The Spiritual Cosmos and It's Encoding in Sacred Geometries

c0mmando · June 24, 2024, 6:01am

Note: I wanted to give this research a new home – all credits due to the amazing Stephen M. Phillips at Sacred Geometries & Their Scientific Meaning

“All things are arranged in a certain order, and this order constitutes the form by which the universe resembles God.” - Dante, Paradiso

This section reveals the Tree of Life map of all levels of reality, proves that it is encoded in the inner form of the Tree of Life and demonstrates that the Sri Yantra, the Platonic solids and the disdyakis triacontahedron are equivalent representations of this map.

Consciousness is the greatest mystery still unexplained by science. This section presents mathematical evidence that consciousness is not a product of physical processes, whether quantum or not, but encompasses superphysical realities whose number and pattern are encoded in sacred geometries.

The Cosmic Tetractys

The 2nd-order tetractys

In Sacred geometry/Tree of Life, we discussed various methods of transforming examples of sacred geometry in order to decode the scientific and spiritual information that they embody. The next level of decoding sacred geometry after the Pythagorean tetractys is through its next-higher version — the 2nd-order tetractys, in which each of the 10 yods of the tetractys is replaced by another tetractys. This generates 85 yods, which is the sum of the first four integer powers of 4:

85 = 4⁰ + 4¹ + 4² + 4³

The yod at the centre of the 2nd-order tetractys denotes Malkuth of the central tetractys, which itself corresponds to this Sephirah. It is surrounded by 84 yods. The 2nd-order tetractys therefore expresses the fact that 84 Sephirothic degrees of freedom in a holistic system exist above Malkuth — its physical form. Of these, (7×7 − 1 = 48*) degrees are pure differentiations of Sephiroth of Construction symbolized by coloured, hexagonal yods in the seven 1st-order tetractyses that are not at the corners of the 2nd-order tetractys. The remaining 36 degrees are denoted by both the 15 white yods at the corners of the 10 tetractyses (these yods formally symbolize the Supernal Triad) and the 21 coloured, hexagonal yods that belong to the tetractyses at the three corners of the 2nd-order tetractys and which, therefore, also refer to the Supernal Triad of Kether, Chokmah & Binah. YAH (יה), the older version of the Godname YAHWEH (יהוה) assigned to Chokmah, has the number value 15 and prescribes the 15 corners of the 10 1st-order tetractyses. ELOHA (אלה), the Godname of Geburah with number value 36, prescribes both the 36 yods lining the sides of the 2nd-order tetractys and the 36 yods just discussed. The number 84 is the sum of the squares of the first four odd integers:

84 = 1² + 3² + 5² + 7²

as

n² = 1 + 3 + 5 + 2n–1

where n is any positive integer, n2 is the sum of the first n odd integers, so that 84 is the sum of (1+3+5+7=16=42) odd integers:

Representation of 84

The Tetrad determines the number of yods surrounding the centre of a 2nd-order tetractys. These yods include 15 corners of 1st-order tetractyses, where

15 = 2⁰ + 2¹ + 2² + 2³ = 1 + 2 + 4 + 8

(the number value of YAH) is the sum of the first four integer powers of 2. There are (85–15=70) hexagonal yods, where 70 = 10×7 = (1+2+3+4)×fourth odd/prime number. These two properties illustrate again how the Tetrad determines properties of the next higher-order tetractys above the 1st-order tetractys. In mathematics, triangular numbers (1, 3, 6, etc) can be represented by triangular arrays of dots and tetrahedral numbers (1, 4, 10, etc) can be represented as a tetrahedral pile of these arrays. The piles representing tetrahedral numbers can themselves be piled up into 4-dimensional “tetrahedral numbers”: 1, 5, 15, 35, 70, etc. The fourth, non-trivial example of these numbers is 70. This is the number of hexagonal yods in the 2nd-order tetractys. Once again, the Tetrad determines both a class of number and a specific member of this class that is a parameter*,* or measure*,* of the Pythagorean representation of Wholeness. It is an example of how the Tetrad Principle governs the mathematical nature of holistic patterns and systems (for more details, see Article 1).

As another illustration of this principle, the four integers 1, 2, 4 & 8 are the first four terms in the geometric series:

1, 2, 4, 8, 16, 32, …

in which each term is twice the previous one. They appear in what is known as Plato’s Lambda. In his treatise on cosmology called “Timaeus,” Plato has the Demiurge marking a strip of the substance of the World Soul into sections measured in length by the numbers 1, 2, 4 & 8 on one side of it and the numbers 3¹ (=3) 3² (=9) 3³ (=27) on its other side (see here). The number of corners of the 10 1st-order tetractyses in the 2nd-order tetractys is

15 = 2⁰ + 2¹ + 2² + 2³ = 1 + 2 + 4 + 8

whilst the number of yods in the 2nd-order tetractys is

85 = 1² + 2² + 4² + 8²

The number of yods surrounding its centre is

84 = 2² + 4² + 8²

and the number of hexagonal yods is

85 - 15 = 70 = (2² - 2) + (4² - 4) + (8² - 8)

This illustrates the power of the integers 2, 4 & 8 to generate properties of the 2nd-order tetractys. In mathematics, there are only four orders of normed division algebras**: the 1-dimensional scalar numbers, the 2-dimensional complex numbers, the 4-dimensional quaternions and the 8-dimensional octonions. Such is its archetypal power as the arithmetic counterpart of sacred geometries, the Lambda and its complete, tetrahedral generalisation (see here) generate not only the tone ratios of the notes of the Pythagorean musical scale (see here) but also the dimensions of the four types of algebras permitting division! The four integers 1, 2, 4 & 8 spaced along the first raised edge of this tetrahedron:

The Tetrahedral Lambda is the generalisation of Plato’s Lambda, which served as the basis of his cosmology.

generate as their sum the 15 corners of the 10 1st-order tetractyses in the 2nd-order tetractys, whilst the four integers 1 (=4⁰), 4 (=4¹), 16 (=4²) & 64 (=4³) spaced along its third raised edge generate as their sum its 85 yods:

1 + 4 + 16 + 64 = 85.

We see that the first raised edge of the tetrahedral array of 20 integers, which we call the Tetrahedral Lambda in the section Plato’s Lambda, generates the number 15 measuring the “skeleton” of the 2nd-order tetractys in terms of a basic, triangular array of 15 points, namely, the corners of 10 1st-order tetractyses. Its third raised edge generates the complete “body” of the 2nd-order tetractys comprising 85 yods. This illustrates the character of the number 15 of YAH, the Godname of Chokmah, as the fifth triangular number.

The sum of the seven integers on the first and third raised edges of the Tetrahedral Lambda = 1 + 2 + 4 + 8 + 4 + 16 + 64 = 99. As the sum of all its 20 integers is 350, the sum of the remaining 13 integers is 251. This is the number of yods in the 1-tree when its 19 triangles are Type A (see here). It is embodied in the UPA as the number of space-time coordinates of points on the 10 whorls as 10 closed curves in 26-dimensional space-time: 10×25 + 1 = 251. Notice that the sum of the squares of the four integers on the first raised edge:

1² + 2² + 4² + 8² = 85

is the same as the sum of the four integers on the third raised edge:

1 + 4 + 4² + 4³ = 85

This means that the number 168 which, being a parameter of holistic systems, always displays the division 168 = 84 + 84 (see here), can be expressed as:

2² + 4² + 8² + 4¹ + 4² + 4³

The holistic parameter 336 = 2×168, which is discussed in numerous places on this website, can be expressed as 4×84 = 4² + 4³ + 4⁴. As 336 = 350 - 14, where 14 is the sum of the integers 2, 4 & 8 on the first raised edge, we see that the sum of the squares of these three integers is 84, which is the sum of the nine integers on the boundary of the first face of the Tetrahedral Lambda, whilst the sum of all its integers except 2, 4 & 8 is 336, which is 4×84, i.e., the sum of the integers 4 assigned to all the 84 yods surrounding the centre of a 2nd-order tetractys.

The correspondences between the 2nd-order tetractys, the 1-tree and the Sri Yantra are discussed here.

Mathematical representation of the hermetic axiom “As above, so below”

The ancient Greeks gave the name of Hermes Trismegistus (“thrice-great Hermes”) to the Egyptian god Thoth, or Tehuti, a god of learning and wisdom, who was the scribe to the other gods. According to legend, Hermes Trismegistus provided the wisdom in the ancient mysteries of ancient Egypt: “He carried an emerald, upon which was recorded all of philosophy, and the caduceus, the symbol of mystical illumination. Hermes Trismegistus vanquished Typhon, the dragon of ignorance, and mental, moral and physical perversion.” Called “The Emerald Tablet,”* its most significant part is within its opening: "That which is above is like that which is below and that which is below is like that which is above, to achieve the wonders of the one thing." Therefore, “This is the foundation of astrology and alchemy: that the microcosm of mankind and the earth is a reflection of the macrocosm of God and the heavens.” This metaphysical statement has been abbreviated to the saying “As above, so below.” As ‘Earth’ (meaning, of course, the physical universe, not just the planet Earth) is Malkuth, symbolized by the hexagonal yod at the centre of a tetractys, this hermetic principle is mathematically implemented in terms of the Pythagorean tetractys by replacing the 1st-order tetractys symbolizing Malkuth (the “below”) at the centre of the 2nd-order tetractys by a 2nd-order tetractys that represents the “above”.

Instead of 49 hexagonal yods symbolizing Sephiroth of Construction (42 in the six tetractyses symbolizing the six Sephiroth of Construction above Malkuth), there are in the modified 2nd-order tetractys (49+42=91) hexagonal yods with seven prismatic colours that symbolize the seven Sephiroth of Construction. Alternatively, circular yods can be replaced by triangles, white triangles referring to Sephiroth that belong to the Supernal Triad and coloured ones referring to Sephiroth of Construction. Either representation is what can be called the “Cosmic Tetractys” that maps the physical and superphysical cosmos. Par excellence, it expresses this famous hermetic principle. The 49 coloured triangles in the central 2nd-order tetractys express the 49 differentiations of the seven Sephiroth of Construction that refer to the Malkuth level of the Cosmic Tetractys. The 42 coloured triangles in the six tetractyses surrounding it express the seven-fold differentiation of the six Sephiroth of Construction above Malkuth.

