The Sacred Geometry of the Platonic Solids

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Ok GLPers this could be the KEY, to the nature of reality, the theory of everything, the most secret occult knowledge that has been hidden from you after the destruction of the school of knowledge ran by pythagoras. (its also possibly the longest sigle post ever on GLP, so be patient)

By Stephen M.Phillips

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1. Introduction
What is ‘sacred geometry’? Whether it refers to Stonehenge, the Egyptian pyramids, Gothic
cathedrals or Tibetan mandalas, this notion is problematic in such contexts because whether one
should regard the geometry of ancient monuments or religious artefacts as sacred depends upon
one's own religious beliefs (or lack of them). Indeed, whether the word ‘sacred’ should be
attributed to anything at all is contingent upon whether one has such beliefs. What is sacred to
Christians or ancient Egyptians is usually not so to Jews or Hindus. However, true sacredness
transcends cultural relativism and religious differences; it must exist in its own right instead of
being merely an attribute projected onto some object by a particular religious frame of mind.
Geometry that is sacred only in the eyes of the believer cannot be truly such. So what makes a
geometrical design sacred? Indeed, does sacred geometry actually exist in this universal sense?
Supposing that God exists (otherwise the word ‘sacred’ is meaningless and such geometry


becomes mere religious doodles), sacred geometry would be that which embodies in a geometrical
representation not just symbolic but quantitative aspects of cosmic reality (spiritual as well as
physical) designed by the mind of God. It must be neither man-made symbolism, such as a cross or
a six-pointed star, nor just a metaphor of universal order expressed in the doctrinal ideas of a
particular religion or spiritual philosophy, which followers of another religion may not accept.
Instead, sacred geometry must be a geometrical pattern or form depicting some quantitative aspect
of the divine design of reality transcending all religious metaphors. It would be sacred because it
encapsulates what actually exists — whether this is known or unknown — and not merely what
fallible human beings speculate about in their mythologies and theologies and incorporate into the
TThhee SSaaccr rreedd GGeeoommeet ttr rryy oof ff t tthhee PPl llaat ttoonni iicc SSool lli iiddss
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architecture of their churches. True sacred geometry does not embody human ideas about the
nature of reality or God; it necessarily transcends them. But how then would we discern the divine
design of reality, which this sacred geometry is supposed to express? Indeed, how would we
recognise its sacredness? For those mathematicians or theoretical physicists who are Platonists,
one of the signatures of mathematical truth is its beauty. However, sacred geometry has to possess
more than just beautiful properties that no human mind could have artfully fabricated. Beauty may
be a necessary attribute but it is not sufficient. Although what is sacred truth must be beautiful,
what is beautiful may not even be true, let alone of divine origin. Sacred geometry must have such
extraordinary properties (and so many of them as to discount the possibility of their being due to
chance) that they can only indicate the existence of transcendental intelligence as well as artistry
behind the sacred object. There must be no rational, conventional explanation for its possession of
so many mathematically miraculous properties. Given this stringent requirement, how many
examples of sacred geometry discussed in so-called ‘New Age’ books would meet this criterion?!
This article will prove that the five Platonic solids (called in mathematics the five ‘regular
polyhedra’ because they are the only 3-dimensional shapes with equal sides) have far deeper
significance than what hitherto has been known to mathematicians. It will show that their
geometry is sacred in the above sense — that is, to say, not because God constructed the world out
of these forms (as the ancient Greeks believed) but because they collectively embody the Divine
blueprint underlying all levels of existence, including the physical universe.
_____________________________________________________________​

The Tree of Life
The Tree of Life (Fig. 1) has claim to be sacred geometry par excellence because, according to
Kabbalah (the Jewish mystical tradition), it is God’s blueprint for Creation. It portrays the ten
divine qualities, or Sephiroth (sing: Sephirah), as spheres arranged on three Pillars and connected
by 22 Paths. The uppermost Supernal Triad of Kether, Chokmah and Binah signify the triple
Godhead outside Creation. The seven Sephiroth of Construction, Chesed, Geburah, Netzach, Hod,
Yesod and Malkuth, represent aspects of God’s immanence in Creation. The ‘Gulf’ or ‘Abyss’ of
Daath separates them from the Supernal Triad, which is not a Sephirah. Malkuth, the lowest
Sephirah of Construction, signifies the material manifestation of the Tree of Life, whether a
subatomic particle, the human body or the whole universe. The Sephiroth manifest in four Worlds
that are stages in the descent of the Divine Life into matter. They correspond to the traditional
Christian divisions: Spirit, soul, psyche and body. The Godname of a Sephirah is its essence or
expression in the highest World of Atziluth (the Archetypal World). Through their many
elaborations by the human mind, it is mostly the Godnames that became anthropomorphized into
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the gods and goddesses of ancient mythologies. (Judaism and Christianity focussed on YAHWEH,
the Godname of Chokmah, although other Godnames appear in the Old Testament).
By means of gematria, wherein numbers are assigned to the 22 letters of the Hebrew alphabet so
that hidden meanings in the texts of scriptures may be extracted from phrases and sentences,
Hebrew words acquire number values that are the sum of their letter numbers. In my book The
Image of God in Matter (1) I proved that the number values of the Godnames of the ten Sephiroth:
SEPHIRAH
KETHER
CHOKMAH
BINAH
CHESED
GEBURAH
TIPHARETH
NETZACH
HOD
YESOD
MALKUTH

