonal yods
needed to construct the 16cell from tetractyses. 336 of the 632 yods surrounding the centre
of the 16cell with Type A triangles are hexagonal yods in its faces, leaving 296 yods
(148 in each half) that are either corners or internal hexagonal yods. 148 is
the number value of Netzach. Here is a conjunction of four numbers associated with the
same Sephirah (Netzach). (96+16=112) points, lines, triangles & tetrahedra
surround its centre, where 112 is the number value of Beni Elohim, the
Order of Angels assigned to Hod. The 16cell has 48 faces (2polytopes) &
tetrahedral cells (3polytopes), where 48 is the number value of Kokab, the
Mundane Chakra of Hod.
3) 280 yods line the 96 tetractyses in its 32 faces. This is
the number value of Sandalphon, the Archangel of Malkuth. Here is a second,
remarkable, chancedefying conjunction in the same context of the gematria number
values of the Godname (65), Archangel (280) & Mundane Chakra (168)
of the same Sephirah (Malkuth). 140 yods line the 48 tetractyses making
up the 16 faces in each half of the 16cell, where 140 is the number value of
Masloth, the Mundane Chakra of Chokmah. 376 yods make up its faces (188 in each
half). 187 yods in the 16 faces in each half surround its axis, where 187 is
the number value of Auphanim, the Order of Angels assigned to Chokmah. As (1) shows
that the 16cell embodies the number 248 of Raziel and the number 26 of
YAHWEH and as (4) shows that it embodies the number 73 of Chokmah, here is a
conjunction of all five numbers associated with the same Sephirah.
4) When its faces are simple triangles, they are lined by (8 + 24×2 =
56) yods; (1 + 8×2 = 17) yods line the 24 internal triangles formed by joining vertices to
the centre of the 16cell. (56+17=73) yods line its (24+32=56) triangles, 72
such yods surrounding its centre. The 16cell embodies the number value 73 of Chokmah
as the number of yods that line all its tetractyses, both externally and internally.
73 is the 21st prime number, showing how EHYEH, the Godname of Kether with
number value 21, prescribes in a minimal way the form of the 16cell. The
number of yods in the 32 faces of the 16cell = 56 + 32 = 88. There are 80 hexagonal
yods, where 80 = 48 (lying on 24 edges) + 32 (at centres of 32 faces). It
should be noticed that this division is the characteristic division of the 80 yods of
the 1tree and the 80 corners of the 94 sectors in its polygonal inner form that was
discussed above. The double presence of this number in both the geometrical and the yod
composition of the 16cell is evidence that it acts as the polytopic "foundation." This
another reason why it should be regarded as fundamental, i.e., the "hydrogen atom" of the
polychorons, even though it is not the polychoron with the least number of
vertices.
5) When its faces and interior triangles are simple triangles, there
are (8+24=32) corners & sides of triangles in its faces and (1+8=9) internal corners
& sides, a total of 41 corners & sides. This is both the 21st odd integer and
the 15th prime number, showing how EHYEH, the Godname of Kether with number value
21 and YAH, the shortened Godname of Chokmah with number value 15, prescribe
the minimal geometry of the 16cell. Collecting together results discussed in (1), (3) &
(4), all six numbers associated with the Sephirah Chokmah are now seen to be naturally
embodied in the 16cell.
6) If we consider an axis passing through two opposite vertices and
its centre, it is composed of three corners and two sides of triangles. Therefore,
(41−3−2=36) corners & sides surround it. This is the number value of ELOHA, the
Godname of Geburah. 632 yods surround the centre of the 16cell. This means
that,
5
given its centre and two "poles" (a pair of vertices at ±1 on the
axis), 630 yods are needed to construct its 168 tetractyses. 630 is the
number value of Seraphim, the Order of Angels assigned to Geburah. The 97
points, lines & triangles include two lines forming this axis. The 16cell contains
95 points, lines & triangles other than these two axial lines, where 95 is
the number value of Madim, the Mundane Chakra of Geburah. Surrounding the centre of
the 16cell with Type A triangles are 264 corners & sides, i.e., 262 geometrical
elements other than these two lines. Each half has 131 such geometrical elements.
This is the number value of Samael, the Archangel of Geburah. The 16cell embodies
all five number values associated with Geburah.
7) The 32 faces of the 16cell contain 336 hexagonal yods (96 at centres of
tetractyses, 240 lining their 120 sides). This number is discussed widely on this website as
a superstring structural parameter embodied in sacred geometries, being the number of turns
in one revolution of each helical whorl of a UPA (subquark state of the
E_{8}×E_{8} heterotic superstring) around its axis of spin. Together
with the above examples of how the Godnames prescribe its properties, here is irrefutable
evidence that the 16cell is a holistic object, for it is highly improbable that a number
appearing in so many sacred geometries would turn up in this context just by chance. Rather,
the number 336 does so in the 16cell because it, too, is holistic, so that this number
must quantify at least one of its properties, viz., how many hexagonal yods are
needed to construct its faces out of Type A triangles.
8) The symmetry group of the ncube and its dual, the ndimensional
crosspolytope, is
B_{n}, and is known as the hyperoctahedral group.
It has order 2^{n}n!. For n = 4, this is 384. B_{4} has the subgroup
D_{4} with order 192. This is the group of rotations of the 4cube and the
16cell. D_{4} has the subgroup D_{3}, which is the