The Cosmic Tetractys is the Pythagorean way of representing the 91 differentiations of the seven Sephiroth of Construction, as symbolized by the seven hexagonal yods in the tetractys. It is a geometrical way of embodying the hermetic principle that the macrocosm is the same as the microcosm — both physical and superphysical. Or, more accurately, they are analogous. The Emerald Tablet expresses a powerful, alchemical formula that, once followed, can accelerate one’s spiritual evolution to enable conscious entry into the Life of God, thereby ending the cycle of birth, death & rebirth. The Cosmic Tetractys maps the stages in that journey. It is a kind of Pythagorean mandala. The sacred geometries of other religions, too, are maps of all levels of reality. The great miracle is that they are equivalent, i.e., they map the same Cosmic Reality in analogous ways, as this website (and, in particular, this section) proves. This conclusion has profound implications, for it is tantamount to being evidence for the existence of a transcendental Mind, the mathematical pattern of Whose thoughts is found to be the same when represented by certain sacred geometries (see The holistic pattern and Wonders of correspondences).

The mathematical scheme expressed in the Cosmic Tetractys has the following natural, Theosophical interpretation: Theosophy teaches that there are seven planes of consciousness (see here). The most material plane is the physical plane. This is the physical universe, the space-time continuum, revealed by the five senses and by science’s instrumental extensions of them, such as the microscope and the telescope. Beyond (or, rather, interpenetrating) it are the astral, mental, buddhic, atmic, anupadaka (monadic) & adi (divine) planes.* Each plane is divided into seven subplanes, so that the seven planes have (7×7=49) subplanes. They constitute the “cosmic physical plane.” Beyond them are six, still higher, cosmic superphysical planes, each composed of seven subplanes, totalling 42 subplanes. This means that the seven cosmic planes comprise (49+42=91) subplanes, where

91 = 1² + 2² + 3² + 4² + 5² + 6²

The lowest seven subplanes constitute the physical plane. There are 84 subplanes in the 12 superphysical planes, where 84 = 1 + 4² + 4³ = 2² + 4² + 8². As shown in Article 16, p. 21), this has a remarkable musical analogue, for the seven diatonic musical scales contain 12 types of notes between the tonic and the octave, and their intervals contain 84 repetitions that belong to the Pythagorean scale.

Comparing this with the 91 coloured, hexagonal triangles making up the Cosmic Tetractys, we see that each such triangle denotes a subplane — a different state of reality. They are all expressions of the seven Sephiroth of Construction, as manifested in the seven cosmic planes of consciousness. This is the Pythagorean representation of the Cosmic Whole — both physical and superphysical reality. The correspondence between the seven Sephiroth of Construction and the seven planes of consciousness is not merely a formal one. The latter are the manifestation of the former:

Plane	Correspondence
Physical plane (space-time)	Malkuth
Astral plane	Yesod
Mental plane	Hod
Buddhic plane	Netzach
Atmic plane	Tiphareth
Anupadaka (Monadic)	Geburah
Adi (Divine)	Chesed

This is the map of all levels of reality that embodies the hermetic axiom “As above, so below” expressed in the Emerald Tablet discussed previously. As any student of Kabbalah understands, the psycho-spiritual aspects of the Sephiroth in Atziluth (Archetypal World), Beriah (Creative World), Yetzirah (Formative World) & Assiyah (World of Action) are found in a human being, whose evolutionary journey to God spans all these planes of consciousness, taking the person potentially far beyond the realm of heaven that Western religions declare awaits the faithful and the good — even beyond the ineffable state of Nirvana that is the goal of Buddhism. All levels are mapped by sacred geometries, as they are representations of the Divine Whole. The 12 types of notes between the tonic and octave of the seven diatonic scales (see here) are the musical counterparts of the 12 superphysical planes of consciousness. The first six types of notes and their six inversions are the respective parallels of the six higher planes of the cosmic physical plane (Astral→Adi) and their six cosmic counterparts. The six Yang meridians and the six Yin meridians familiar to students of acupuncture are another parallel (see Article 32 for the analogy between the 12 types of notes and the 12 meridians).

A triangular array of 153 points, 17 points per side, is the basis of the template that generates the Cosmic Tetractys. It shows how ELOHIM SABAOTH, the Godname of Hod with number value 153, arithmetically prescribes the Cosmic Tetractys: 153 is the 17th triangular number. Each tetractys array of 10 triangles consists of 15 points, 30 lines and 16 triangles, i.e., 61 geometrical elements. The Godname YAH with number value 15 prescribes the basic unit, as does the Godname EL with number 31, because it has 31 points & triangles, whilst 61 is the 31st odd integer.

There are 48 points on the sides of the triangular array. This is the number value of Kokab, the Mundane Chakra of Hod.

36 points line the boundary of the triangular array that are not corners of the tetractys arrays of triangles. 36 is the number value of ELOHA, which is the Godname of Geburah, the Sephirah directly above Hod located on the Pillar of Judgement of the Tree of Life.

The 48 points therefore divide up into a set of 36 points and 12 corners of the ten triangular arrays of points forming a tetractys. Their counterparts in the seven regular polygons making up one half of the inner Tree of Life (see here) are the 36 corners of the first six separate polygons and the 12 corners of the dodecagon. All holistic structures display analogous patterns, such as this 36:12 division. For example, dividing a straight line into 48 segments is the initial step in one of the ways of constructing the Sri Yantra (see here).

Type B hexagon encodes CTOL

As an example of the hidden knowledge that is revealed by the tetractys to be embodied in sacred geometries and various polygons, the Type B hexagon contains 91 yods, where

91 = 1² + 2² + 3² + 4² + 5² + 6²

As the hexagon is the fourth of the regular polygons, the Tetrad Principle picks out the very polygon whose yod population measures all the levels of consciousness. For each yod symbolizes one of the 91 subplanes of the seven cosmic planes. The 91 yods are made up of 49 black yods, either on sides of sectors of the hexagon or at the centres of its 18 tetractyses, and 42 red hexagonal yods on sides of the latter. The former symbolize the 49 subplanes of the cosmic physical plane and the latter symbolize the 42 subplanes of the six cosmic superphysical planes. The centre of the hexagon and its six corners denote the seven subplanes of the physical plane. The remaining 42 black yods denote the 42 subplanes of the six higher planes in the cosmic physical plane. The same 49:42 pattern is created by the Star of David/Sign of Vishnu shape formed by yods in the hexagon. The black yod at the centre of the hexagon denotes the lowest subplane of the physical plane, the six black yods at its corners denote the six other subplanes of the physical plane as discussed in Theosophy and the remaining 84 yods denote the 84 subplanes of superphysical planes (black yods denote cosmic physical subplanes; red yods denote cosmic superphysical subplanes). The Type B hexagon is the single, polygonal counterpart of the Cosmic Tetractys. It is discussed also here. Properties of the Type A and Type B hexagon are discussed here. As one might expect, the number 91 is one of the defining parameters of holistic systems. Its embodiment in the Type B hexagon illustrates the Tetrad Principle formulated in Article 1, for the hexagon is the fourth of the regular polygons, and holistic parameters are always either the fourth member of a class of numbers or numbers that are embodied in the fourth member of a class of geometrical objects. The number 6 is the fourth smallest integer that can be represented by the corners of a polygon, and the fact that the shape of the hexagon manifests so often in nature, e.g., it is the shape of ice crystals, the lattice of atoms in graphite and the ring of carbon atoms in the benzene molecule that forms the basis of aromatic hydrocarbons, bears powerful testimony to the ubiquitous working of the Tetrad Principle. Indeed, the carbon atom itself is composed of six electrons bound to a nucleus made up of six protons and six neutrons! Curiously, carbon is the sixth most abundant element in the universe (see here). The number 6 plays a pivotal role in determining not only the number of subplanes of consciousness but also the electron and nuclear structure of the very chemical atom that is the basis of life on Earth.

The Cosmic Tree of Life

The ten yods in the tetractys symbolize the ten Sephiroth, each of which can be represented by a Tree of Life. Suppose, then, that we regard as a Tree of Life each of the 91 triangles (all ‘yods’ of tetractys arrays of ten triangles) in the Cosmic Tetractys representing differentiations of the Sephiroth of Construction. How many Sephirothic levels, or ‘SLs,’ are there in the 91 overlapping Trees of Life? The formula for the number N(n) of SLs in n overlapping Trees of Life is*:

N(n) = 6n + 4.

Therefore, N(91) = 550. As will be proved in this section, the number 550 is embodied in all sacred geometries because they are equivalent maps of all levels of reality — physical and superphysical. The set of 91 overlapping Trees with 550 SLs (see opposite) will be called the “Cosmic Tree of Life,” or “CTOL.” This ladder-like structure is the basis of what is referred in the Old Testament as “Jacob’s ladder” (see Book of Genesis 28: 10-19).

91 is the sum of the squares of the first six integers:

91 = 1² + 2² + 3² + 4² + 5² + 6²

It is prescribed by the Godname EHYEH with number value 21 because 91 is the sum of a triangular array of 21 integers 1-6 (see diagram). As

n² = 1 + 3 + 5 + … + 2n-1

i.e., n2 is the sum of the first n odd integers, 91 is also the sum of a triangular array of 21 odd integers. The sum of the 15 integers on the boundary of the array is 65 and the sum of the six integers in its interior (indicated in the diagram opposite by the grey triangle in the array) is 26. The number 91 is, therefore, the gematria number value of ADONAI (65) TETRAGRAMMATON (26), where ADONAI is the Godname of Malkuth and the word that Jews use, when they recite their scriptures, as a substitute for TETRAGRAMMATON, the sacred Name of God that they are forbidden to pronounce. Its proper pronunciation is unknown except to a few, although there have been many speculations.

In view of the significance of the tetractys as a symbol for the Decad (10), being the fourth triangular number:
… … 1
… . .1 1
. … 1 1 1 = 10
… 1 1 1 1

the number 550 has a particularly Pythagorean character because it is the sum of the first ten multiples of 10:

550 = 10×55 = 10(1+2+3+4+5+6+7+8+9+10)

… … .10 … …
… … 20 30 …
=…40 50 60 …
…70 80 90 100.