GODNAME

EHYEH
YAH, YAHWEH
ELOHIM
EL
ELOHA
YAHWEH ELOHIM
YAHWEH SABAOTH
ELOHIM SABAOTH
EL CHAI
ADONAI

NUMBER VALUE

21
15, 26
50
31
36
26 + 50 = 76
26 + 103 = 129
50 + 103 = 153
49
65

(All Godname numbers appearing in the text are written in bold-face type)

prescribe — geometrically as well as arithmetically — the Tree of Life topography of all possible
levels of reality, including the 4-dimensional space-time domain of physical (brain) consciousness.
Godname numbers also define the dynamics and structure of the basic building blocks of matter,
which my book proves is what particle physicists call a ‘superstring’ (see Article 2). The reason
for this is that the superstring is the microscopic manifestation of the Tree of Life blueprint, whose
geometry is defined by Godname numbers. The Godname numbers associated with the Sephiroth
are potent ‘master numbers’ quantifying the Tree of Life nature of reality, this prescription
becoming ever more concrete, further down the Tree the Sephirah is located.
The presence of all ten Godname numbers defining properties of a geometrical pattern or set of
geometrical objects such as the Platonic solids is a necessary and sufficient condition for it to
constitute ‘sacred geometry.’ However, a mathematical pattern or object possesses sacred
geometry not just because it embodies one or more numbers of cosmic significance (although this
invariably turns out to be the case) but because the ten Godname numbers defines its structure. In
the case of the 3-dimensional Tree of Life, my book shows that its projection on a plane is
minimally generated from 21 points arranged in rows of 1, 2, 3, 4, 5 & 6 points, where 21 is the
number value of EHYEH, the Godname of Kether. The Tree of Life consists of 10 corners of 16
triangles assembled by the joining of 34 of their 48 sides to create 22 sides or ‘Paths,’ of which 12
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are the edges of two tetrahedra. 26 sides disappear in their joining, and this is the number value of
YAHWEH, whilst 50 (the number of corners, edges, triangles and tetrahedra comprising the Tree,
i.e., its geometrical elements) is the number value of ELOHIM, the Godname of Binah expressing
the most abstract archetypes about divine form. How the remaining Godnames prescribe the
geometry of the Tree requires consideration of the tetractys, which is discussed next.
___________________________________________________________

3. The Tetractys
At the heart of the Pythagorean philosophy based upon the power of numbers was the tetractys, a
triangular array of ten dots, or ’yods’ (2), arranged in four rows (Fig. 2). With no insight into what
Pythagoras taught other than what is provided by the few remnants of his teachings distorted by
later generations of commentators, scholars of ancient Greek mathematics assume that the tetractys
represents the numbers 1, 2, 3 & 4 summing to 10, which, as their sacred Decad, the Pythagoreans
regarded as the perfect number. But it meant to them far more than this. According to
H.P. Blavatsky: “The ten Points inscribed within that ‘Pythagorean triangle’ are worth all the
theogonies and angelologies ever emanated from the theological brain” (3). Partly the reason for
this is that the tetractys signifies the same as the Tree of Life, namely, the ten-fold nature of God
(Fig. 3 shows this equivalence in detail). But — more important for the present discussion — the
main reason for the Pythagorean reverence for the tetractys is that numbers expressing information
about the nature of reality (space-time and beyond) manifest in objects possessing sacred geometry
when they are re-assembled from tetractyses. In other words, the tetractys is the key that unlocks
information about reality encoded in sacred geometry. This is the real reason why the followers of
Pythagoras esteemed the symbol at the heart of their master’s teachings, swearing by the oath:
I swear by the discoverer of the Tetractys,
Which is the spring of all our wisdom,
The perennial root of Nature’s font (4).
The numbers generated by triangulating objects with sacred geometry and transforming the
resulting triangles into tetractyses are of three types: 1. number of yods belonging to tetractyses, 2.
number of yods at corners of tetractyses (as Figure 3 indicates, these symbolise the Supernal Triad
of Kether, Chokmah and Binah), and 3. number of yods arranged at the corners and centre of a
hexagon within each tetractys (symbolising the seven Sephiroth of Construction, these will be
called ‘hexagonal yods’).
________________________________________________________