Figure 2.
When its 10 sectors are constructed from 2ndorder
tetractyses, the decagon contains 620 hexagonal yods (310 hexagonal yods
in each 5fold array of sectors). The 1st 10 regular polygons have 75 sectors.
One corner, two sides & a triangle are associated with each sector, i.e., 4
geometrical elements. Including their centres, the 1st 10 separate polygons are
composed of (75×4 + 10 = 310) geometrical elements. The 1st (10+10) separate
polygons comprise (310+310=620) geometrical
elements.

Figure 3.
When its 16 triangles are Type A, the Tree of Life contains 214 yods. 26
yods are corners and 188 are hexagonal yods. The two hexagonal yods on the
GeburahChesed Path coincide with the centres of the two triangles in the inner
Tree of Life. Four hexagonal yods on each side pillar are shared with it.
(188−2−4−4=178) hexagonal yods are unshared with the inner Tree of Life, which
has 444 hexagonal yods (442 outside the root edge, which has two hexagonal
yods). The number of hexagonal yods outside the root edge in the combined Trees
of Life = 178 + 442 = 620.

6
group of rotations of the cube and octahedron. The holistic parameters
384 and 192, which show the divisions:
384 = 48 + 336
and
192 = 24 + 168,
signify in the present context the 384 rotations+reflections of the
4cube and 16cell, the 48 rotations+reflections of their 3dimensional counterparts
— the cube and the octahedron, the 192 proper rotations [1]
of the 4cube and 16cell and the 24 proper rotations of the cube and octahedron. The
appearance of the holistic parameters 384 & 192 as orders of the hyperoctahedral group
for 4polytopes does not mean that all sacred geometries that embody these numbers depict
only the symmetries of 4‑dimensional objects. For example, the five Platonic solids embody
them (see here), as
do the seven separate polygons making up the inner form of the Tree of Life (see
here). What it
does mean is that such objects, in which these numbers uniquely appear, are holistic
in character, exhibiting the complete pattern of the Whole, even though they may be
only components of it.
9) 632 yods surround the centre of the 16cell constructed from Type A
triangles. There are four hexagonal yods lining the axis. Surrounding the axis are
(632−4−8=620) yods that are not vertices. In other words, 620 yods need to be
added to create its shape from Type A triangles. This is the number value of Kether
("Crown"). It is:
1. the
number of hexagonal yods in a decagon with 2ndorder tetractyses as sectors;
2. the
number of geometrical elements in the first (10+10) polygons;
3. the
number of hexagonal yods in the combined outer & inner Trees of Life constructed from
Type A polygons.
(Figs. 2 & 3).

Figure 4. A Type B Ngon has (15N+1)
yods that comprise (2N+1) corners & 13N hexagonal yods. The number of
hexagonal yods in the 7 Type B polygons with 48 corners = ∑13N =
13×48 = 624. This is the number of yods needed to construct the 16cell
from Type A triangles, starting with its 8 vertices.

