Abstract
No doubt because of their fear of peer ridicule,
mathematicians have taken little professional interest in the ‘sacred
geometries’ of various cultures and mystical traditions. It may, therefore,
surprise them to learn that Leibniz, the great German mathematician and
philosopher who discovered calculus at the same time as Sir Isaac Newton, chose
to analyze the table of hexagrams used in Chinese divination and found that it
was a binary number representation of the vertices of a cube and the lines that
join them. He did not pursue the matter further, and apart from studies by a few
architects and mystically minded geometers, little crosscultural study of
sacred geometries like the Kabbalistic Tree of Life and the Tantric Sri Yantra
have been made with the aim of elucidating what (if any) is the nature of the
information hidden in these purported blueprints for all existence. Academics
are loath to examine such a topic partly because of the farreaching (indeed,
paradigmshifting) implications that they would have to confront if they ever
found significant, but rationally unexplainable, similarities between purported
representations of God’s design for His universe, even though these geometrical
systems are separated in provenance by thousands of miles and years!
This article reveals mostly in pictorial form what the
archetypal nature of these sacred geometries is and what they imply for particle
physics (for more details, refer to Articles 1939 on the author’s website). The
latter has to be tentative as yet because this material summarizes work still in
progress. The former, however, can be established with the rigour of a proof of
a theorem of Euclidean geometry and does not require modeldependent
interpretation. The present purpose is not to discuss the implications –
religious and philosophical – of a demonstration that certain sacred geometries
are isomorphic to one another. Rather, it is to assemble and then to compare
research results from other articles by the author so as to identify the
essential properties shared in different ways by the sacred geometries of the
Tree of Life, the I Ching & the Sri Yantra. This research article will prove
conclusively that the fundamental information that they contain relates to the
superstring nature of matter. In his various books, the author proved beyond
reasonable doubt that superstrings were described over 100 years ago with the
aid of a yogic siddhi called ‘anima.’ This mental faculty generates highly
magnified images of microscopic objects as they exist in real time, i.e., they
are NOT merely symbolic. The information embodied in the sacred geometries
discussed here relate unequivocally to the details of this description of the
basic constituents of matter. However amazing it may be, such a conclusion
should come as no surprise, given that these geometries are isomorphic
blueprints that determine the very nature of reality, including physical matter.
A supersymmetric generalization of the Standard Model used by particle
physicists will be hypothesized in order to interpret the common characteristics
of these geometries. Previous articles by the author have shown that their
properties also appear in the Catalan solid called the ‘disdyakis
triacontahedron,’ which was found to be the polyhedral counterpart of both the
Tree of Life and the Sri Yantra. Like them, this polyhedron possesses amazing
properties that are unique to itself. However, it also shares with them equally
amazing features that indicate that the essential meaning of its spectacular
geometry is identical to theirs, as, indeed, it has to be because they are ALL
expressions of the one universal blueprint. Despite differences in morphology,
their essential similarity can only be regarded as irrefutable evidence of a
universal archetype pervading the sacred geometries of East and West that can
now be shown to manifest in the mathematics of music. Moreover, it is one that
is beginning to appear in the research journals of particle physics. This is the
true ‘theory of everything,’ for the blueprint in sacred geometries applies not
only to matter but to ALL holistic systems.

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Figure 1
At the heart of Kabbalah, the Jewish mystical tradition, is the glyph called
the Tree of Life (Otz Chiim). It is a geometrical representation of Adam Kadmon
(Heavenly Man), the divine paradigm forming the basis of all holistic systems.
The Sri Yantra is the most famous and revered of the yantras used in India
for meditation. So old that its origin is unknown, it depicts the nature of Creation.
The I Ching table has been used for hundreds of years in China for the
purpose of divination. Its 64 hexagrams consist of pairs of trigrams, each a set of three
parallel lines that are either Yang (unbroken) or Yin (broken).
The disdyakis triacontahedron is the most complex of the Catalan solids–
generated from the Archimedean solids by interchanging vertices and faces.
Over a century ago, the leading Theosophists, Annie Besant and C.W.
Leadbeater, claimed to observe the subatomic unit of matter with a yogic siddhi . The author
has proved that this object is the E_{8}×E_{8}' heterotic superstring
constituent of up and down quarks.
Article 40 shows that the Tree of Life, the Sri Yantra, the I Ching table and the
disdyakis triacontahedron embody the same universal blueprint as that which manifests in the
smallest subatomic particle.
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Figure 2
The tetractys is at the core of the mathematical philosophy of Pythagoras. A
symbol of holistic systems, it is equivalent to the Kabbalistic Tree of Life. Each of its ten
yods symbolizes one of the ten Sephiroth of the Tree of Life. The three yods at its corners
symbolize the Supernal Triad (the triple Godhead). The seven hexagonal yods at its centre or at
the corners of a hexagon denote the seven Sephiroth of Construction.
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Figure 3
Four circles arranged in a vertical line and overlapping
centretocircumference generate the positions of the ten Sephiroth of the Tree of Life.