4. The inner form of the Tree of Life
The 3-dimensional Tree of Life with its 16 triangles turned into tetractyses comprises 70 yods. But
my book reveals for the first time that the Tree has an inner form made up of two identical sets of
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seven enfolded, regular polygons (triangle, square, pentagon, hexagon, octagon, decagon and
dodecagon), which share what I call the ‘root edge’ (Fig. 4). This extends between Daath and
Tiphareth, the centre of the Tree of Life in both a physical and a metaphysical sense. Remarkably,
these 14 enfolded, regular polygons have 70 corners — the same number as the number of yods in
the outer form of the Tree when assembled from tetractyses. This is not coincidental but the
manifestation of a hidden regularity or pattern of order and design. Just as a DNA molecule
encodes the development of a living organism from a single cell, so this inner Tree is found to
encode in a geometrical way the self-replication of its outer form to map all possible levels or
states of consciousness attainable by evolution. The resulting ‘Cosmic Tree of Life’ (CTOL)
consists of 91 overlapping Trees of Life, where
91 = 12 + 22 + 32 + 42 + 52 + 62.
They correspond to the 91 subplanes of the seven cosmic planes described in Theosophical
literature (the cosmic physical plane comprises seven planes with 49 subplanes (49 is the number
value of the Godname EL ChAI assigned to Yesod) and the six cosmic superphysical planes
comprise 42 subplanes — see Article 1). Calling the Sephirah of each Tree a ‘Sephirothic Level’
(SL), CTOL is found to consist of 550 SLs, where
55
55 55
550 = 55 55 55
55 55 55 55
and
10
9 8
55 = 7 6 5 = 12 + 22 + 32 + 42 + 52.
4 3 2 1
This is the first indication of the central role played by the Pythagorean tetractys in mathematically
representing the properties of CTOL.
As these equations indicate, this map of all levels of being is a geometrical object the beauty of
whose mathematical proportions reflects its divine design. As another example of its beautiful
properties, this ‘Jacob’s Ladder’ consists of 3108 triangles and their corners and sides, where
3108 = 14 + 34 + 54 + 74.
My book discusses many other examples. With their triangles turned into tetractyses, the lowest 49
Trees of CTOL mapping the cosmic physical plane have 2480 yods, which is the number of yods
in 248 tetractyses. Compare this with the prediction made by superstring theory that the unified
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force between superstrings in ordinary matter is transmitted by 248 particles (so-called ‘gauge
bosons’). Each is described by a mathematical function called a ‘gauge field’ having ten
independent components because superstring theory predicts that space-time has ten dimensions
(the components are measured along the directions of these dimensions). This demonstrates that
the number characterising the kind of perfect, unbroken, mathematical symmetry of superstring
forces described by the so-called ‘E8 group’ is encoded in the map of the cosmic physical plane,
the ten yods of the equivalent 248 tetractyses denoting the ten components of each of the 248
gauge fields. Another 248 particles are predicted to mediate interactions between superstrings
making up what theoretical physicists call ‘shadow matter.’ This is an as yet undetected, invisible
kind of matter that may comprise some of the ‘dark matter’ believed by astronomers to make up
about 90% of the mass of the universe. Its invisibility is due to the prediction that only the force of
gravity acts between superstrings of ordinary matter and shadow matter. Most of these particles
play no part in the physics of the cooled-down universe today because they are too massive to be