These examples demonstrate how the number 620 of Kether (the first
Sephirah) is a parameter of holistic systems. 310 (=31×10) such yods in each half of
the 16cell surround this axis. 31 is the number value of EL ("God"), the Godname of
Chesed. 62 corners of tetractyses surround the axis. This is the number value of
Tzadkiel, the Archangel of Chesed. We found in (4) that 72 yods line the 56
tetractyses in either its faces or interior when its faces and internal triangles are
tetractyses. 72 is the number value of Chesed. 432 geometrical elements surround its
centre. The axis is composed of two vertices, two sides and the centre. Surrounding it are
(432−2−2=428) geometrical elements, where 428 is the number value of
Chasmalim, the Order of Angels assigned to Chesed. Here is another remarkable
conjunction of gematria number values (this time, four) associated with the same
Sephirah.
10) The number of yods in the 16cell other than the five internal yods on its axis = 633 − 5 =
628. Each half of the 16cell has 314 yods other than yods inside it on its axis. The
number 314 is the number value of Metatron, the Archangel of Kether. The number
of yods in the 16cell other than vertices = 633 − 8 =
[1] Proper rotations are pure rotations about an axis unaccompanied by
reflections. Improper rotations are rotations about an axis followed by reflection in a
mirror perpendicular to that axis. For more details, see here.
7
625 = 5^{4}. There are 624 such yods surrounding its centre.
This is the number of hexagonal yods in the seven separate Type B polygons that make up the
inner form of the Tree of Life (Fig. 4).
11) The 16cell has 568 hexagonal yods. Four of them line the axis, so
that (568−4=564) hexagonal yods surround it. 282 such yods in each half of the
16cell surround its axis. The number value of Aralim, the Order of Angels assigned
to Binah, is 282. Surrounding its centre are 464 yods lining sides of
168 tetractyses. Six yods apart from the centre lie on the axis.
(464−6=458) boundary yods surround the axis, 229 such yods in each half. 229 is the
50th prime number, showing how ELOHIM, the Godname of Binah with number value
50, prescribes the number of yods shaping the faces and interior of the
16cell constructed from Type A triangles. 632 yods surround its centre, 316 yods being in
each half. Including the centre, which is shared by both halves, there are
317 yods in each half. 317 is the number value of Shabathai,
the Mundane Chakra of Binah. Excluding the two hexagonal yods on each half of the axis and
the four vertices in each half of the 16cell, there are (317−2−4=311) yods in
each half sharing the centre. 311 is the number value of Tzaphkiel, the
Archangel of Binah. The axis has one corner and two sides inside the 16cell. When they are
simple triangles, the 32 faces comprise 32 corners & sides. The faces & axis
comprise nine corners, 26 sides & 32 triangles, i.e.,
67 geometrical elements, where 67 is the number value of Binah. The
16cell embodies all five number values associated with this Sephirah.
The fact just proven above that all
the Kabbalistic gematria number values of the 10 Sephiroth emerge naturally
from analysis of the geometrical and yod compositions of the 16cell is
powerful evidence of the holistic nature of this polychoron. Had the appearance
of these numbers been manufactured artificially by adding numbers that referred
to unrelated contexts, the five numbers associated with the
same Sephirah would be scattered about randomly in unconnected
contexts. Instead, they often appear in the same, mathematical context
with a frequency that exceeds chance. This is indicative of a
coherent design or conceptual
order that exists because the 16cell is a holistic object. The
gematria numbers quantify 10 sets of properties of the 16cell, each set
expressing the mathematical aspect of a Sephirah functioning in the four
Kabbalistic Worlds of Atziluth, Beriah, Yetzirah & Assiyah. As we shall
discover below, the 16cell displays the same universal patterns
in its structural parameters as those found in sacred geometries.

2. 36:48 divisions in the 16cell and
sacred geometries
Each half of the 16cell has (28×3=84) sectors of 28 Type A triangles.
The 16 faces in each half have


The 84 yods up to the top of the 1tree
consist of 48 red yods up to Chesed (the 1st Sephirah
of Construction) and 36 black yods above it up to
Kether.

The 3dimensional Sri Yantra has 42 triangles
surrounding its central one with 84 corners. The 1st & 2nd
layers with 18 triangles
have 36
corners; the 3rd & 4th layers with 24 triangles
have 48 corners.

84 yods in the 2ndorder tetractys surround
its black centre. 36 green yods are either corners of
tetractyses or in the 3 corner
tetractyses. 48 brown hexagonal yods in the 7
tetractyses surround the black centre.

The sum of the 9 integers in the Lambda
Tetractys that surround its central integer 6 is 84. The sum of
the black integers at the 3 corners
is 36. The sum of the six red integers
is 48.