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Figure 4
Two similar, overlapping circles form a Vesica Piscis (shown shaded) as
their region of overlap. Its apices are the centres of two new circles. The four circles have
18 centres and endpoints of their vertical and horizontal diameters. The 16 points shown are
sufficient to generate the “inner” form of the Tree of Life.
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Figure 5
The inner form of the Tree of Life is generated from its outer form by
joining pairs of points that belong to the set of 16 points. Straight lines joining pairs of
points intersect at the 70 corners of two similar sets of seven regular polygons enfolded in
one another and sharing one edge (the “root edge”).
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Figure 6
The 3dimensional, outer Tree of Life can be projected onto the plane of the
two sets of enfolded polygons that constitute its inner form. Six of their corners outside
their shared edge are shared with Sephiroth of the Tree of Life. Their corners on the root edge
coincide with Tiphareth and with Daath. The outer and inner forms of the Tree of Life have
73 corners, where 73 is the number value of Chokmah
(“Wisdom”), the second Sephirah.
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Figure 7
The number 137 is embodied in the blueprint of the inner Tree of Life. Its
(7+7) enfolded polygons have 94 sectors. When they are each divided into three tetractyses,
1370 yods are generated, that is, the number of yods in 137 tetractyses. This proves beyond
question that the number 137 is a basic structural parameter of the Tree of Life, in keeping
with its central status in physics as a number which determines one of the fundamental
constants of nature – the socalled 'finestructure constant,' whose magnitude sets the scale
of the energies of electrons in atoms.
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Figure 8
496 extra yods are needed to turn into three tetractyses
each sector of the last four enfolded polygons of the inner Tree of Life.
248 yods belonging to the hexagon and dodecagon symbolize the
248 particles transmitting the superstring force with the symmetry of the
gauge symmetry group E_{8} and 248 yods belonging to the octagon
and decagon symbolize the 248 particles associated with E_{8}'
(identical to E_{8}). The remarkable, natural division of the yod populations of the
four polygons into two sets of 248 shows how the universal blueprint of the
inner Tree of Life embodies the dynamics of the E_{8}×E_{8}' heterotic
superstring. It also illustrates how the Pythagorean Tetrad (4) defines parameters of
scientific significance, for it is the last four regular polygons that embody the
number 496 characterizing the unified force between superstrings.
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Figure 9
16 separate triangles with 48 corners join together to form
the Tree of Life. It has 10 vertices and 22 edges of 16 triangles – a total of
48 geometrical elements. The reappearance of this number is not a coincidence
because it is a basic parameter of all geometrical structures that conform to the archetypal
Tree of Life pattern and possess sacred geometry. 48 is the number value of
Kokab, the Mundane Chakra of Hod.
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Figure 10
The number 48 also characterises the inner form of the Tree
of Life. It is a set of seven enfolded polygons which, when separate, have 48
corners.
The Sri Yantra is difficult to draw accurately because a slight error at any
stage of its construction can seriously distort the diagram. A template that achieves this task
is a vertical, straight line with 48 spaces of the same width marked out on
it.
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Figure 11
The parameter 48 characterising sacred geometry is
displayed in the Sri Yantra, the I Ching table and the tetrahedron. The first layer of
triangles in the Sri Yantra consists of eight triangles joined corner to corner. They have 16
vertices, 24 edges and eight triangles, totally 48 geometrical elements as two
sets of 24. The eight hexagrams (shown black) along the diagonal of the I Ching table have
48 unbroken and broken lines as two sets of 24. The tetrahedron – the simplest
Platonic solid – has four faces. When each face is divided into its three sectors and the
latter then turned into tetractyses, there are 48 hexagonal yods in the 12
tetractyses. They comprise 24 red hexagonal yods inside each face and 24 black hexagonal yods
on the edges of the tetrahedron. This division into two sets of 24 of the 48
degrees of freedom embodied in the generative, or germinal aspect of a holistic system is
characteristic of such systems.
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Figure 12
The number 48 is the sum of the first six odd integers
after 1. 24 is the sum of the first four odd integers after 1. Arithmetically, therefore, the
number 48 divides naturally into a pair of number 24s. The encircled
selections are the only ones that add up to 24. As we shall see, this division always appears
in holistic systems that display sacred geometry.
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Figure 13
The octagon constructed from tetractyses displays the same 24:24 division of
the 48 yods surrounding its centre. There are 24 yods on its boundary and 24
inside it surrounding its centre.
When constructed from “2ndorder” tetractyses, in which each yod of a
tetractys is replaced by a tetractys, the octagon has 496 hexagonal yods. They
symbolize the 496 spin1 particles that superstring theory predicts transmit
the unified superstring force. Surrounding the centre of the octagon are 80
corners of tetractyses. 80 is the number value of Yesod, the penultimate
Sephirah of the Tree of Life. 496 is the number value of Malkuth, the final
Sephirah.