created by the typical energies with which subatomic particles interact. The lighter particles of
shadow matter are predicted to form a kind of parallel universe that co-exists with the one visible
to human sight but which is ever beyond the ability of the five human senses to detect.
The superstring parameters 248 and (248 + 248 = 496) are also encoded in the inner form of the
Tree of Life whose 49-fold replication maps the cosmic physical plane. (It is remarkable and no
coincidence that 496 is the number value of Malkuth, the tenth Sephirah, signifying the physical
universe. My book explains why this is so). The seven separate, regular polygons of the inner Tree
have 48 corners (Fig. 5). Together with the two endpoints of the root edge considered separately
(which formally correspond to corners) they constitute 50 corners. This is how the Godname
ELOHIM with number value 50 prescribes this pattern of sacred geometry. With their 48
triangular sectors turned into tetractyses, the seven polygons have 55 corners of tetractyses and
240 hexagonal yods, so that the two sets of polygons contain 480 hexagonal yods. But the numbers
240 and 480 are part of the group-theoretical description of the mathematical symmetry exhibited
by the unified superstring force, for 240 is the number of non-zero roots of the gauge symmetry
group E8 and 480 is the number of non-zero roots of the superstring symmetry group E8×E8! So we
see that a parameter characterising the group-mathematics of the unified superstring force is
encoded in the inner form of the Tree of Life. This is just one example among many discussed in
my book of how its sacred geometry incorporates the mathematical structure of superstring theory
and embodies numbers of significance to particle physics.
Inspection of the list of Godname numbers given above shows that 240 is the sum:
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240 = 21 + 26 + 50 + 31 + 36 + 76
of the Godname numbers of the first six Sephiroth of the Tree of Life, which are located at the
corners of a hexagon or, equivalently, a six-pointed star. The encoding of the number 240 in the
inner form of the Tree of Life has its counterpart in the 49 trees representing the cosmic physical
plane. As we have seen, they contain 2480 yods. The lowest tree in CTOL contains 80 yods. There
are therefore 2400 yods above the lowest tree in this section of CTOL, i.e., the yods in 240
tetractyses. The 248 roots of E8 consist of eight zero roots and 240 non-zero roots. This is
paralleled in, respectively, the eight and 240 tetractyses whose yod populations are the number of
yods in the lowest tree and the 48 trees above it mapping the cosmic physical plane.
It is also remarkable that there are 67 yods below Binah of the lowest tree and 73 yods up to
Chokmah (Fig. 6), for the former is the number value of the Hebrew word ‘Binah’ and the latter is
the number value of the Hebrew word ‘Chokmah.’ This obviously cannot be coincidence but
demonstrates a profound connection between the names of the Sephiroth (as well as their
Godnames) and numbers generated in the Tree of Life when its triangles are changed into
tetractyses. Correspondence between the inner and outer forms of the tree is also demonstrated by:
1. the seven separate regular polygons making up the former have 48 corners, whilst there are 48
yods in the lowest tree of CTOL up to Chesed, the first Sephirah of Construction (Fig. 5),
2. both sets of enfolded polygons have 70 corners, whilst the Tree of Life contains 70 yods
(Fig. 7);
3. both sets of enfolded polygons have 80 corners of their 70 tetractys-converted sectors, whilst
the lowest tree in CTOL contains 80 yods (Fig. 8).
Other remarkable correspondences are discussed in my book.