Figure 5. The 16cell
is holistic because the 36:48 division of its triangles in each half exists in
sacred geometries.
8
(3×16=48) sectors and its 12 internal Type A triangles have
(12×3=36) sectors. This 36:48 division in the 84 simple triangles
making up each half of the 16cell is characteristic of holistic systems that embody the
number 84 as one of their defining parameters. Figure 5 displays examples.
7 Diatonic scales as the musical counterpart of the
16cell [1] between the notes of the seven diatonic musical
scales making up the eight Church Modes. The four Authentic Modes have
48 repetitions of Pythagorean intervals [1] between their notes, whilst the
three Plagal Modes have 36 repetitions of Pythagorean intervals between
their notes (for proof, see the fourth comment under the table of
intervals here). One half of
the 16cell with 84 triangles is analogous to the 84 repetitions of rising,
Pythagorean intervals; its opposite half with 84 triangles corresponds to the 84 repetitions
of falling, Pythagorean intervals (N.B. if a rising interval has a magnitude of
N (N>1), its falling counterpart has a magnitude of 1/N). The 48 triangles
making up the 16 faces in each half of the 16cell are analogous to the
48 repetitions of Pythagorean intervals between the notes of the 4 Authentic
Modes, whilst the 36 triangles inside each half of the 16cell correspond to the
36 repetitions of Pythagorean intervals (rising or falling) between the notes of
the three Plagal Modes. The distinction between the exterior and interior of the 16cell is
what corresponds to the difference between the four Authentic Modes and the three Plagal
Modes in the seven diatonic scales.

The 16cell.
Each half has 28 triangles with 84 sectors.

Figure 6. The 16cell embodies the same structural parameter 168
as the UPA (subquark state of the
E_{8}×
E_{8}
heterotic superstring) because both are holistic systems that
must be parameterised
by the gematria number values of the 10 Sephiroth in the four
Kabbalistic Worlds (in this case, the number value 168 of
Cholem Yesodoth, the Mundane Chakra of
Malkuth).
The 84:84 division in the UPA/superstring
The 84 sectors in each half of the 16cell has their counterpart in
the UPA/superstring (Fig. 6). Each of its

Figure 7. The 84:84 division in a halfrevolution of a whorl of the UPA has its
counterpart in the 7 enfolded Type A polygons as the 84 geometrical elements
outside the root edge in either the triangle, square, pentagon & dodecagon
or the hexagon, octagon & decagon that surround the 5 centres that are not
corners of polygons. It appears in the (7+7) enfolded Type B polygons as the 84
corners of the 141 triangles associated with each set of 7
polygons.