55 (the tenth triangular number) is the sum of the squares of the first five integers:

55 = 1² + 2² + 3² + 4² + 5²

550 is therefore the sum of (10×5=50) squares, showing how ELOHIM, the Godname of Binah with number value 50, prescribes this number quantifying the Cosmic Tree of Life. The number 550 is the sum of numbers of the Godnames of all the Sephiroth except the first Sephirah (Kether) and the last one (Malkuth):

550 = 26 + 50 + 31 + 36 + 76 + 129 + 153 + 49.

Alternatively, it is the sum of the numbers of the Godnames of all the Sephiroth except Binah and Geburah:

550 = 21 + 26 + 31 + 76 + 129 + 153 + 49 + 65.

This number is the sum of the number 474 of Daath and the number 76 of YAHWEH ELOHIM. The 550 Sephirothic levels that span CTOL are therefore the measure of the “knowledge of God the Creator.”

The number 55 is the number of yods lining the sides of two nested pentagrams:

This is remarkable because this number is the tenth number in the infinite sequence of Fibonacci numbers:

1,1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …

the ratio of whose pairs of successive integers converges to the Golden Ratio Ø = 1.618…, which determines the ratio of red & blue sides of either the pentagram, its enclosing pentagon or its internal triangles:

phi in pentagram

91 overlapping Tree of Life consist of 3108 points, lines & triangles.* This number is the sum of the fourth powers of the first four odd integers:

3108 = 1⁴ + 3⁴ + 5⁴ + 7⁴

This beautiful property reveals the unique status of this number of Trees of Life in CTOL as a representation of the 91 subplanes of consciousness. Such mathematical design cannot be, plausibly, dismissed as the product of chance. Instead, it is undeniable evidence of the transcendental origin of CTOL. Other remarkable, geometrical properties of CTOL are discussed in the author’s new book. Its embodiment of this number is an illustration of the Tetrad Principle, formulated in Article 1, whereby defining parameters of holistic systems are numbers expressed by either the fourth member of a class of mathematical object or its first four members. Another example of this Pythagorean principle at work can be found here when the cubes of the first four odd integers in the 4×4 array shown opposite are replaced by their squares, generating the holistic parameter 496:

496 = 1³ + 3³ + 5³ + 7³

This is the mysterious number at the heart of superstring theory whose sacred-geometrical basis is demonstrated in Superstrings as sacred geometry.

The inner form of the Tree of Life is the set of (7+7) enfolded regular polygons:

triangle, square, pentagon, hexagon, octagon, decagon, dodecagon.

They have 444 hexagonal yods (see here). As 3108 = 7×444, the (49+49) regular polygons making up the inner form of seven overlapping Trees contain 3108 hexagonal yods:

In terms of the formal correspondence between the Tree of Life and the tetractys, the seven hexagonal yods in the latter correspond to the seven Sephiroth of Construction. Amazingly, the 3108 such yods in the inner form of seven Trees, each expressing one of these Sephiroth, denote the 3108 points, lines & triangles needed to construct the 91 Trees of Life that map the seven cosmic planes of consciousness, each formally corresponding to a Sephirah of Construction. This is how EL ChAI, the Godname of Yesod with number value 49, prescribes the geometrical composition of CTOL. It is a powerful example of how analogous sections of CTOL are characterised by the same parameters. The physical universe mapped by seven Trees of Life is analogous to CTOL itself mapped by 91 Trees of Life because each plane of consciousness is an expression of one of the seven Sephiroth of Construction, the physical plane being just their material manifestation. In fact, every plane is simply that aspect of their collective reality which is experienced from the perspective of the particular Sephirah of Construction that corresponds to that plane.

Another way in which EL ChAI determines the number 3108 is now explained: The section Polygonal numbers introduces the concept of polygonal numbers. Pⁿₙ, the nth polygonal number of order N, is the number of dots needed to construct n N-gons nested inside one another with n dots equally spaced along each side of the outermost N-gon:

Polygonal number

Pⁿ₁ = 1. Writing P³ₙ ≡ Tₙ, P⁴ₙ ≡ Sₙ, P⁵ₙ ≡ Pₙ, P⁶ₙ ≡ Hₙ, P⁸ₙ ≡ Oₙ, P¹⁰ₙ ≡ Dₙ & P¹²ₙ ≡ dₙ, the sum of the first seven non-trivial polygonal numbers (i.e., numbers larger than 1) of orders corresponding to the seven regular polygons of the inner Tree of Life is:

₈
Σ (Tₙ + Sₙ + Pₙ + Hₙ + Oₙ + Dₙ + dₙ) = 3108.
ⁿ=²

(see the entry for c8 in Table 2 here). Therefore, the 49 polygonal numbers in this 7×7 array add up not only to the number of geometrical elements in CTOL but also to the number of hexagonal yods in the seven sets of seven regular polygons (i.e., 49 polygons) that make up the inner form of seven overlapping Trees of Life. The property demonstrates the remarkable harmony between the arithmetic and geometric aspects of the inner Tree of Life. The cosmic Whole (CTOL), composed of 3108 points, lines & triangles and mapping the seven cosmic planes, and its physical counterpart (the seven subplanes of the physical plane) embody the same number! This is simply because they bear a formal correspondence to each other, being different differentiations of the seven Sephiroth of Construction.

Consider n overlapping Trees of Life. Starting from its lowest SL, six more SLs are present in successive Trees up to Chesed of each one. The number of SLs up to Chesed of the nth Tree = 6n + 1.

Consider an n-sided polygon (it need not be regular). Turn each of its sectors into tetractyses. Six yods are added per sector. Including its centre, which is shared by all n tetractyses, the n-sided polygon is constructed from n tetractyses with (6n+1) yods. This establishes that such a polygon is equivalent to n overlapping Trees of Life up to Chesed — the first Sephirah of Construction — of the nth Tree. Enclose the polygon in a square and it becomes the counterpart of the complete n-tree, the four corners of the square corresponding to Kether, Chokmah, Binah & Daath of the nth Tree of Life. The centre of the polygon corresponds to Malkuth of the first Tree. The 2n hexagonal yods on its sides correspond to the n pairs of Chesed & Geburah of the n Trees and the n corners and n centres of tetractyses correspond to their Yesods & Tiphareths.

Encoding of CTOL in the inner Tree of Life

The table lists the number of yods in the 14 separate regular polygons making up the inner Tree of Life, starting at the bottom with the simplest one — the triangle (red numbers are gematria number values of the Sephiroth in the four Worlds of Atziluth, Beriah, Yetzirah & Assiyah. Each set of seven polygons is separated by the root edge, which has four yods. The order of the second set is reversed for reasons which will be become apparent later. Also listed is the number of yods in the polygons as a running sum, again starting from the bottom.

Let us ask the following question: what combinations of polygons have total yod populations that are equal to the number (6n+4) of SLs in n overlapping Trees? Running totals that satisfy this are ticked. Either 18 or 91 overlapping Trees of Life have the same number of SLs as their corresponding combinations of polygons have yods. The first four polygons are the counterpart of 18 overlapping Trees and the first 12 polygons (ordered in the sequence shown) are, when the root edge is included, the counterpart of 91 overlapping Trees. As 91 is larger than 18, the latter number is of no interest, for we are seeking what polygons are the counterpart of the largest possible number of overlapping Trees. We find that 12 of the 14 polygons are the exact equivalent of 91 Trees of Life. The Cosmic Tree of Life is therefore encoded in this subset of the polygons making up the inner Tree of Life.

That this is not just a coincidence is demonstrated by the fact that the first seven polygons have 295 yods. This is the number of SLs up to Chesed of the 49th Tree of Life in CTOL. Including the four yods of the root edge makes 299 yods. This is the number of SLs in the 49-tree. This is amazing because it tells us that one half of the inner Tree of Life encodes the 49 Trees mapping the 49 subplanes of the cosmic physical plane and that the next five polygons up to the pentagon with 251 yods encode the 42 Trees with 251 SLs that map the 42 subplanes of the six cosmic superphysical planes. The two halves of the inner Tree of Life define the division between the cosmic physical and superphysical planes. That both numbers 49 and 91 could appear in this context by coincidence can be dismissed as highly implausible.

Notice that the 26-tree has the same number (161) of SLs as the first five polygons have yods. Only one other set of polygons is the counterpart of an n-tree, namely, the first 13 polygons, which are the counterpart of the 95-tree. However, this is of no interest, because only a number of overlapping Trees of Life has significance, an n-tree being always part of a larger number of overlapping Trees. We should expect only the former, not the latter, to be encoded in the inner Tree of Life. What is amazing is that there exists a combination of polygons whose yod population is equal to the number of SLs in CTOL. This encoding will next be shown to be unique.

Let us satisfy ourselves that a different (but still sequential) ordering of polygons would not also give rise to an encoding of either CTOL or N overlapping Trees, where N is larger than 91. The table lists the yod populations of the polygons in the inner Tree of Life and their running totals when the second set of seven polygons is reversed, so that they end with the dodecagon instead of the triangle. As before, the first seven polygons and the root edge have the same number of yods as the 49-tree has SLs. The first 12 polygons up the octagon now correspond to 76 overlapping Trees. This is uninteresting (hence the cross against this number) because 76 is smaller than 91, so it has no meaning in the wider context of CTOL. The first 13 polygons correspond to the 86-tree. This is permissable, being less than 91, but uninteresting because it is not 86 overlapping Trees.There is, therefore, no combination of polygons that is the counterpart of N overlapping Trees, where N>91. It is straightforward to confirm (see p. 394 in the author’s new book) that neither of the two remaining possible orderings of polygons:

dodecagon-triangle-triangle-dodecagon
dodecagon-triangle-dodecagon-triangle

are satisfactory, the first case because it generates the same results as before, the second case because, whilst it leads to 91 overlapping Trees, a subset of sequential polygons is also equivalent to 36 overlapping Trees, which makes no sense in the context of CTOL with 91 overlapping Trees. Only one ordering of polygons leads to a meaningful set of 91 overlapping Trees. We conclude that the encoding of CTOL in the inner Tree of Life is unique, as one would expect.

