20. The 24-cell embodies the holistic parameter
672
We found in #4 that, when they are Type A triangles, the faces of
the 24-cell have 672 corners, sides & triangles other than their 24 vertices & 96 edges:
Numbers of geometrical elements other than
vertices & edges surrounding the centre of the 24-cell
Corners
Sides
Triangles
Total
Faces
96
3×96=288
3×96=288
672
Interior
96
24 + 3×96 = 312
3×96=288
696
Total
192
600
576
1368
Rotating 24-cell
1368 yods are intrinsic to the (7+7) Type B
polygons enfolded in each overlapping Tree of Life; 12 black yods in each set of 7 polygons→12 sides
of48 internal Type A triangles in each half of the 24-cell; 672 coloured yods in each set of 7 enfolded polygons→672 geometrical
elements in each half of the 24-cell.
Given how this number is embodied in the various sacred geometries discussed in previous pages,
the fact that 672 extra points, lines & triangles are needed to construct the faces of the 24-cell from Type A
triangles is strong evidence for its holistic nature. This is confirmed by the fact that, including the internal
corners and sides of its internal triangles, 1368 geometrical elements in addition to its vertices & edges
surround the centre of the 24-cell, where
1368 = 3 + 5 + 7 + 9 +... + 73
The number (1368) of points, lines & triangles
other than vertices & edges needed to construct the 24-cell is the sum of the first
36 odd integers after
1 up to73.
The Type A dodecagon
has73 yods.
Star of David representation of the
number73 of Chokmah.
The number 73 of Chokmah ("Wisdom"), the second Sephirah of the Tree of
Life, determines the geometrical composition of the 24-cell as the last of the 36 odd
integers after 1 that add up to the number of geometrical elements in addition to its vertices & edges that are
needed to construct the interior and exterior of the 24-cell. ELOHA, the Godname of Geburah with number value
36, prescribes in this arithmetic way how many extra geometrical elements compose the 24-cell
constructed from Type A triangles. The number 73 is embodied in the Type A dodecagon as its
73 yods. It is also the number of yods needed to construct a Star of David/Sign of Vishnu
from 12 tetractyses. Including the centre of the 24-cell, the number of yods on the 24 internal sides of the 96
interior Type A triangles = 1 + 24×3 = 73. As confirmation of the holistic character of the
24-cell, these extra 1368 geometrical elements are symbolised in the inner Tree of Life by the 1368
yods that are intrinsic (i.e., belonging solely) to the (7+7) Type B polygons enfolded in successive
Trees of Life (see picture above). The 24 black yods that are either centres of the 14 polygons or yods lining the
vertical axes of the two hexagons that are shared with the outer Tree of Life correspond to the 24 internal sides
of the 96 Type A triangles inside the 24-cell formed by its 96 edges. It means that the (70+70) Type B polygons
enfolded in 10 Trees of Life have 13680 yods (see Fig. 3 in #2). This is the number of yods that line the 6720 edges of the
421 polytope with 240 vertices when its 60480 faces become tetractyses: 240 + 2×6720 = 13680. It
is also the number of corners, sides & triangles other than their 1200 vertices & edges that surround
the common centre of the ten 24-cells in the compound of two concentric, different sized 600-cells that is the
4-dimensional projection of the 421 polytope. The 10 Trees are the Tree of Life counterpart of
the ten 24-cells, the compound of the five 24-cells that form one 600-cell corresponding to the five Trees
mapping the uppermost five Sephiroth of the Tree of Life and the compound of the five other 24-cells forming the
second 600-cell corresponding to the five Trees that map the lowest five Sephiroth. The question that arises
from this correspondence is whether these Trees individually map a single half-revolution of all
10 whorls of the UPA or whether they each map the 10 half-revolutions of a single whorl.
Previously, we concluded that the 24-cells correspond to the half-revolutions of all 10 whorls because we
presented evidence that the two 600-cells generate the outer and inner halves of the UPA. In either case,
however, the 24-cell is corresponding to a set of 10 things — either all 10 whorls or 10 half-revolutions of a
single whorl. This means that we should expect the 24-cell to display a 10-fold pattern in either its
geometrical or yod composition (or in both). In the former case, it is trivial, for the 24-cell is composed of
24 vertices, (4×24=96) lines, (4×24=96) triangles & 24 octahedra, i.e., 10 sets of 24 geometrical
elements.