[1] "Pythagorean intervals" are intervals that are also tone
ratios of the 6 notes between the tonic and octave of the Pythagorean scale.
9
10 whorls is a helix with 1680 circular turns. The helix winds five
times around its axis of spin, so that every halfrevolution comprises 168 turns (84
per quarterrevolution).
The 84:84 division in the inner Tree of Life
As evidence that the 16cell is a holistic object containing features
analogous to those found in certain sacred geometries, consider the inner Tree of Life when
its polygons are either Type A or Type B (Fig. 7).
Type A polygons
The seven enfolded Type A polygons have 176 corners, sides &
triangular sectors (see Table 3 here). Their shared root
edge is a side with a corner at each end, i.e., it is composed of three geometrical
elements, so that (176−3=173) geometrical elements are outside the root edge. The centres of
the hexagon and decagon are also corners of, respectively, the triangle and the pentagon.
Therefore, five centres (shown as black dots in the picture above) are
not corners of polygons. They are surrounded by (173−5=168) geometrical
elements outside the root edge. The centre of a Type A ngon divided into its n sectors is
surrounded by 4n geometrical elements. The separate Type A triangle, square, pentagon &
dodecagon with 24 corners have (4×24=96) geometrical elements surround their centres.
Enfolded, they have (96 − 3×3 = 87) such geometrical elements, i.e., (87−3=84)
geometrical elements surrounding centres of polygons are outside the root edge, where
87 is the number value of Levanah, the Mundane Chakra of Yesod. The
hexagon, octagon & decagon with 24 corners have (168−84=84) geometrical elements
outside the root edge that surround their centres.
Type B polygons
The seven enfolded Type B polygons have 141 triangles with 88 corners.
The (7+7) enfolded Type B polygons making up the inner Tree of Life have 282
triangles with 174 corners, where 282 is the number value of Aralim, the Order
of Angels assigned to Binah. Three corners lying on the vertical axis of the hexagon
coincide with corners of triangles belonging to the outer form of the Tree of Life (coloured
green in Fig. 8). (174− 2×3 = 168) corners are intrinsic to the inner Tree of
Life. Each endpoint of the root edge can be associated with one set of seven enfolded
polygons. This means that 84 intrinsic corners can be associated with each set. They are the
counterpart of the (84+84) triangles in the two halves of the 16cell. As the set of seven
enfolded Type A polygons have 47 sectors with 41 corners, (41−2−3=36) corners outside
the root edge are intrinsic to each set. They are coloured dark blue or light blue in Figure
8.
Figure 8
The remaining 48 corners consist of the corners at the centres
of the 47 sectors and one of the two endpoints of the root edge; they are coloured either
dark red or light red. The 36 light or dark blue corners correspond to the 36
triangles making up the 12 Type A triangles inside each half of the 16cell, whilst the
48 dark red or light red corners correspond to the 48 triangles making up the
16 faces in each half when they are Type A triangles. We find that the 84 triangles making
up each half of the 16cell are the counterpart of these 84:84 divisions exhibited by the
two halves of the inner Tree of Life. Furthermore, each set of 84 triangles displays the
36:48 division that is characteristic of this holistic parameter, as
illustrated above for various sacred geometries, including the inner Tree of Life. It is
evidence that the 16‑cell is the polychoron version of the archetypal structure of a
holistic system.
3. 36:48 & 84:84
divisions in the square representation of 624 &168
As
13^{2} − 1 = 168 = 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 +
19 + 21 + 23 + 25,
the holistic parameter 168 is the sum of the first 12 odd integers
after 1. As
25^{2} − 1 = 624 = 3 + 5 + 7 +... + 49 = 168 + 27
+ 29 + ... + 49 = 168 + 456,
and a Type A square has 25 yods, the number of yods other than vertices
(624) surrounding the centre of
10
a 16cell constructed from Type A triangles is the sum of the 24 odd
integers 349 that can be assigned to the 24 yods surrounding the
centre of a Type A square (Fig. 9).