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5. The Platonic solids
Plato propounded in his Timaeus the Pythagorean doctrine that particles of the four Elements Fire,
Air, Earth and Water had the shapes of, respectively, the tetrahedron, octahedron, cube and
icosahedron (Fig. 9). The Pythagoreans associated the fifth regular polyhedron, the dodecahedron,
with the cosmic sphere, which came later to be identified with the fifth Element, Aether. But the
five regular polyhedra have a more profound significance than this — one that truly justifies their
geometry being called ‘sacred.’ In order to uncover it, one must regard the tetractys as the
universal template out of which all sacred geometry is built. We found in the last section how,
when CTOL was considered as assembled from tetractyses, a number of significance to
superstring theory appeared in a region of CTOL having ‘physical meaning’ (the word ‘physical’
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is used in its widest possible sense, pertaining not only to space-time, which Article 2 indicates is
mapped by the lowest seven trees of CTOL, but to its cosmic counterpart — the lowest 49 trees).
Finding such a scientific parameter of cosmic relevance is evidence of the sacred geometry of the
Tree of Life. But it requires reconstructing the Tree of Life from tetractyses so as to make manifest
the information encoded in its geometry. Likewise, to uncover information encoded in the five
Platonic solids, we must imagine their faces constructed from tetractyses. A triangular face
transforms into three tetractyses, a square face is made up of four tetractyses and a pentagonal face
is composed of five tetractyses (Fig. 10). Notice that 5, 4 & 3 are the sizes of the well-known,
right-angled triangle illustrating Pythagoras’ Theorem. Notice also that the centre of a solid and
any two of its adjacent corners form a triangle in its interior. Each polyhedron is built also from
such triangles in its interior. A Platonic solid will be called ‘type A’ if its internal triangles are
turned into single tetractyses and ‘type B’ if they are divided into three tetractyses (internal
triangles divided into three tetractyses will also be called ‘type B’ in order to distinguish them
from the internal triangles of type A solids).
Table 1 displays the numbers of corners (C) edges (E) and faces (F) of the five Platonic solids and
the yod populations of their faces and interiors. The former are related by Euler’s equation for a
regular polyhedron:
C – E + F = 2.
Inspection of the Godname numbers listed earlier tells us that the most obvious sign that the
Platonic solids constitute sacred geometry is that they have 50 corners and 50 faces, i.e., their
shapes are collectively prescribed by the number value of ELOHIM, the Godname of Binah, which
is the third member of the Supernal Triad in which the possibility of form— the abstract notion of
spatial relationship — first arises. Indeed, I provide in my book many examples of how
geometrical forms or patterns of numbers expressing parameters of superstring theory or the Tree
of Life are always defined arithmetically by the ten Godname numbers. Below are listed examples
of how Godname numbers define the geometrical composition and yod population of each solid,
together with other properties illustrating their correspondence with the outer and inner forms of
the Tree of Life:
TETRAHEDRON
21 yods other than polyhedral corners on sides of 6 internal tetractyses (type A);
26 yods in 6 internal tetractyses other than polyhedral corners surround centre (type A);
50 internal yods on sides of 18 internal tetractyses surround centre (type B). 55 yods on sides
of 18 internal triangles;
31 yods in 6 internal tetractyses (type A);
36 hexagonal yods in 18 sides of 12 tetractyses in 4 faces;
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44 yods on sides of 12 tetractyses in faces, of which 36 are hexagonal; also 44 corners and
centres of 30 tetractyses of type B solid surrounding centre; 80 hexagonal yods on 40 sides
of 30 tetractyses of type B solid;
76 yods other than 48 hexagonal yods in faces surround centre (type B). 50 are on edges of 18
internal tetractyses, 26 are centres of internal tetractyses and corners of tetractyses in faces:
76 = 26 + 50 = YAHWEH ELOHIM;
48 hexagonal yods in faces  48 corners of 7 separate polygons  48 yods up to Chesed of
Tree of Life. The sums of the numbers of their corners = (3+4+5) + (6+8+10) + 12 = 12 +
24 + 12  12 hexagonal yods on solid edges, 24 hexagonal yods on tetractys sides and 12
tetractys centres;
70 yods surround centre (type A)  70 yods of Tree of Life; 23 internal yods or tetractys
corners  23 yods above Chesed of Tree of Life;
Transformation into type A tetrahedron generates 67 yods; transformation into type B generates 73
yods other than 48 hexagonal yods in faces (also, 73 internal yods and corners of solid); 120 new
yods surround centre, where 120 = 22 + 42 + 62 + 82; 95 yods on edges of 30 tetractyses (type B),
of which 91 are generated by transformation;