Those visitors to this website who are Theosophists need to realise that this proof of the encoding of CTOL in a unique subset of the set of 14 regular polygons is tantamount to a mathematical proof of the Theosophical doctrine of the seven planes of consciousness, each divided into seven subplanes. The fact proven above that the seven polygons making up one half of the inner Tree of Life encode the 49 subplanes of the cosmic physical plane is remarkable evidence supporting this proof and refutes the suggestion that the encoding could arise by chance. The two halves of the inner Tree of Life express the distinction between the words “physical” and “superphysical” — not in their normal sense, in which the former refers to the physical universe and the latter denotes non-material realms of existence, but in a much more profound sense that will be familiar only to students of mystical traditions. The proof confirms the elaboration of the teaching by Alice Bailey and others that the seven planes of consciousness discussed in the early Theosophical literature constitute but the lowest plane of seven cosmic planes. The five largest polygons in the other half of the inner Tree of Life encode the Tree of Life/tetractys map of the six superphysical cosmic planes. Their nature can be understood only in a faint, intuitive sense by means of the hermetic principle of correspondence: “As above, so below,” although Bailey’s writings may help to provide insight.

CTOL, the Tree of Life representation of all levels of reality, is encoded in 12 of the 14 polygons making up the inner Tree of Life. The complete growth of the Cosmic Tree of Life out of a single Tree of Life is encoded within the inner form of the latter. Like a hologram of an object, any piece of which in principle contains all the information needed to construct a complete holographic image of it, “the part contains the whole.” Like the DNA molecule in a living cell, the 14 polygons encode how the generic Tree of Life replicates itself until it becomes the Cosmic Tree of Life. The root edge and the set of seven separate polygons have 299 yods symbolising the 299 SLs of the 49-tree mapping the cosmic physical plane. The remaining five polygons have 251 yods symbolising the 251 SLs in CTOL above the 49-tree.

How some gematria number values of the Sephiroth, their Godnames, etc prescribe the 12 enfolded polygons encoding CTOL

The number of Binah is both the number of yods below Binah in the 1-tree and the number of corners of the 12 enfolded polygons encoding CTOL.

YAHWEH (26 ) ELOHIM (50 ) prescribes the 12 enfolded polygons with 76 corners of 87 sectors.

Enfolded, the 12 polygons that encode CTOL have 67 corners. Amazingly, this is both the number value of Binah, the third member of the Supernal Triad, and the number of yods below Binah of the 1-tree, when it is constructed from tetractyses. This illustrates the profound connection between the properties of sacred geometry — not just the Tree of Life but any object possessing sacred geometry — and the Kabbalist names of the Sephiroth, their Godnames, etc. There are 34 red corners outside the root edge of one set of seven polygons and 33 blue corners in the remaining five polygons of the set of 12. The former are the counterpart of the 34 red yods up to the level of Tiphareth and the latter are the counterpart of the 33 blue yods between Tiphareth and Binah. As the centre of the Tree of Life in both a geometrical and metaphysical sense, Tiphareth marks the transition in a human to transpersonal levels of awareness. The Lower Face of Malkuth-Yesod-Hod-Netzach-Tiphareth representing the human soul connects at Tiphareth to the Upper Face of Tiphareth-Binah-Chokmah-Kether — truly, cosmic levels of being. This is flagged geometrically by how the root edge separates two sections of the 12 enfolded polygons whose yods exactly mirror those in the Lower Face and the remainder of the 1-tree. Its extra 13 black yods correspond to the three corners of the missing triangle and square outside the root edge and to the 10 centres of the 14 polygons that are not also corners of polygons (the centre of the hexagon is a corner of the triangle and the centre of the decagon is a corner of the pentagon).

The (7+5) enfolded polygons encoding CTOL have 87 sectors with 76 corners, where 87 is the number value of Levanah, the Mundane Chakra of Yesod. 76 is the number value of YAHWEH ELOHIM, the Godname of Tiphareth. The nine enfolded polygons consisting of one set of seven polygons, the pentagon & the hexagon in the other set have 50 black corners, where 50 is the number of ELOHIM; the remaining octagon, decagon & dodecagon have 26 white corners, where 26 is the number of YAHWEH. See page 6 in Article 4 for how the Godnames of the 10 Sephiroth prescribe the (7+5) enfolded polygons.

Binah as the ‘Great Mother’ of CTOL

The connection between the number 67 of Binah and the 550 SLs in CTOL is shown by the amazing fact that the endpoints of the root edge coincide with the projections onto the plane of the (7+7) enfolded polygons of Daath (number value 474) and Tiphareth, whose Godname number is 76, so that 550 is their sum, whilst the sum of the numbers 31 & 36 of the Godnames assigned, respectively, to Chesed (EL) and Geburah (ELOHA) is 67. Daath and these three Sephiroth are the endpoints of two mutually perpendicular straight lines that divide in half the overlapping grey area — a shape known as the “Vesica Piscis” — of two circles of the same radius that overlap so that the centre of one lies on the circumference of the other. This central area of the 14 enfolded polygons of the inner Tree of Life (see here) determines the very numbers that define the cosmic whole, namely, CTOL.

The Vesica Piscis has been the subject of considerable speculation. Its significance in Kabbalah is that the Tree of Life is generated from four similar circles that overlap, centre-to-circumference, creating three such shapes:

The four circles symbolize the four Worlds of Atziluth, Beriah, Yetzirah & Assiyah. A set of overlapping Trees of Life is a chain of Vesica Piscis, so that it can be regarded as the basic building block of this particular kind of sacred geometry. However, its true meaning is more profound: all existence — physical & superphysical — emerges from Daath like a baby passing through the birth canal from the womb, and this is represented in the geometry of the inner Tree of Life as the two sets of seven regular polygons growing out of their shared side — the “root edge,” which connects Daath outside phenomenal existence to Tiphareth at the centre of the Tree of Life:

outer & inner Trees of Life

The outer and inner Trees of Life.*

One of the titles of Binah in Kabbalah is “The Great Mother” (Hebrew: aima, or “mother”). The number 67 of this Sephirah (see here) embodying the cosmic feminine principle quantifies the 67 corners of the 12 enfolded polygons that map CTOL (see previous page):

In other words, this number selects that subset of the set of 14 polygons which encodes the Tree of Life representation of all levels of reality! It is, truly, an amazing, mathematical reason for why Binah should have this title.

YAHWEH prescribes the yod population of the (7+5) enfolded polygons encoding CTOL

 1¹  2¹  3¹  4¹  	  	 
 1²  2²  3²  4²
 1³  2³  3³  4³
 1⁴  2⁴   3⁴  4⁴
  	 494 =

There are 490 (49×10) yods outside the root edge, 260 (=26×10) yods being in the set of seven enfolded polygons on one side of this edge and 230 yods belonging to the five polygons enfolded on the other side of it (see diagram above). This shows how EL CHAI, the Godname assigned to Yesod with number value 49, prescribes this set of polygons. As 494 = 26×19, where 26 is the number value of YAHWEH, the Godname of Chokmah, and 19 is the number of yods in a Type A triangle (see here), the number 494 has the representation shown below:

triangular representation of 494

The sum of the nine red 26s on the sides of the outer triangle is 234, which is the number of yods in the last five enfolded polygons. The sum of the ten blue 26s inside the triangle is 260, which is the number of yods outside the root edge in the seven enfolded polygons on the other side. Ways in which other Godnames mathematically prescribe the holistic parameter 494 are discussed in Article 4 (Web, PDF).

Outer & inner Tree of Life representation of CTOL

The outer Tree of Life contains 70 yods when its 16 triangles become tetractyses. Seven yods lining each side pillar and two hexagonal yods on the Path connecting Chesed and Geburah (that is, the 16 black yods in the diagram opposite) are shared with the (7+7) enfolded polygons, which have 524 yods. Therefore, (70−16=54) yods belonging to the outer Tree are intrinsic to it. They include four yods located at the Sephiroth Kether, Tiphareth, Yesod & Malkuth, leaving 50 intrinsic yods that are not located at Sephiroth, where 50 is the number value of ELOHIM, the Godname of Binah. The (7+7) enfolded polygons have (524−16=508) yods that are intrinsic to them. (4+4=8) of these yods are yellow centres of polygons (the centres of the two decagons are not included because they are also corners of the pentagons and we want to count here only intrinsic yods that surround the centres of their own polygons; the centres of the triangles belonging to the inner Tree coincide with the two hexagonal yods on the Chesed-Geburah Path and are not among the 508 yods, being shared with the outer Tree). Therefore, (508−8=500=50×10) intrinsic yods surround the centres of the polygons. The outer and inner Trees of Life have (50+500=550) intrinsic yods that are neither Sephiroth nor centres of polygons. Outside the root edge are 260 yods in each set of seven enfolded polygons. Of these, (260−8−4=248) yods are intrinsic yods that surround their centres, that is, both sets have 496 such yods. 248 is the number value of Raziel, the Archangel of Chokmah, and 496 is both the number value of Malkuth (the physical universe, in the cosmic context of this Sephirah) and the dimension of the two possible superstring gauge symmetry groups SO(32) and E₈×E₈. The mirror symmetry of the two sets of seven enfolded polygons of the inner Tree of Life is responsible for the existence of two identical groups E8 present in one unified symmetry group as a direct product.

The 550 yods other than Sephiroth and centres of polygons that are intrinsic to the outer and inner Trees of Life symbolize the 550 SLs of CTOL. The 50 intrinsic yods other than Sephiroth in the outer Tree denote the lowest 50 SLs up to Yesod of the ninth Tree. It is the 501st SL from the top of CTOL. The complete Godname of Kether is:

Ehyeh Asher Ehyeh

Its traditional translation is “I am that I am.” The gematria number value 543 of this Godname measures the 543 SLs in CTOL down to Yesod of the second Tree — the last SL before the first Sephirah of Construction of the lowest Tree (see here). The number value 501 of ASHER is the number of SLs down to Yesod of the ninth Tree. The 500 SLs above it are symbolized by the 500 intrinsic yods surrounding the centres of all the polygons except the decagons, whilst the 50 SLs up to this SL are symbolized by the 50 intrinsic yods other than Sephiroth that belong to the outer Tree of Life and are coloured green in the adjacent diagram. The interpretation of what the number value of EHYEH ASHER EHYEH means in terms of CTOL is consistent with the geometries of the outer and inner Trees of Life because the 50 intrinsic yods of the former correspond to its lowest 50 SLs and the 500 intrinsic yods of the latter correspond to its highest 500 SLs.