Figure 9

The sum of the first 12 odd integers after 1 assigned to the 12 yods
on the boundary of the square is 168, which is the number of hexagonal yods at the
centres of its 168 tetractyses. The sum of the remaining 12 odd integers assigned to
the 12 internal yods surrounding the centre of the square is 456, which is the number of
yods other than vertices that line the sides of the 168 tetractyses. The central
integer 1 denotes the centre of the 16cell. Here is a remarkable arithmetic representation
of the number of yods needed to construct the 16cell from Type A triangles. What is also
remarkable is that the red and blue, Lshaped sides of the square array each add up to 84,
the integers in either one naturally adding up to 36 and 48 to generate
arithmetically the 36:48 division that is characteristic of this holistic
parameter. Moreover, 36 is the sum of the first 6 odd integers, whilst 48 is
the sum of the first 6 odd integers after 1. The number of yods needed to construct the
16cell from Type A triangles is not only the number of hexagonal yods in the 7 separate
Type B polygons making up the inner Tree of Life but also the sum of the 24 odd integers
after 1 that can be assigned to the 24 yods surrounding the centre of a Type A square.
Instead of these congruencies between number and sacred geometry being dismissed as just a
coincidence, it is more plausible to see them as yet more evidence for the holistic nature
of the 16cell.
4. 16cell embodies the holistic parameters
137, 248, 384 & 496
A remarkable property of the yod population (633) of the 16cell
is:
137 + 496 = 633.
Many physicists regard the number 137 as one of the most mysterious
numbers in physics because its reciprocal measures approximately what they call the
"finestructure constant,"
which, in electrostatic c.g.s. units, is defined as α = e^{2}/ħc ≅ 1/137. Many sceptics of superstring
theory view the dimension 496 of E_{8}×E_{8} and SO(32) (the only two
gauge symmetry groups that leave superstring interactions free of quantum anomalies) as equally
mysterious because no one has ever convincingly proposed a theory that derives either number
from fundamental principles. The latter emerges only from imposing consistency of these
interactions with quantum mechanics. Both numbers are parameters of sacred geometries (e.g.,
the yod population of the inner Tree of Life with Type B polygons is 1370, which is the
number of yods in 137 tetractyses; see here). There
is, therefore, no longer any mystery about either number because both of them characterise
all holistic systems, including superstrings. For examples of their presence in sacred
geometries, see The holistic
pattern under the headings "137" and "496 = 248 + 248."
Another remarkable property of the number 633 is that:
385 + 248 =
633,
where
385 = 1^{2} + 2^{2} + 3^{2} +... +
10^{2}
is the 10th square pyramidal number and 248 is the dimension of
E_{8}. The first 10 Type A polygons have 75 sectors lined by 385 yods (see Table 2
in Article 58). As
496 = 248 + 248,
∴ 137 +
248 = 385.
11
Figure 10
The number 385 is another holistic parameter, although the number 384 is, usually, the
more visible signature of its presence, the extra "1" denoting merely the centre of the system.
As 632 yods surround the 16cell,
384 + 248 = 632.
The number of yods surrounding the centre of the 16cell is the sum of
the holistic parameters 384 & 248.
Let us investigate what the first highlighted equation means for a
16cell constructed from tetractyses. Figure 10 depicts a Type A triangular face of an upper
and a lower half of the 16cell, together with a Type A internal triangle with an edge as
one side and the centre of the polychoron as its interior corner.
137
The upper half has 16 faces with four green vertices. Its 16
triangular faces have orange centres. Its 12 internal Type A triangles has black centres,
three violet hexagonal yods at the centres of the sectors of each one and six dark grey
hexagonal yods lining their sides. The lower half has 16 faces with four dark green vertices
and a brown yod at the centre of each face. Its internal Type A triangles have light grey
centres, three yellow hexagonal yods at the centres of their sectors and six mauve hexagonal
lining their sides. Each half has (3×12 + 3×16 = 84) tetractyses with 32 corners and
36 internal hexagonal yods (violet/yellow). Together with the centre of the 16cell,
there are [1 + 2(32+36) = 137] yods that are either corners (65) or hexagonal
yods (72) at the centres of internal tetractyses.
496
Four vertices in each half of the 16cell are joined to the centre of
the 16cell by 4 sides with eight hexagonal yods (light green/turquoise). 72
hexagonal yods (dark grey/mauve) line the three tetractyses in each internal Type A triangle
and 168 hexagonal yods (red or dark green) either line the tetractyses in the faces
or are at their centres. Each half comprises (8+72+168=248) yods that
are not either corners or centres of internal tetractyses. Both halves contain
(248+248=496) such yods. They correspond to the

Figure 11.
Correspondences between
the 248 roots of
E_{8} and the 248 yods in each half of the 16cell
that are neither corners nor centres of
tetractyses.

248 hexagonal yods
→ 248 roots of E_{8};

8 hexagonal yods
on internal sides joining 4 vertices to the
centre
of the 16cell → 8
zero (simple) roots of E_{8};

72 hexagonal yods
on 36 sides of 36 internal tetractyses
→ 72
roots of
E_{6};

168 hexagonal yods lining 60 sides
of 48 tetractyses or at
their
centres in the 16
faces → 168 roots of E_{8} that are not roots of
E_{6};

240 hexagonal yods
in 16 faces or lining tetractyses
inside
internal Type A
triangles → 240 nonzero roots of E_{8}.