49 corners, sides and tetractyses (type A);
49 polyhedral corners and internal yods other than centres and corners of 18 internal
tetractyses;
65 yods on faces or on sides of 6 internal tetractyses (type A);
26 corners and sides of 12 tetractyses in 4 faces;
31 corners and sides of 18 tetractyses in type A.
OCTAHEDRON
26 corners, edges and faces of polyhedron;
15 corners of 36 tetractyses (type A); 26 corners of 60 tetractyses surround centre (type B);
36 sides of 24 tetractyses in 8 faces;
50 corners and sides of 24 tetractyses in 8 faces;
129 yods other than polyhedral corners (type A);
96 hexagonal yods in faces  96 corners of (7+7) separate polygons;
14 corners of 24 tetractyses in faces  14 centres of (7+7) separate polygons;
110 yods in faces  110 corners of tetractyses in (7+7) separate polygons.
CUBE
26 corners, edges and faces of polyhedron;
36 sides of 24 tetractyses in faces;
50 corners and sides of 24 tetractyses in faces;
15 corners of 36 tetractyses (type A); 26 corners surround centre (type B);
96 hexagonal yods in faces;
247 yods (type B) (7 in centres of faces and centre of cube)  240 hexagonal yods and 7
centres of 7 polygons;
220 hexagonal yods (type B), where 220 = 22 + 42 + 62 + 82 + 102.
ICOSAHEDRON
240 hexagonal yods in faces;
10
50 corners of 150 (15×10) tetractyses other than polyhedral corners surround centre (type B);
260 (26×10) yods in 20 faces other than polyhedral corners;
597 yods (type B)  597 hexagonal yods outside root edge of 7 enfolded, type B polygons.
FIRST 4 PLATONIC SOLIDS
128 corners, edges and faces, where 128 = 21 + 26 + 50 + 31;
Number of yods on edges of 4 solids = 150 = 15×10;
Average number of yods (including centres) in 1st 4 solids (type A) = 168;
Average number of yods inside 1st 4 solids (type A) = 31;
Average number of yods in faces of 1st 4 solids = 137 (1370 yods in (7+7) enfolded polygons with
type B triangular sectors);
150 (15×10) yods on polyhedral edges, of which 30 are corners and 120 are hexagonal, where 30 =
12 + 22 + 32 + 42, and 120 = 22 + 42 + 62 + 82;
1st 4 solids have 248 corners and sides of 120 triangles in faces, where 120 = 22 + 42 + 62 + 82.
DODECAHEDRON
50 corners and edges;
70 yods in interior surround centre;
Number of yods = 343 = 73 (type A). This is the sum of the 26 combinations of the first ten
integers arranged in a tetractys:
sum of combinations
1 1
2 3 10
4 5 6 60
7 8 9 10 272
TOTAL = 343
Numbers 1, 70 and 272 denote, respectively, the yod at the centre, the number of internal yods
surrounding centre and number of yods in faces;
Number of internal hexagonal yods (type B) = 310 = 31×10;
Number of hexagonal yods (type A) = 310 = 31×10;
Number of hexagonal yods (type B) = 550;
Number of yods without faces divided into tetractyses = 91;
Number of yods surrounding centre other than polyhedral corners (type B) = 592 number of
yods in two sets of 7 separate polygons + root edge other than its corners → number of yods inside
type B triangles of Tree of Life;
260 (26×10) yods in 12 faces other than their centres  260 yods outside root edge in 7 enfolded
polygons;
251 yods of internal (type B) triangles other than centres of tetractyses  251 yods in lowest tree
of CTOL with type B triangles  251 yods outside root edge of 7 enfolded polygons not either
Sephirothic corners of Tree of Life or centres of polygons.
70
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COLLECTIVE PROPERTIES OF 5 SOLIDS
50 corners and 50 faces;
720 hexagonal yods in faces  720 yods surrounding centres of 7 separate type B polygons;
910 hexagonal yods  91 trees of CTOL; 26-tree has 910 corners of enfolded polygons unshared
with hexagon enfolded in 27th tree;
Number of yods surrounding centres (type A) = 1010, where 101 = 26th prime number = 50th odd
integer after 1;
Number of yods surrounding centres (type B) = 1820 = 70×26;
Number of yods in 5 solids = 1825
D2
D3 D4
= D5 D6 D7
D8 D9 D10 D11,
where Dn is the nth decagonal number (D1 = 1);
Average number of yods on edges of solids and internal triangles = 67 = number value of Binah;
Average number of yods other than corners on edges of solids = 36;
Average number of yods on edges of internal triangles = 31;
Average number of internal yods on edges of internal triangles = 21;
5 solids have 550 corners, edges and triangles in their 50 faces;
Number of yods in 5 solids not corners of type B tetractyses other than polyhedral corners = 1680.
NESTED SOLIDS
First 4 type B solids have 1080 hexagonal yods surrounding their common centre; 1081 = number
of Tiphareth;
31 polyhedral corners + centre of first 4 solids;
Number of yods surrounding their centre on sides of internal triangles and on edges of 4 solids =
210 = 21×10;
Number of corners of tetractyses in first 4 type B solids = 129;
Number of corners of triangles inside 5 type B solids = 91;
1820 yods surround the centre of 5 solids; this is the number of yods in 36 overlapping trees and in
26 separate trees;
1680 yods (including 50 polyhedral corners) surround the centres of 5 solids other than corners of
tetractyses.