Through its letter values, EHYEH (Hebrew: AHIH) specifies the ten Sephiroth (H=10) of the Tree of Life, Daath (A=1) and the five centres of each set of seven polygons (H=5) that do not coincide with any of their corners (see here). This means that the number 550 refers to the yods unshared by the combined outer and inner Trees of Life other than those specified by EHYEH.

What does the switch between the outer and inner Trees of Life mean in terms of CTOL? The 50th SL (Yesod of the eighth Tree), counting from its lowest point, is the 26th tree level (see Article 2). The lowest 25 tree levels signify the 25 spatial dimensions of the 26-dimensional space-time predicted by the quantum mechanics of spinless strings. The 26th tree level signifies the dimension of time. The switchover, therefore, represents the transition from the space-time continuum to non-temporal realms — from that part of CTOL where time exists to those levels of reality beyond space and time. The 248 intrinsic yods outside the root edge of one set of seven enfolded polygons correspond to the 248 SLs in CTOL down to Binah of the 50th Tree — the very Sephirah whose Godname number specifies the Tree to which it belongs! The 496 intrinsic yods outside the root edge of both sets of enfolded polygons correspond to the 496 SLs in CTOL down to Chesed of the ninth Tree. The remaining 54 yods consist of the four yods on the root edge and the 50 intrinsic yods in the outer Tree other than Sephiroth; they correspond to the four SLs from Geburah of the ninth Tree to Hod of this Tree and to the lowest 50 SLs up to Yesod of this Tree.

An alternative view
The fact that the outer Tree of Life has 54 intrinsic yods unshared with its inner form and that

54 + 496 = 550

allows an alternative analysis that retains as part of the set of 550 yods the four unshared Sephirothic points (Kether, Tiphareth, Yesod & Malkuth) and excludes the four yods making up the root edge from the set of 496 yods. Outside the root edge in each set of seven enfolded polygons are 260 yods that comprise eight shared yods, four centres of polygons and the centre of the decagon that coincides with a corner of the pentagon. Because of this dual character, it should be included in any count of the total number of yods that surround the centres of the polygons to which they belong. This means that the number of unshared yods outside the root edge that truly surround centres of each set of seven polygons = 260 − 8 − 4 = 248, both sets having (248+248=496) such yods. Hence, the number 550 is embodied in the combined Trees of Life as the number of yods unshared between them that are outside the root edge and surround centres of polygons. This viewpoint seems more natural than the one discussed above because it does not exclude from the set of 550 yods for no good reason the four Sephiroth of the outer Tree of Life that are unshared with its inner form. Instead, it excludes the four yods of the root edge. This is easier to justify because of their unique status in being the root source of the outer Tree that connects it to the microbiological level of its “DNA” — its inner form.

The Tetrad expresses the outer & inner Trees of Life
The total number of yods in the combined Trees of Life = E₈×E₈ = 54 + 524 = 578 = 2 + 576 = 2 + 24² = 2 + 2×288, where

288 = 1!× 2!×3!×4!.

It comprises the two endpoints of the root edge and 24² yods, of which 1!×2!×3!×4! yods belong to each half of the combination. This illustrates the Pythagorean character of its mathematical beauty as an archetypal object. Surrounding the centres of the seven separate, regular Type A polygons in each half of the inner Tree of Life are 288 yods. The permutational meaning of this number is as follows: arranged in a tetractys, ten different objects have 1! (= 1) permutations of its apex object, 2! (=2) permutations of the two objects in the second row, 3! (=6) permutations of the three objects in the third row and 4! (=24) permutations of the four objects in the fourth row. Every permutation generates a different tetractys. The total number of tetractyses that can be generated by rearranging the objects in each row = 1×2×6×24 = 288. The 288 yods surrounding the centres of the seven polygons can be regarded as corresponding to every possible tetractys that can be created from 10 objects by re-arranging them within each row. Rows 1 & 3 generate the factor 1!×3! = 6, which corresponds to the six yods per sector of a polygon that surround its centre, and rows 2 & 4 generate the factor 2!×4! = 48, which corresponds to the 48 sectors of the seven polygons.

As 17² = 289, this is the number of yods in each half of the combined Trees of Life: 578 = 2×289 = 2×17²

17² = 1 + 3 + 5 + … + 33,

and 33 is the sum of the number of permutations of the objects in each row:

1! + 2! + 3! + 4! = 33,

The outer Tree has 70 yods (35 yods associated with each half), of which eight black yods are shared with its inner form (see below), leaving 27 green yods in each half. There are two white endpoints of the root edge and 522 yods in the (7+7) enfolded, Type A polygons. Eight yods (the black yods in the diagram shown above) in the 261 yods associated with each half of the inner Tree of Life are shared with its outer form, leaving 253 red or blue intrinsic yods. Associated with each half is one endpoint and (27+253+8=288) yods. The total number of yods in the combined Trees of Life = 2(1+288) = 2 + 576 = 578 = 2 + 24².

The number of yods in the n-gon whose sectors are 2nd-order tetractyses = 72n + 1, where 72 is the number value of Chesed and “1” denotes its centre. A square (n=4) has 289 yods, where 289 = 17². The two squares present in the inner Tree of Life have (2×289=578) yods when constructed from 2nd-order tetractyses. In the identity highlighted above, the first “2” denotes the two centres of the pair of squares, 24² denotes the number of yods surrounding them and 17² denotes the number of yods in each square. The fact that the yod population of the combined Trees of Life is the yod population of the two squares present in the inner Tree when its sectors are 2nd-order tetractyses is a remarkable illustration of the Tetrad Principle at work (this is discussed in Article 1). It is further illustrated by the fact that

 	                   3 	5 	7 	9
	                  11 	13 	15 	17
17² − 1 = 288 =       19 	21 	23 	25
  	                  27 	29 	31 	33

i.e., the number of yods other than the endpoints of the root edge that are associated with each half of the combined Trees of Life is a 4×4 array of the first 16 odd integers after 1. Notice that 24 yods surround the centre of a Type A square, so that 24² is the sum of the integers 24 assigned to these 24 yods. Symbolising the Tetrad, the square generates out of itself the yod population of the combined forms of the Tree of Life.

The identity highlighted above is also expressed by the two sets of separate, Type A polygons of the inner Tree of Life when partitioned by the root edge with its two endpoints. 288 yods surround the centres of each set, so that both sets have 24² such yods. Including the endpoints of the root edge separating the two sets, 17² yods are associated with each set.

Of the 578 yods in the combined Trees of Life, four are corners of triangles in the outer Tree that are unshared with its inner form because they are located at the Sephiroth on the Pillar of Equilibrium, whilst 80 are corners of the (47+47=94) sectors of the (7+7) enfolded polygons. Hence, the (16+47+47=110) tetractyses in the combined Trees have 84 corners, where

84 = 1² + 3² + 5² + 7²,

and (578−84=494) hexagonal yods, where

  	 11 	 21 	 31 	 41
  	  	  		 
  	 12 	 22 	 32 	 42

 494 =
	  	  	  	 
  	 13 	 23 	 33 	 43
  	  	  	  	 
  	 14 	 24 	 34 	 44

This is another illustration of the power of the Tetrad to determine properties of holistic systems. It is worth noting that two of the 578 yods are unique in that they behave as both hexagonal yods and corners, i.e., they have a dual character. This is because, when all triangles in the outer Tree of Life are tetractyses, the two hexagonal yods on the Path connecting Chesed and Geburah coincide with the centres of the two Type A triangles in the inner Tree of Life. Whether we regard these shared yods as hexagonal yods of tetractyses in the outer Tree of Life or as corners of tetractyses in the inner Tree is, purely, a matter of context. If we count them as corners, then the total yod population is

578 = 84 corners + 494 hexagonal yods.

If we count them as hexagonal yods, then there are (2+494=496) hexagonal yods and (84−2=82) corners, so that

578 = 82 + 496 = 2 + 80 + 496.

“2” denotes the two endpoints of the root edge, 80 is the number value of Yesod (“Foundation”) and 496 (“KIngdom”) is the number value of Malkuth. In other words, the yod population of the combined Trees of Life (excluding the two endpoints of the root edge) is the sum of the gematria number values of the last two Sephiroth! Here is the foundation of the mathematical domain or territory that is the cosmic blueprint.

The number 550 is the number of sides of the sectors of the triangles in the 10-tree

ADONAI (Hebrew: אדני, ADNI), the Godname of Malkuth with number value 65 , prescribes the 10-tree with 65 SLs. In fact, its very letter values: A = א = 1, D = 4 = ד, N = 50 = נ and I = 10 = י are the numbers of SLs of various types in the 10-tree (see diagram). The 10-tree is equivalent to a decagon with tetractys sectors drawn inside a square. The number 65 is the sum of the first 10 integers after 1:

65 = 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11.

The 10-tree contains 127 triangles, where 127 is the 31st prime number. This shows how EL, the Godname of Chesed with number value 31, prescribes the 10-tree. The 127 triangles have 169 sides. Their (3×127=381) sectors have (65+127=192) corners with (3×127 + 169 = 550) sides. Embodied, therefore, in the 10-tree are two numbers, 192 and 550, that are characteristic parameters of holistic systems (see The holistic pattern). The 10-tree embodies the very number of SLs in CTOL! As we shall discover in the section Superstrings as sacred geometry/Tree of Life, it also embodies the primary structural parameter 1680 of the subquark state of the E8×E8 heterotic superstring. This shows how ADONAI prescribes the helical form of each of the ten whorls of the UPA/subquark, the ten overlapping Trees of Life representing the ten spatial dimensions in 11-dimensional space-time. It is an example of a sacred-geometrical context manifesting the gematria number values of both the Godname (65) and Mundane Chakra (168) of the same Sephirah. It is implausible in the extreme to dismiss this as due to chance, particularly when this very number 1680 appeared in the century-old description (see here) by C.W. Leadbeater of the whorls of the UPA (another coincidence, according to this view!). It is tantamount to appealing to the miraculous to argue that the possession by the 10-tree of the holistic parameter 550 could be coincidental as well. Clearly, it is an untenable way of accounting for such remarkable properties in order to avoid having to admit that the Tree of Life is, indeed, the cosmic blueprint, as Kabbalists have always claimed, although, of course, not on the basis of any of the hard, mathematical evidence presented in this website.