12
(248+248=496) roots of
E_{8}×E_{8}. Figure 11 shows correspondences between the 248 yods and
the 248 roots of E_{8} (the properties refer to each half of the
16cell).
The hexagonal yod composition of the 16cell is isomorphic to the 8:72:168
root composition of each E_{8} in E_{8}×E_{8}, each half of the 16cell
containing 248 yods that correspond to the 248 roots in E_{8}. This is
amazing but not unexpected, for the 16cell is, after all, a holistic object, so that it
must display all the parameters that characterise such objects, one of which is the
number 496, whose origin is still so mysterious to superstring physics. Here we find it
embodied in the 16cell, along with the equally enigmatic number 137, whose reciprocal
physicists have long known is approximately equal to the finestructure constant, as discussed
earlier. Their mystery vanishes once it is understood that they belong to a class of numbers
that parameterise those geometrical objects (or, more generally, holistic systems) that
embody the divine archetypes, such as certain sacred geometries of the world's religions.
The numbers 137 and 496 appear in atomic and particle physics because they are signatures of
holistic systems. The problem of their appearance there exists only for the philosophies of
materialism and atheism. They refuse to countenance the possibility of the transcendental as
the ultimate source and cause of all things, and they have no room for the notion of
"holistic" as they confuse what they deem irrational with what is beyond the bounds of science
because it is both suprarational and the divine source of all rationality.
Figure 12
The identity:
384 + 248 = 632
has the following interpretation in terms of the various classes of
yods that make up the 632 yods surrounding the centre of the 16cell (see Fig.
12):
384 = [16
centres of faces + 4×2 hexagonal yods] + 168
hexagonal yods in faces in one half
+ [16 centres of faces + 4×2 hexagonal yods] +
168 hexagonal yods in faces in the other
half
= (24+168) + (24+168) = 192 +
192.
248 = 8 vertices + 24×3
hexagonal yods at centres of internal
tetractyses
+ [24 centres of internal Type A triangles + 24×3×2
hexagonal yods on sides of internal tetractyses]
= 8 + 72 + 168.
We find that the classes of yods in the 16cell
naturally make up a set of 384 yods (192 yods in each half) whose distribution in
4dimensional space conforms to the archetypal pattern of division of a holistic system
(see here).
We also find that the remaining 248 yods naturally divide into a set of eight yods
corresponding to the eight simple roots of E_{8}, a set of 72 hexagonal yods
corresponding to the 72 roots of E_{6} and a set of 168 yods
corresponding to the remaining 168 roots of E_{8}. There are too many
correspondences to be attributable to chance. Its embodiment of the dimension 248 of
E_{8}, the dimension 496 of E_{8}×E_{8} and the 192:192
pattern governing holistic systems found in sacred geometries amounts to convincing evidence
for the holistic character of the 16cell.
13
5. Noncoincidental, paranormallyderived
numbers
Let us now examine the 16cell composition of the compound of two
600cells that is the E_{8} Coxeter plane projection of the 4_{21} polytope.
Each 600cell is a compound of five 24cells, each 24cell being a compound of three
16cells. We found above that 56 yods line the 24 edges of a 16cell when its faces are
tetractyses. Hence, (3×56=168) yods line the (3×24=72) edges of three 16cells
in a 24cell and 1680 yods line the 720 edges of the 30 16cells in 10 24cells (840 per
five 24cells or 15 16cells in a 600cell). The number of boundary yods shaping
all the 16cells in two 600cells is the very number of circular turns in each helical whorl
of the UPA, which has an outer half comprising 840 turns and an inner half comprising 840
turns! Can anyone believe that this is merely a coincidence? Well, perhaps one could do
so with some justification if two 600cells had no connection whatsoever to the symmetry
groups describing superstring forces. But they do have a connection, for their
compound is the E_{8} Coxeter plane projection of the 240 vertices of the
4_{21} polytope, which mathematicians know represents the 240 roots of
E_{8}, the very symmetry group that appears (twice) in
E_{8}×E_{8} heterotic superstring theory! Moreover, the factorisation of 840
here is 5×168, where "5" denotes the 5 24cells. Compare this with Leadbeater's
account (summarised here), in which each whorl
of the UPA revolves five times around its axis of spin, making 5 halfrevolutions in its
outer spiralling and five halfrevolutions in its inner windings. It would seem that the
natural conclusion is that the outer and inner halves of the UPA are the string
manifestation of the E_{8} gauge charges associated with the 120 vertices of
each 600cell, the five halfrevolutions of the 10 whorls in each half being the string
manifestation of gauge charges associated with five 24cells. However, this cannot be the
entire explanation, because the UPA has 16800 turns in its 10 whorls, whereas the number of
boundary yods in the 30 16cells is only 1680 (the number of turns in
one whorl). The reader should understand that it is not being claimed here that
this correlation amounts to the whole story, for the dynamic connection between circularly
polarised, stringlike oscillations and the yod composition of 16cells has still to be
elucidated. It is therefore more accurate to say that the 1680 boundary yods
correspond to the 1680 turns that make up — not a whorl — but a