_________________________________________________________

6. Discussion
It is remarkable (as well as indicative of their supernatural/archetypal design) that the four Platonic
solids thought by the ancient Greeks to be the shapes of particles of Earth, Water, Air and Fire are
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made up of 128 polyhedral corners, edges and faces, where
128 = 21 + 26 + 50 + 31
is the sum of the Godname numbers of the first four Sephiroth.
Let us now compare the hexagonal yod populations of these solids with properties of the outer and
inner forms of the Tree of Life shown in, respectively, Figures 1 and 4:
1. The surface of the tetrahedron has 48 hexagonal yods. This is the number of corners of the
seven separate, regular polygons (as well as the number of their tetractyses) constituting the
inner form of the Tree. 48 is also the number of corners, sides and triangles in its outer form
and the number of yods in the Tree of Life up to the horizontal Path connecting Geburah and
Chesed, i.e., the part of the Tree spanned by the seven Sephiroth of Construction contains 48
yods. The tetrahedron embodies the minimum number of geometrical elements defining both
the outer and the inner forms of the Tree;
2. The faces of the octahedron and cube each contain 96 hexagonal yods. This is the number of
corners of the two sets of polygons;
3. The surfaces of the tetrahedron, octahedron and cube comprise 240 hexagonal yods, which
compares with the 240 hexagonal yods in the 48 tetractyses of the seven polygons;
4. The surface of the icosahedron has 240 hexagonal yods, which compares with the 240
hexagonal yods of the 48 tetractyses in the second set of seven polygons.
5. The surfaces of the four Platonic solids, whose shapes the ancient Greeks thought were those
of the particles of the four Elements, contain (240 + 240 = 480) hexagonal yods, which
compares with the 480 hexagonal yods in both sets of seven polygons. This correspondence
with the inner form of the Tree and their prescription by Godname numbers are evidence of the
sacred geometry of these solids; they embody numbers of universal physical significance,
namely the (240 + 240 = 480) gauge fields corresponding to the (240 + 240 = 480) non-zero
roots of E8×E8 mediating the unified superstring force. The mirror symmetry of the inner form
of the Tree explains why these 480 gauge fields are divided into two similar sets of 240. It
confirms the basic Pythagorean insight that the tetrahedron, octahedron, cube and icosahedron
defined the physics of the cosmos, if not the sense of the teaching attributed to them.
The interior of the type A dodecahedron has 70 yods surrounding its centre. As it has 20
polyhedral corners, (1 + 70 + 20 = 91) yods are needed to construct its shape from tetractyses.
Type B comprises 550 hexagonal yods. Symbolising for the Pythagoreans the cosmic sphere, the
dodecahedron therefore embodies the number (91) of Trees in CTOL and the number (550) of its
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SLs, i.e., it represents the cosmic version of the Tree of Life. The Godname ELOHIM prescribes
this representation because the dodecahedron has 50 corners and edges.
More evidence for the sacred geometry of the five Platonic solids comes from considering their
yod populations. Remarkably, in view of the central significance of the number 10 for the
Pythagoreans, who regarded it as ‘all perfect,’ the number of yods inside the five type B solids
surrounding their centres is 1000, i.e., 103. The total number of yods surrounding their centres is
1820. As 1820 = 26×70, they have as many yods as 26 separate Trees of Life, showing how the
Godname YAHWEH with number value 26 prescribes their yod population. It can be proved that
1820 is the number of yods in 36 overlapping Trees with their triangles turned into tetractyses,
illustrating how ELOHA, the Godname of Geburah with number value 36, also prescribes this
number. 1820 is the number of yods surrounding the common centre of five type B Platonic solids
nested one inside another. Furthermore, they have the property that their 50 faces comprise 550
corners, sides and tetractys triangles, i.e., 550 geometrical elements define their shapes, whilst
their interiors have 91 corners of tetractyses. This is remarkable in view of the fact that the 91
Trees of CTOL have 550 SLs. Their embodiment of the basic numbers characterising the Tree of
Life map of all levels of reality is evidence of the sacred geometry of the five Platonic solids.
Perhaps the most spectacular property of the five Platonic solids with Type B internal triangles is
the following: there are 180 tetractyses in their 50 faces and 270 internal tetractyses, a total of 450
tetractyses. Of the 1820 yods surrounding the centres of the solids, 450 yods are centres of
tetractyses, leaving 1370 yods lining their edges. This is the number of yods in the inner form of
the Tree of Life with Type B triangles as the sectors of the 14 enfolded polygons (Fig. 11))! What
more convincing evidence is needed of the Tree of Life character of the five Platonic solids? The
number of yods needed to create the shapes of their external and internal tetractyses is that of 137
tetractyses. This is how the five Platonic solids embody the number 137 defining the fine-structure
constant. We saw in Section 4 during the discussion of the collective properties of the first four
Platonic solids that they have on average 137 yods in their faces. Now we see that this number also
defines the shape of all five solids.
My book Extra-sensory Perception of Quarks (5) and its sequel, ESP of Quarks and Superstrings