The embodiment of the number 550 in the geometry of the 10-tree as the 550 sides of the 381 sectors of its 127 triangles is discussed here as one of the examples of how it manifests in objects possessing sacred geometry. See case (g) in the discussion.

Representation of CTOL parameter 550 by two joined squares

550 yods outside their root edge surround the centres of two joined squares with 2nd-order tetractyses as their sectors.

A 2nd-order tetractys contains 85 yods, where

85 = 4⁰ + 4¹ + 4² + 4³.

13 yods line each side, so that (85−13=72) yods are added by each sector of an n-gon when it is a 2nd-order tetractys. The number of yods in an n-gon constructed from 2nd-order tetractyses ≡ N(n) = 72n + 1, where “1” denotes its centre. The number of corners of 1st-order tetractyses = 10n + 1 and the number of hexagonal yods = 62n. Notice that 72 is the number of Chesed and that 62 is the number of Tzadkiel, the Archangel of this Sephirah. For the square (n=4), the number of yods surrounding its centre is

N(4) = 4×72 = 288 = 1¹ + 2² + 3³ + 4⁴ = 1!×2!×3!×4!.

They comprise 40 corners (35 outside one side) and 248 hexagonal yods (240 outside one side), where 248 is the number of Raziel, the Archangel of Chokmah. For two joined squares, (288−13=275) yods outside their shared root edge surround the centre of each square. Symbol of the Tetrad, the square determines the number of SLs in CTOL because (275×2=550) yods outside their root edge surround the centres of two joined squares. They comprise (35+35=70) black corners of (40+40=80) 1st-order tetractyses and (240+240=480) coloured hexagonal yods, each square having 240 hexagonal yods outside the shared side. Hence, the two separate squares contain (248+248=496) hexagonal yods. This is the square representation of the (248+248=496) roots of E₈×E₈, one of the two symmetry groups known to describe superstring forces that are free of quantum anomalies. The two sets of eight hexagonal yods that line two sides of the separate squares that become the root edge of the joined squares symbolise the two sets of eight simple roots of E₈×E₈. We discover that the square provides a natural connection between the number (550) of Sephirothic emanations in CTOL and the root composition of E₈×E₈. The Tetrad determines all properties of the pair of squares. For example, it determines their 70 corners outside the root edge because 70 is the fourth, 4-dimensional, tetrahedral number* after 1, whilst 35 (number of corners of 1st-order tetractyses in each square outside the root edge) is the fourth tetrahedral number after 1. It determines their 80 2nd-order tetractyses because 80 = 10×8, where 10 = 1 + 2 + 3 + 4 and 8 = 4th even integer. It determines their 480 hexagonal yods because

480 = 16×30 = 4²×(1²+2²+3²+4²) = 4² + 8² + 12² + 16².

It determines their 550 yods outside the root edge that surround their centres because 550 = 10×55, where 10 = 1 + 2 + 3 + 4 and 55 is the fourth, square pyramidal number** after 1. Including its centre, each square has 36 corners of 40 1st-order tetractyses outside the root edge, where 36 = sum of the first four odd integers and the first four even integers:

36 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = (1+3+5+7) + (2+4+6+8).

The inner form of 10 Trees of Life embodies the CTOL parameter 550

The seven separate Type A polygons in each half of the inner Tree of Life have 48 corners. Their 48 sectors have (48+7=55) corners. The 70 separate Type A polygons in the inner form of 10 Trees of Life have 480 sectors with 550 corners. Each corner corresponds to an SL of CTOL. The seven enfolded Type A polygons in each Tree have 47 sectors with 41 corners. Hence, the 470 sectors of the 70 polygons enfolded in 10 Trees have 401 corners. As the Godname ADONAI with number value 65 prescribes the 10-tree with 65 SLs (see #12), this Godname prescribes the number of SLs in CTOL in two ways:

the sectors of the 127 Type A triangles in the 10-tree have 550 sides (see #12);
the 70 separate Type A polygons making up the inner form of the 10-tree have 480 sectors with 550 corners.

The complete inner form of the 49-tree mapping the 49 subplanes of the seven planes of consciousness consists of (2×7×49=686) polygons whose (2×48×49=4704) sectors have (49×2×55=5390) corners, where 5390 = 10×539 and 539 is the sum of the number values of the Godnames of the seven Sephiroth of Construction:

539 = 31 + 36 + 76 + 129 + 153 + 49 + 65.

As these seven planes are the manifestation of the seven Sephiroth of Construction, the sum of the Godname numbers assigned to the latter determines how many corners create the sectors of the polygons in the inner form of the 49-tree. This is an amazing property of these numbers. Notice that 539 = 474 + 65, where 474 is the number of Daath (“knowledge”) and 65 is the number of ADONAI, the Godname of Malkuth that expresses the form of the Tree of Life (in this case, the inner form of the 49-tree mapping the cosmic physical plane). The sum of the gematria number values of Daath and ADONAI aptly measures the inner geometry of the 49-tree as the “cosmic physical” expression of the seven Sephiroth of Construction.

The number 539 is the number of SLs in CTOL above the 1-tree, which has 11 SLs:

11 + 539 = 550.

As 11 + 65 = 76, the CTOL parameter 550 is the sum of the numbers of Daath and YAHWEH ELOHIM:

474 + 76 = 550.

It is the sum of numbers of the Godnames of all the Sephiroth except the first Sephirah (Kether) and the last one (Malkuth):

550 = 26 + 50 + 31 + 36 + 76 + 129 + 153 + 49.

Alternatively, the number 550 is the sum of the numbers of the Godnames of all the Sephiroth except Binah and Geburah:

550 = 21 + 26 + 31 + 76 + 129 + 153 + 49 + 65.

As 11 + 15 = 26, where 15 is the number of YAH, the older version of YAHWEH, 539 is the sum of the following Godname numbers of all Sephiroth except the first and last ones:

539 = 15 + 50 + 31 + 36 + 76 + 129 + 153 + 49.

The CTOL parameter 91 is the sum of the number values of ADONAI and YAHWEH:

65 + 26 = 91.

The Platonic solids

The five Platonic solids

The tetrahedron, octahedron and icosahedron have triangular faces, the cube has square faces and the dodecahedron has pentagonal faces. Consider each face of a Platonic solid divided into its sectors. The table below lists their numbers of corners, sides & triangular sectors.

Number of vertices = V; number of edges = E; number of faces = F; number of sectors in a face = m.

Number of corners = C = V + F; number of sides = e = E + mF; number of triangles = T = mF (m = 3 for tetrahedron, octahedron & icosahedron; m = 4 for cube; m = 5 for dodecahedron).

Polyhedron	V	E	F	m	C	e	T	Total = C + e + T
tetrahedron	4	6	4	3	8	18	12	38
octahedron	6	12	8	3	14	36	24	74
cube	8	12	6	4	14	36	24	74
icosahedron	12	30	20	3	32	90	60	182
Subtotal	30	60	38	-	68	180	120	368
dodecahedron	20	30	12	5	32	90	60	182
Total	50	90	50	-	100	270	180	550

Embodiment of the holistic parameter 550

Here is the converted text in Markdown format:

a) Geometry of faces

The lowest row in the table indicates that there are 550 corners, sides & triangles in the 50 faces of the five Platonic solids. The Divine Name ELOHIM with number value 50 that is associated with Binah, the third Sephirah in the Tree of Life, prescribes the five regular polyhedra with 50 vertices and 500 ( = 50 × 10) other geometrical elements. This demonstrates par excellence the formative, or shape-determining, nature of the archetypes embodied in this Sephirah heading the Pillar of Judgement (one of the Kabbalistic titles of Binah is “Aima,” the divine mother). The shapes of the regular polyhedra require 550 geometrical elements to create them, where 550 = 10(1+2+3+4+5+6+7+8+9+10) = 10(1²+2²+3²+4²+5²), i.e., this number is the sum of 50 squares of integers. They include 100 corners of 180 triangles, where 100 (the 50th even integer) = 10² = 1³ + 2³ + 3³ + 4³. This illustrates how the Pythagorean Decad (10=1+2+3+4) and the integers 1, 2, 3 & 4 symbolized by the four rows of dots in the tetractys symbolizing the Decad express the geometry of the Platonic solids. Their holistic character is demonstrated by the fact that their faces are composed of 550 geometrical elements, for this is the number of SLs in the 91 Trees of Life that make up CTOL.

(SL = Sephirothic level, denoted by the black dots in the diagram above). Every single, geometrical element composing the faces of the five Platonic solids corresponds to an SL of CTOL. Notice also that 550 = 10×55, where 55 is the tenth number after the beginning of the famous Fibonacci sequence of numbers:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …

550 is, therefore, not only ten times the sum of the first ten integers after 0 but also ten times the tenth Fibonacci number after 0! This beautiful property points to a deep involvement of the Fibonacci numbers in the sacred geometry of the Platonic solids. This is confirmed in Article 50 (Parts 1 & 2) (WEB, PDF). As 550 = 10 + 20 + 30 + 40 + 50 + 60 + 70 + 80 + 90 + 100, the tetractys array of the first ten integers multiplied by 10:

	10
	20	30							20	30
40	50	60		=			+	40	50	60	=	180 + 50 + (20+30=50) + (40+60+80+90=270),
70	80	90	100		70	100			80	90

naturally reproduces the geometrical composition of the triangles making up the five Platonic solids! This is because:

the sum of the integers 10, 70 & 100 at the corners of this tetractys is 180, which is the number of triangular sectors in their faces;
the number 50 at its centre is the number of their vertices;
the sum of the integers 20 & 30 in the second row is 50, which is the number of corners of these sectors inside their edges;
the sum of the remaining four integers 40, 60, 80 & 90 is 270, which is the number of sides of their 180 sectors. These 270 sides comprise 90 polyhedral edges (the penultimate number 90 in the sequence of integers 10-100) and (40+60+80=180) other sides of sectors. Alternatively, the 270 sides comprise the 90 sides of the 60 sectors in the faces of the dodecahedron and the 180 sides of the 120 sectors of the 38 faces of the first four Platonic solids.