The 24-cell has 576 triangles with 600 sides other than edges. The number of sides &
triangles in the 24-cell other than edges = 600 + 576 = 1176 = 1 + 2 + 3 +...+ 48, where
48 is the number of Kokab, the Mundane Chakra of the Sephirah Hod. As
576 = 242 = 1 + 3 + 5 +...+ 47,
600 = 2 + 4 + 6 +...+ 48.
In other words, the number of sides in the 24-cell other than edges when its faces and interior
triangles are Type A triangles is the sum of the first 24 even integers and the number of its triangles is the sum
of the first 24 odd integers. This property illustrates the beautiful harmony between number and
sacred geometry. As 1176 = 7×168 and as a circular turn in a minor whorl of the
UPA comprises seven circular 2nd-order spirillae spaced evenly along its circumference (see here), this is the number of 2nd-order spirillae in every half-revolution of
a minor whorl. This is not true for major whorls because every nth-order spirilla in them is composed of 7.04
(n+1)th-order spirillae, not exactly 7 (n+1)th-order spirillae. The number of sides & triangles other than
edges in the ten 24-cells of the two 600-cells = 1176×10 = 7×1680, which is both the number of 2nd-order
spirillae in a minor whorl and the number of 1st-order spirillae in the seven minor whorls.
Including its vertices & edges, the total number of geometrical elements surrounding the
centre of the 24-cell constructed from Type A triangles = 24 + 96 + 1368 = 1488 =
31×48. Assigning the number 31 of the Godname EL ("God") to
the 48 corners of the seven separate, regular polygons of the inner Tree of Life generates
the number of points, lines & triangles making up the 24-cell. As the compound of five 24-cells, there are
(5×1488=7440=155×48=31×240) geometrical elements surrounding the
common centre of the 600-cell. In other words, assigning the number 31 of EL, the Godname of
Chesed, to the 240 yods in the 1-tree that are not "Sephirothic levels" generates the number of geometrical
elements in the five 24-cells whose compound is the 600-cell:
1-tree representation of the 7440 geometrical elements in five
24-cells whose compound is the 600-cell.
This is how EL prescribes the number of geometrical elements making up these five 24-cells.
As 2×7440 = 14880 = 62×240, assigning the number 62 of
Tzadkiel, the Archangel of Chesed, to these 240 yods generates the number of geometrical elements in the
compound of two 600-cells whose 240 vertices are the 4-dimensional projection of the 240 vertices of the
421 polytope. Both the Godname and Archangel of Chesed determine the geometry of the two 600-cells!
As 62×240 = 620×24 = 620×1×2×3×4, the number of geometrical
elements in the ten 24-cells is 24 times the number 620 of Kether, the number of hexagonal
yods in the 10-sided decagon:
Decagonal representation of the 14880 geometrical elements in the ten 24-cells in the compound of
two 600-cells.
Weighted with 24 (the sum of the first 4 odd integers), the 620 hexagonal
yods in the decagon with 2nd-order tetractyses as sectors generate the number of geometrical elements in the ten
24-cells of the compound of two 600-cells. Assigning the number of Kether to the 24 vertices of the 24-cell also
generates this number. As 14880 = 30×496, the 30 yods lining the sides of a pentagram and weighted
with the number 496 of Malkuth generate the total number of geometrical elements in the ten
24-cells:
496
=
The 30 yods lining a pentagram and weighted with the
number496 of Malkuth generate the total number (14880) of geometrical elements
that surround the centre of the ten 24-cells in
the compound of two 600-cells.
496 is the sum of 16 cubes of 1,
2, 3 & 4.
496 is the sum of the squares of 1, 3, 6, 9, 12
&15.
496 is the sum
of the first31 integers arranged at the
centre and on the boundary of a pentagram.