halfrevolution of all 10 whorls of the UPA. After all, its 10 whorls are what
constitute the E_{8}×E_{8} heterotic superstring, not a
single whorl. Despite two 600cells being a compound of 10 24cells, a single
24cell is not the basis in some still unexplained way for each whorl of the
UPA. After all, the 4_{21} polytope merely represents through its 240 vertices
the 240 roots of E_{8} in an 8dimensional space; it does not map as well the
3dimensional structure of the UPA. Rather, because each whorl twists 10 times
through an angle of 180°, every one of the 10 24cells is associated with a rotation through
a halfcircle of all 10 whorls of the UPA, each one having the topology of a 1torus. What
is significant here is that, just as the E_{8} Coxeter plane projection of the
4_{21} polytope is a compound of 10 24cells, so, too, the 10 whorls of the UPA
make 10 halfrevolutions around its axis of spin. Both systems are 10fold because they
are holistic objects. In fact, the UPA is the source in 10dimensional
spacetime of the E_{8}symmetric superstring force whose 240 gauge charges are
mapped by the 240 vertices of the 4_{21} polytope.
The crucial point to be noted is that numbers like 168, 336,
840, 1680, 3360 & 6720, which the author points out quantify the vibratory form
of what he claims are E_{8}×E_{8} heterotic superstrings, remoteviewed
over a century ago, naturally appear among the properties of objects known to be
mathematically connected to the symmetry group E_{8} describing the unified force
between these very superstrings! Moreover, they do so with a frequency that any reasonable
person can see renders coincidence highly improbable as the reason for their appearance. For
the 16‑cell, it is as obvious as the 168 sectors of its 56 Type A
triangles (32 triangular faces & 24 internal triangles). How credible is it to
attribute the manifestation of just this number to chance, let alone all the other gematria
number values in Table 1? How, otherwise, can we account for the equally natural
appearance in the 16cell of all the Kabbalistic numbers except in terms of
transcendental, mathematical design, as revealed through gematria applied to
the ancient Hebrew alphabet? There is no alternative, plausible explanation for
how all these numbers quantify the properties of the 16cell in such an uncontrived way.
They appear for the simple reason that — like E_{8}×E_{8} heterotic
superstrings — the objects that embody them are all examples of holistic
systems that express the divine archetypes — a concept which, of course, most
scientists deliberately exclude from consideration, although they do so for no better
reason than that they do not accept as a matter of principle that such systems can
exist. Let us be plain: there is only one sensible explanation for why certain
numbers allegedly obtained by remoteviewing subatomic particles turn up in the
4_{21} polytope, E_{8} and its exceptional subgroups so often as to make it
highly improbable that this could happen by chance. It is that the UPA described by
Leadbeater really is an E_{8}×E_{8} heterotic superstring. It is as
simple and as obvious as that! The alternative explanation requires all these numerous
appearances of numbers supposedly obtained by paranormal means to be miraculous coincidences
— a possibility which, statistically speaking, is so highly improbable that no one with any
common sense would take it seriously. Is it any more rational or more
reasonable to reject an ideologically problematic explanation, viz., UPAs are
remoteviewed E_{8}×E_{8} heterotic superstrings, in favour of one that is
more acceptable to one's materialistic philosophy but which requires having to believe that
comments 111 listed
14
above reveal nothing more significant than the chance appearance of numerous, Kabbalistic
numbers? Why, for the sake of scientific propriety, should we have to accept that this highly
implausible explanation is "better," scientifically speaking, when sceptics prefer it for
no better reason than that it avoids their having to accept that the paranormal ability to
remoteview subatomic particles exists because of overwhelming, irrefutable evidence that
it was successfully demonstrated by the Theosophists Annie Besant and C.W. Leadbeater? Why
should a personal ideology trump common sense as to what can, reasonably, be
attributed to coincidence?! Well, of course, it should not do so. But it will for
materialists who, faced with evidence that they cannot, plausibly, debunk, try to salvage
their cherished philosophy by any means they can find, whether it is credible or not. However,
a more sensible person unbiased by a priori beliefs or philosophical viewpoints such
as atheism or materialism will see the desperate appeal by sceptics to chance for what it
clearly is, viz., blatant confirmation bias exhibited by those who habitually and perversely
refuse to recognise any evidence — however strong — that discredits their ideological
aversions towards the paranormal and the transcendental and who resort to unreasonable
explanations requiring multiple miracles of coincidence to occur in order to provide themselves
with an excuse — however flimsy — to reject such evidence. In science, prima facie
evidence for a phenomenon — whatever the implications of its physical reality — must
always trump personal presuppositions about what is (and what is not) possible, otherwise
its assessment becomes subjective, hinging upon one's personal outlook or religious
belief, and ceases to be scientific. But that, of course, is not the logical way every
scientist thinks....
15