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Figure 14
The holistic parameter 48 exists in the 144 Polyhedron (see
Article 23 for details), the polyhedron which – together with the disdyakis triacontahedron –
constitutes the polyhedral version of, respectively, the inner and outer forms of the Tree of
Life. It is generated by attaching tetrahedra to the 48 faces of the disdyakis
dodecahedron, which has 26 vertices and 72 edges. The latter
is generated from the rhombic dodecahedron by sticking rhombic pyramids onto its 12 faces. The
resulting 48 faces comprise 24 faces and their 24 mirror images. This means
that the 48 peaks of the tetrahedra consist of 24 vertices and their 24 mirror
images. This is how the 144 Polyhedron displays the 24:24 division of the parameter
48. Their counterpart in the I Ching table are the 24 lines & broken lines
in the eight upper and eight lower trigrams making up the eight diagonal hexagrams. Each half
of a disdyakis dodecahedron consists of six sets of four faces, i.e., four sets of six faces.
The grouping of the 48 faces into eight sets of six means that
48 of the (48+26=74) vertices of the 144
Polyhedron are similarly grouped. They correspond to the eight diagonal hexagrams of the I
Ching table, each with six lines/broken lines, and to the eight sets of geometrical elements
making up the eight triangles in the first layer of the Sri Yantra. They are related to the 24
rotational symmetries and the 24 rotation/reflection symmetries of the octahedral group.
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Attaching tetrahedra to the 48 faces of the disdyakis
dodecahedron with 26 vertices generates the 144 Polyhedron with
(48+26=74) vertices. The 8 groups of 6 peaks of these
tetrahedra are the counterparts of the 8 diagonal hexagrams of the I Ching diagram, each with 6
lines/broken lines. The 24 peaks in each half of the 144 Polyhedron correspond to the 24
lines/broken lines in one set of 8 trigrams and their Yang/Yin opposites in the other set
making up the 8 diagonal hexagrams. They signify the (24+24=48) symmetries of
the octahedral group.
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Figure 15
The correspondence between the polygonal and polyhedral versions of the Tree
of Life is revealed here. As the generator of the disdyakis triacontahedron, the 144 Polyhedron
has 144 faces creating its shape. This is the number of yods needed to create the
48 formgenerating edges of the seven separate, regular polygons of
the inner Tree of Life when the sectors of the latter are turned into tetractyses. It is also
the number of yods inside the seven enfolded polygons and surrounding their centres.
This corresponds to the 144 Polyhedron being the inner form of the polyhedral Tree of Life. It
should therefore be thought of as inside the disdyakis triacontahedron — the outer form of the
polyhedral Tree of Life. Just as there are 120 faces shaping the latter, so there are 120 yods
on the 42 edges of the seven enfolded polygons that delineate their shapes. The polyhedral
version of the outer and inner Trees of Life displays the same 144:120 division as
their polygonal counterpart. This is because they are both isomorphic representations of
holistic systems.
The Pythagorean Tetrad (4) defines the numbers 120 and 144 as square arrays
of powers of the integers 1, 2, 3 & 4 symbolized by the four rows of dots in the
tetractys.
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Figure 16
There are 84 yods up to the level of the lowest Tree of Life when it is
constructed from tetractyses. Of these, ten are Sephiroth, leaving 74 yods other than
Sephiroth. This is the number of nonSephirothic degrees of freedom hidden within the outermost
form of the Tree of Life. They comprises 26 yods down to the level of Daath
and 48 yods below it. These yods correspond, respectively, to the
26 vertices of the disdyakis dodecahedron underlying the 144 Polyhedron and to
the 48 vertices of the tetrahedra attached to its 48
faces.
The inner form of the lowest Tree of Life is two similar sets of seven
enfolded polygons with 70 corners. They share seven corners with Sephiroth of the Tree of Life
and one corner with Daath (both denoted by black yods), leaving 62 corners.
These 62 independent degrees of freedom correspond to the 62
vertices of the disdyakis triacontahedron. The 31 corners of each set of
polygons are the counterpart of the 31 vertices of this polyhedron and their
mirror images. The 30 corners of the two sets of pentagons, hexagons & dodecagons
correspond to the 30 A vertices, the 12 corners of the two octagons correspond to the 12 C
vertices (vertices of an icosahedron). The 20 corners of the square & decagon correspond to
the 20 A vertices (vertices of a dodecahedron).
The 144 Polyhedron and the disdyakis triacontahedron are therefore implicit
in the geometry of the outer and inner forms of the Tree of Life as their polyhedral
counterparts.
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Figure 17
Turned into tetractyses, the 16 triangles of the Tree of Life have 70 yods.
This is the number of vertices (including the central bindu point) of the 2dimensional Sri
Yantra. This demonstrates the isomorphism between these representations of holistic
systems.
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Figure 18
The four groups of triangles in the 2dimensional Sri Yantra are composed of
236 geometrical elements. (The central triangle does not count formally as a triangle because
of the bindu, or point, at its centre).
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Figure 19
236 yods lie on the 83 edges of the two identical sets of seven enfolded
polygons constituting the inner Tree of Life. This the number of geometrical elements that
compose the 42 triangles of the 2dimensional Sri Yantra. Different representations of holistic
systems embody the same structural parameter.
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