This demonstrates the fundamental connection between the pentagram and the geometrical
composition of the ten 24-cells within the two 600-cells whose compound is the 4-dimensional projection of the 240
vertices of the 421 polytope defining the root vectors of E8.
As 14880 = 30×496 = 60×248, assigning the number
248 of Raziel, the Archangel of Chokmah, to the 60 hexagonal yods in either the Tree
of Life or the dodecagon generates the total number of geometrical elements surrounding the centre of the ten
24-cells:
Weighted with the number 248 of Raziel, the Archangel of
Chokmah, the 60 hexagonal yods in the Tree of Life and in the dodecagon generate the number (14880) of
geometrical elements in the ten 24-cells in the compound of two 600-cells.
The number of geometrical elements other than vertices in the faces of the 24-cell = 96 + 672 =
768 (see table above). As
Each half of the 24-cell with 48 edges and 336 other geometrical elements
conforms to the division: 384 = 48 + 336 that is characteristic of holistic systems (see
The holistic pattern). The number of geometrical elements other than
vertices in the faces of the five 24-cells (the compound of which is the 600-cell) = 5×768 = 3840 =
10×(22+32+42+52+62+72+82+92+10
2). This can be represented by a 10-fold array of the squares of the integers 2-10, each set of nine squares
representing the number of geometrical elements other than vertices in the 48 faces of half a
24-cell:
The sum of the 10-fold array of the squares of the integers 2-10 is
the number (3840) of geometrical elements other than vertices in the faces of the five
24-cells, the compound of which is the 600-cell. The blue pentagram denotes the upper halves of the
five 24-cells and the red pentagram denotes their lower halves.
This demonstrates how the Decad determines the geometrical composition of the 600-cell as the
compound of five 24-cells.
Given that the 4-dimensional projection of the vertices of the 421 polytope is
the compound of two 600-cells, each of which is the compound of five 24-cells, it is amazing — no,
wonderful — that the number of edges (6720) of the 421 polytope is the same as the
number of corners, sides & triangles in addition to their 240 vertices and 960 edges that are required to
construct the 960 faces of the ten 24-cells making up the two 600-cells. As we have shown in previous pages that
the number 672 is a global parameter of sacred geometries, the conclusion is inescapable that the 24-cell and the
421 polytope are holistic systems described by the universal pattern embodied in these sacred
geometries, thereby indicating that the E8×E8 heterotic superstring is its microscopic
manifestation, the UPA being its subquark state.
Conclusion What has emerged from this
analysis of the 24-cell is its holistic character. The fact that it is composed of 1368 geometrical elements other
than its vertices & edges when its faces are Type A triangles and the fact that this is the number of yods
intrinsic to the inner form of successive, overlapping Trees of Life are astounding pieces of evidence of this
character that cannot be disputed. Whether the 24-cell is the geometrical basis of each whorl of the UPA or (more
plausibly) each of the ten half-revolutions of all 10 whorls, the fact remains that the faces of the 24-cell
contain 672 geometrical elements other than vertices & edges and that this number is a parameter of sacred
geometries, being found in the first four Platonic solids, the 2-dimensional Sri Yantra and the disdyakis
triacontahedron. This, alone, bestows archetypal significance to the 24-cell. That the 421 polytope
has precisely ten times as many edges and that the 4-dimensional projection of its 240 vertices
determining the 240 root vectors of E8 coincide with the vertices of two 600-cells, each a compound
of five 24-cells, is what rigorously establishes the clear link between sacred geometries and
E8×E8 heterotic superstrings. That the UPA paranormally described over a century ago by
Besant & Leadbeater has 6720 plane wave components in every revolution of its ten whorls is what
rigorously establishes its clear link to E8×E8 heterotic superstrings. For
sceptics to dismiss this connection as the result of chance is to replace the extraordinary power of micro-psi with
a hypothetical coincidence so improbable as to amount to a miracle. It is not a more sensible or more rational,
alternative explanation. Instead, let us see their invocation of chance for what it is, namely, an unrealistic,
untenable way that debunkers often resort to in desperation in order to protect at all costs their ideological
disbelief in anything that is paranormal.