(6), provided a wealth of evidence indicating that the Theosophists Annie Besant and
C.W. Leadbeater used a form of ESP to describe quarks, the constituents of the protons and
neutrons making up atomic nuclei (see Article 2). They also proved that the ‘ultimate physical
atom’ (UPA), which Besant and Leadbeater claimed are the basic constituents of atoms, are
fundamental particles making up the up and down quarks in nuclei. According to Leadbeater, there
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is a ‘positive’ and a ‘negative’ UPA (Fig_12), one the mirror image of the other. Each comprises
ten closed curves, or ‘whorls,’ which spiral 2½ times towards the base of the UPA, where three
thicker whorls separate from the other seven, both strands then spiralling separately 2½ times in
opposite senses around the axis of the UPA towards its top. Each whorl is a helix in which
Leadbeater counted 1680 circular turns. In my new book The Image of God in Matter I identified
the UPA with the superstring constituents of up and down quarks and showed how the number
1680 — a structural parameter of the superstring — is encoded in the Tree of Life and is
mathematically defined by the ten Godname numbers. Using Table I, which shows the yod
populations of the Platonic solids, it is easy to calculate that the five nested, type B Platonic solids
contain 1680 yods which, apart from those that are the 50 polyhedral corners, are not corners of
tetractyses.
This astounding embodiment of a number of cosmic significance illustrates the sacred character of
the geometry of the Platonic solids, for the Godname ELOHIM prescribes just that number of
points whose distribution in 3-dimensional space forms five regular polyhedra, the construction of
which from tetractyses requires (including these points) 1680 yods other than corners of
tetractyses. The kernel 168 of this structural parameter of the superstring constituents of quarks is
in fact the number value of Cholem Yesodeth (lit: “breaker of the foundations”), the Hebrew name
of the Mundane Chakra of Malkuth, which is the physical, cosmic manifestation of this Sephirah.
So Kabbalah provides remarkable, independent confirmation of the number clairvoyantly
determined by Leadbeater! That this might be coincidental is implausible because the number 168
refers appropriately to the very Sephirah signifying the physical manifestation of the Tree of Life.
Furthermore, using the numbers listed in the table, it is straightforward to calculate that 168 is the
average number of yods (including their centres) in the four type A Platonic solids representing
Earth, Air, Fire and Water. This is an amazing embodiment of the number characterising the
structure in ordinary space of the subquark state of the superstring. The 120 tetractyses from which
the 38 faces of the first four solids can be assembled have 248 corners and sides. In other words,
248 geometrical elements are needed to shape the surfaces of the Platonic solids corresponding to
the four Elements! It was mentioned earlier that superstring theory predicts that 248 particles
transmit the unified force operating between superstrings very briefly after the Big Bang, later
remnants of which bound (according to my analysis of Besant’s and Leadbeater’s observations)
three superstrings into quarks and then three quarks into the protons and neutrons that eventually
built up through stellar nuclear synthesis the atomic nuclei of the 92 naturally occurring elements!
In this sense, therefore, these four Platonic solids do determine the fundamental substance of the
material world through their shapes — just as the ancient Pythagoreans claimed they did, although
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not, of course, as the actual form of the particles making up the four Elements. Basic parameters of
superstring theory (one obtained by psychic means and awaiting confirmation by theoretical
developments in particle physics) are therefore embodied in these four Platonic solids as their
structural properties.
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7. Conclusion
This article has demonstrated the seemingly miraculous way in which the geometry of the Platonic
solids embodies numbers (240, 480 and 248) predicted by a scientific theory to describe the basic
forces operating in the natural world and numbers (168 and 1680) predicted by psychic
observations of superstrings to be their structural parameters. Another example of the latter is the
fact that the UPA comprises 50 revolutions of its ten whorls, each one making five revolutions.
This compares with the 50 corners of the five regular polyhedra. Both properties exist because
both are Tree of Life structures prescribed by the mathematical archetype embodied in the Divine
Name ELOHIM, whose number value is 50. The article has also demonstrated the remarkable
parallels between the stages of unfolding of the inner form of the Tree of Life and the sequence of
Platonic solids, reaching its culmination in the dodecahedron. This embodies two parameters of
CTOL (91 and 550), showing that it is an analogous representation, just as the other solids are
analogous to distinct stages of development of the inner form of the Tree of Life. For the
Pythagoreans, mathematics could not be separated from theology because they taught, “the
essence of the Gods is defined by Number.” Their teaching is amply illustrated by the five Platonic
solids, whose beautiful, sacred geometry is prescribed by the number values of the ten divine
names as the counterpart of the outer and inner forms of the Tree of Life, the universal blueprint.