The 550 geometrical elements making up the faces of the five Platonic solids consist of their 50 vertices and 500 other elements. This 50:500 division manifests in the superposed outer and inner Tree of Life as the 50 intrinsic yods other than Sephiroth that belong solely to the former and as the 500 yods belonging only to the (7+7) enfolded polygons that surround their centres (see here). The 50 vertices of the five Platonic solids correspond to the outer Tree of Life with 50 intrinsic yods and the 500 extra geometrical elements needed to construct the faces of the Platonic solids correspond to the inner Tree of Life with 500 intrinsic yods. Four of these extra elements are the centres of the faces of the tetrahedron. They correspond to the four yods on the root edge shared by the (7+7) enfolded polygons.

The 550 SLs in CTOL correspond to the 550 geometrical elements making up the faces of the 5 Platonic solids when they are divided into their sectors.

The picture above shows how the 550 geometrical elements correspond to the 550 SLs in CTOL. Seven overlapping Trees of Life have 46 SLs. This is also the number of SLs below the top of the seventh Tree of Life in CTOL. They correspond to the 46 vertices of all the Platonic solids except the tetrahedron. The remaining 504 geometrical elements comprise 252 more elements in the lower halves of the five Platonic solids and 252 elements in their upper halves. The former correspond to the 252 SLs in the next 42 Trees up to the top (298th SL) of the highest Tree in 49 overlapping Trees of Life; counting the top of the 7-tree, there are 252 SLs up to the top of the 49-tree. The latter correspond to the 252 SLs in the next 42 Trees up to the top (550th SL) of the 91st Tree. The upper halves (excluding vertices) correspond to the six cosmic superphysical planes with 42 subplanes mapped by 42 Trees, whilst the lower halves (again, excluding vertices) correspond to the six superphysical planes with 42 subplanes mapped by 42 Trees. The 46 vertices correspond to the 46 SLs of seven Trees mapping the physical plane. As will be revealed in the discussion of the Sri Yantra in the next section, this 46:252:252 division exists as well in the representation of the seven cosmic planes/CTOL by the Sri Yantra.

b) Yod composition of tetractys sectors of faces

When the mF sectors of the faces of a Platonic solid with (E+mF) sides become tetractyses, they have [2(E+mF) = 2E + 2mF] hexagonal yods on their sides and (V + F = 2 + E) corners, using Euler’s formula for a convex polyhedron:

V − E + F = 2.

Therefore, the number of yods lining the sides of the mF tetractyses in the faces of a Platonic solid = 2E + 2mF + 2 + E = 2 + 3E + 2mF. “2” denotes two diametrically vertices (two adjacent vertices, in the case of the tetrahedron). For the five Platonic solids, ∑E = 90 and ∑mF = 180 (see table above). The number of yods lining the sides of the 180 tetractyses in the 50 faces of the five Platonic solids = ∑(2+3E+2mF) = 5×2 + 3×90 + 2×180 = 640, where 640 is the number value of Shemesh, the Mundane Chakra of Tiphareth. They include 50 vertices and 50 corners of sectors at the centres of their faces. Hence, (640−50−50=540) hexagonal yods line the sides of their 180 tetractyses. (10+540=550) yods line these sides other than the 40 vertices and the centres of the 50 faces that surround axes passing through the two diametrically opposite vertices of each Platonic solid (any two adjacent vertices, in the case of the tetrahedron). The 550 boundary yods correspond to the 550 SLs in CTOL. The 10 vertices lying on the axes of the five Platonic solids (their “poles”) correspond to the 10 SLs that belong to the highest Tree in CTOL. The 540 hexagonal yods on the sides of the 180 tetractyses making up their 50 faces correspond to the 540 SLs in the 90 Trees below it. This is another way in which the five Platonic solids represent CTOL.

The number of yods in a Type B n-gon = 15n + 1 (see Section 2 in Power of the polygons/General view). 30n yods surround the centres of two separate, Type B n-gons. The first four separate, regular polygons of the inner Tree of Life are the triangle (n=3), square (n=4), pentagon (n=5) & hexagon (n=6). They have (3+4+5+6=18) corners. The number of yods surrounding the eight centres of the first (4+4) separate, Type B polygons = ∑30n = 30×18 = 540. They are the counterpart of the 540 hexagonal yods that line the 270 sides of the 180 tetractyses in the 50 faces of the five Platonic solids when these faces are Type A polygons. The holistic nature of the first (4+4) regular polygons is discussed in Article 48. This examplifies the Tetrad Principle discussed in Article 1 when it is applied to the regular polygons.

91 corners of triangles inside 5 Platonic solids
Suppose that the centre of each Platonic solid is joined to its vertices. This creates E internal triangles, where E is the number of its edges. When each triangle is Type A, the Platonic solid has 3E sectors of its internal triangles. They have (E+1) corners. Noting that the five Platonic solids have 90 edges, the number of sectors of their internal triangles = 3∑E = 3×90 = 270. The number of their corners = ∑(E+1) = 90 + 5 = 95. This is the number value of Madim, the Mundane Chakra of Geburah. Next, suppose that the five Platonic solids lie inside one another, sharing only a common centre. The number of internal corners = 91. This is the number of Trees of Life in CTOL:

The interiors of the five Platonic solids sharing the same centre have 91 corners that correspond to the 91 Trees in CTOL. The tetrahedron has six edges that are sides of six internal Type A triangles. Their 18 sectors have seven corners. Inside the remaining four Platonic solids are 252 sectors with (91−7=84) corners. This 7:84 division in the 91 corners corresponds to the 7-tree mapping the physical plane and to the 84 Trees above it in CTOL. It also corresponds to the 84 coloured hexagonal yods in the Cosmic Tetractys that are outside the seven coloured hexagonal yods in its central tetractys. The octahedron and the icosahedron have 126 sectors of 42 internal Type A triangles with 42 corners other than their centres. Likewise, the cube and the dodecahedron have 42 corners in their interiors other than centres. One set of 42 corners corresponds to the 42 Trees of Life above the 7-tree in the 49-tree mapping the cosmic physical plane and to the 42 coloured hexagonal yods in the six smaller 1st-order tetractys in the Cosmic Tetractys; the other set of 42 corners corresponds to the 42 Trees above the 49-tree and to the 42 coloured hexagonal yods in the six larger 1st-order tetractyses in the Cosmic Tetractys. Remarkably, we see that the tetrahedron corresponds to the 7-tree mapping the physical plane, the octahedron & icosahedron correspond to the 42 Trees mapping the six superphysical planes and the cube & dodecahedron corresponds to the next 42 Trees mapping the six cosmic superphysical planes. There are three other possible combinations. However, the combination scheme just discussed seems the correct one, intuitively speaking, as the cube & dodecahedron have more external corners than the three other pairs of Platonic solids with 42 internal corners, a property that makes their correspondence with the cosmic superphysical planes more appropriate than these others pairs. The essential point that needs to be made here is the amazing fact that, constructed from Type A triangles, the five Platonic solids embody the same 7:42:42 pattern that exists for the physical plane with seven subplanes, the six superphysical planes with 42 subplanes and the six cosmic superphysical planes with 42 subplanes. This pattern must exist in the set of Platonic solids, the Cosmic Tetractys and CTOL because, being holistic systems, they must display analogous structural parameters.

The table in Sacred geometry/Platonic solids indicates that there are 550 hexagonal yods in either the sectors of the faces of the dodecahedron constructed from tetractyses or its internal triangles divided into their sectors. The table calls Platonic solids “Type A” if they have tetractyses as internal triangles and “Type B” if their internal triangles are divided into their sectors. Regarded by Plato as the representation of the celestial sphere because, of all the five regular polyhedra, it most approximates the perfect sphere, the dodecahedron embodies the number 550 as one of the defining parameters of holistic systems, of which this mathematically complete set of 3-dimensional, regular polyhedra is an example. There are 240 hexagonal yods in its 12 faces and 310 (=31×10) hexagonal yods in its interior. The counterparts of these in the 550 yods making up the root edge and the (7+5) regular polygons encoding CTOL are the 240 hexagonal yods in one set of seven polygons and the remaining 310 yods in the root edge and all 12 polygons (see here). The dodecahedron has 12 faces with 20 vertices & 30 edges, so that its (12×5 + 30×3 = 150 = 15×10) tetractyses have (20+12+30=62) corners surrounding its centre, where 15 is the number value of YAH, the (older) Godname of Chokmah, and 62 is the number value of Tzadkiel, the Archangel of Chesed (see here), and the 31st even integer. The dodecahedron has (20+12=32) vertices & faces, where 32 is the 31st integer after 1. These properties clearly demonstrate how the Godname EL (“God”) of Chesed with number value 31 prescribes the Platonic solid that represents the completion of the sequence of regular polyhedra. EL also prescribes all five solids because, starting with their 50 vertices as unconnected points in space, (50+90+180=320=32×10) more points & lines are needed to construct their 50 faces with 50 centres, 90 edges & 180 sides of sectors, 32 being the 31st integer after 1. Alternatively, 310 (=31×10) points & lines in their faces surround axes that join pairs of vertices diametrically opposite each other and pass through the centres of the polyhedra.

The table in Sacred geometry/Platonic solids also indicates that the five Platonic solids have 910 (=91×10) hexagonal yods when they are Type A. This number is the number of hexagonal yods inside the five Platonic solids when they are Type B. They embody not only the 550 SLs of the Cosmic Tree of Life (CTOL) as the 550 geometrical elements in their faces but also its 91 Trees of Life. This, again, is not a coincidence, because all representations of holistic systems are characterized by the same set of defining parameters, such as 550 & 91.

Each half of the dodecahedron contains (550/2=275) hexagonal yods. Given that either half comprises 10 vertices, 15 edges & 6 pentagonal faces, the various types of these yods are shown below:

Types of hexagonal yods in dodecahedron

Each half of the dodecahedron has:

Types of hexagonal yods

There are 20 violet hexagonal yods on the 10 sides of internal triangles that join the centre of the dodecahedron to the 10 vertices in each half of it. There are (275+20=295) hexagonal yods either in the lower half or on sides joining the centre to the 20 vertices. This leaves 255 hexagonal yods in the upper half that do not line internal sides joining the centre to vertices.

295 = 20 + 20 + 60 + 30 + 30 + 90 + 45.

255 = 60 + 30 + 30 + 90 + 45.

Suppose that the dodecahedron is orientated so that its vertical axis passes through its centre and two diametrically opposite vertices. Then two violet hexagonal yods lie on each half of the axis, which is surrounded by (9+9=18) vertices, so that the number of violet hexagonal yods in its lower half is:

20 = 2 + (9×2=18).

Therefore,

295 = 2 + 18 + 45 + 20 + 60 + 30 + 30 + 90

= (2+45) + (20+60) + (18+30+30+90)

= 47 + 80 + 168,

where

47 = 2 + 45,

80 = 20 + 60,

and

168 = (18+30+30) + 90 = 78 + 90.

There are:

47 hexagonal yods either on the lower half of the axis or at centres of tetractyses in the 15 Type A triangles inside the lower half of the dodecahedron. They correspond to the 47 SLs in the 7-tree that maps the seven subplanes of the physical plane (see previous page);
(80+168=248) more hexagonal yods in the lower half. 80 is the number value of Yesod, 248 is the number value of Raziel, the Archangel of Chokmah, and 168 is the number value of Cholem Yesodoth, the Mundane Chakra of Malkuth. The number value 78 of Cholem is the number of red (30), yellow (30) & violet (18) hexagonal yods in the lower half of the dodecahedron that surround its axis; the number value 90 of Yesodoth is the number of turquoice hexagonal yods in this half:

Number value of Cholem Yesodeth

The 248 hexagonal yods correspond to the 248 SLs above the 7-tree as far as Chesed of the 49th Tree mapping the highest subplane of the seventh plane. The 80:168 division of 248 is characteristic of holistic systems embodying the number 248 as the dimension of E8, the rank-8, exceptional Lie group that plays a central role in E8×E8 heterotic superstring theory. For example, it shows up in the way the Godname EL prescribes this number in five overlapping Trees of Life (see here).

255 hexagonal yods in the upper half of the dodecahedron that do not line sides joining vertices to its centre; they correspond to the 255 SLs above Chesed of the 49th Tree (295th SL) up to the top of CTOL. The unique nature of this SL is that it is the first Sephirah of Construction of the first Tree of Life expressing this Sephirah in the 49 Trees that map the cosmic physical plane.

Remarkably, we see that the complete set of 550 hexagonal yod contains subsets that differentiate between the 47 SLs of the 7-tree mapping the physical plane, the next 248 SLs up to Chesed of the 49th Tree and the remaining 255 SLs of CTOL. The 47 hexagonal yods either line the axis or are at centres of tetractyses that formally correspond to Malkuth. In the same way, the physical plane corresponds to Malkuth, being the lowest of the seven planes that correspond formally to the seven Sephiroth of Construction. The 295 SLs up to Chesed of the 49th Tree correspond to the 295 hexagonal yods that are either in the lower half of the dodecahedron or on sides of internal Type A triangles that join vertices in its upper half to its centre. The 255 SLs spanning the 42 Trees that map the six cosmic superphysical planes correspond to the 255 other hexagonal yods in its upper half.

What emerges from the analysis on this page, the previous page and the next page is the existence of a profound and beautiful property of the five Platonic solids that mathematicians have not known about because:

they do not know how to recognize real sacred geometry, even supposing that some of them, perhaps being Platonists, might accept that the concept of ‘sacred geometry’ is legitimate;
they have had little insight into the power and significance of Pythagoras’ tetractys as the template of sacred geometries that reveals how they embody numbers of universal (and, therefore, scientific) significance. Mathematicians have understood that, as the representation of the fourth triangular number, the tetractys symbolises the numbers 1, 2, 3 & 4. But their understanding extends little further than that;
they have, of course, been unaware of the mathematical map of reality that the author calls “CTOL”;
they have not realised how various objects that religions traditionally regarded as ‘sacred geometry’ have properties that, although overtly different, are characterized by the same set of parameters, such as the number 550. Such characterisation is inevitable, given that sacred geometries must be analogous, or isomorphic, being merely alternative versions of the same, universal blueprint. However, mathematicians have never recognised this fact. Instead, they have assumed that no mathematical equivalence or isomorphism underlies sacred geometries because their atheism or agnosticism made them presuppose that no religious geometry can have any significance beyond that of being a symbolic representation of religious beliefs, let alone any connection to science.

The five Platonic solids are the regular polyhedral representation of CTOL. Its 550 SLs are symbolized by the 550 hexagonal yods making up the dodecahedron; they have their geometrical counterpart in the 550 corners, sides & triangles that make up the faces of the five Platonic solids. The icosahedron does not possess this property, despite being the dual of the dodecahedron (see the table). This is because it is one of the four Platonic solids associated with the four “Elements” of the physical universe. The ancient association of the dodecahedron with the celestial sphere has a more profound rationale than the superficial one known to historians of science, i.e., its approximate resemblance to a sphere, although it is true that, for the ancient mind, the sky, or celestial sphere, represented Heaven, the realm of the gods. We now see that the dodecahedron is, truly, the regular polyhedral representation of all levels of reality, both physical and superphysical. But it takes its construction from the tetractys — the template of sacred geometry — to reveal this profound and beautiful property, as well as the Decad — the number that the tetractys symbolizes — to express it arithmetically. Only this particular Platonic solid embodies through its hexagonal yods the number that measures the geometrical composition of the faces of all five Platonic solids, as well as the the number of SLs in the Cosmic Tree of Life. It is, therefore, fitting that the ancient Greeks should have associated it with the fifth Element Aether as the source of the material universe, the particles of whose Elements they believed had the shape of the other four Platonic solids that can be fitted within the dodecahedron.

550 embodied in 5 Platonic solids with Type A triangles

Suppose that the 50 faces of the five Platonic solids are divided into their 180 sectors. Then, suppose that their 50 vertices and the centres of their 50 faces with 90 edges are joined to their centres. This generates (90+180=270) internal triangles. When the (180+270=450) triangles in their faces and interiors are each divided into their three sectors, i.e., regarded as Type A triangles, the resulting (3×450=1350) simple triangles have (50+50+450=550) corners. (50+50+180=280) corners are in their faces and 270 corners are in their interiors. 280 is the number value of Sandalphon, the Archangel of Malkuth. ELOHIM, the Godname of Binah with number value 50, prescribes the five Platonic solids because they have 50 vertices, 50 faces and 500 (=50×10) corners of triangles that are not polyhedral vertices.

The 550 corners of the 1350 triangles in the 5 Platonic solids constructed from Type A triangles correspond to the 550 SLs in CTOL.

Suppose that the vertices of the five regular polyhedra are coloured violet, the centres of their faces are coloured white, the centres of the sectors of their faces are coloured brown, the centres of the internal triangles formed by their edges are coloured yellow and the centres of internal triangles formed by sides in the faces that are not edges are coloured purple. The table shows the numbers of corners of each colour. There are 280 corners in the faces and 270 inside the polyhedra, where 280 is the number value of Sandalphon, the Archangel of Malkuth. The former include (24+60=84) brown corners in the octahedron & icosahedron (dark red cells) and (24+60=84) brown corners in the cube & dodecahedron (dark blue cells). The latter includes (24+60=84) purple corners in the octahedron & icosahedron (light red cells) and (24+60=84) purple corners in the cube & dodecahedron (light blue cells). All four of these Platonic solids have (12+12+30+30=84) yellow corners, leaving (8+6+20+12=46) white corners, which, added to the 38 corners in the tetrahedron, creates a sixth set of 84 corners. We find that the 504 corners in the five Platonic solids other than the 46 vertices of the octahedron, cube, icosahedron & dodecahedron (the two pairs of regular polyhedra that are duals of each other) naturally group into six sets of 84.

Compare this with the distribution of SLs in CTOL:

46 SLs up to top of 7-tree mapping the physical plane;
84 dark red SLs on the Pillar of Judgement up to the top of the 49-tree;
84 dark green SLs on the Pillar of Equilibrium up to (but not including) the top of the 49-tree;
84 dark blue SLs on the Pillar of Mercy up to the top of the 49-tree;
84 light red SLs on the Pillar of Judgement above the 49-tree up to the top of CTOL;
84 light green SLs on the Pillar of Equilibrium from the top of the 49-tree to the top of CTOL;
84 light blue SLs on the Pillar of Mercy up to the top of CTOL.

The numbers tabulated in cells of a given colour add up to the number of SLs of the same colour. Notice that (84+84=168) light & dark green SLs line the central pillar from the top of the 7-tree to the top of CTOL. In other words, the number value 168 of Cholem Yesodoth, the Mundane Chakra of Malkuth, specifies the very beginning in CTOL of the physical plane that is the space-time continuum! The 252 internal corners in the octahedron, cube, icosahedron & dodecahedron correspond to the 252 SLs that span the 42 Trees of Life mapping the six cosmic superphysical planes; the 252 corners in either their faces or in the tetrahedron correspond to the 252 SLs that span the 42 Trees mapping the six superphysical planes of the cosmic physical plane; the 46 remaining vertices correspond to the 46 SLs up to, but not including, the top of the 7-tree, which maps the physical plane. An analogous pattern is exhibited by the Sri Yantra (see here for details).

There are (280+18=298) corners in either the tetrahedron or the faces of the four other Platonic solids. This is the number of SLs up to (but excluding) the top of the 49-tree mapping the 49 subplanes of the cosmic physical plane.

The 550 points that surrounding the centres of the five Platonic solids when they are constructed from Type A triangles symbolize the 550 SLs of CTOL. More generally, their embodiment of this parameter of holistic systems demonstrates their holistic nature. The fact that they are composed of 1350 triangles is another indication of this because this number, too, is a holistic parameter, being the number of yods outside the root edge in the (7+7) enfolded Type B polygons containing 1370 yods that are not shared with the outer Tree of Life:

Their 50 faces have 180 triangular sectors with (90+180=270) sides. When the triangles are Type A, (3×180=540) sides are added, making a total of 810 sides. Joining the (50+50=100) vertices & face-centres to the centres of the five Platonic solids adds 100 sides, whilst the 270 internal triangles have (3×270=810) other sides when they are Type A. Hence, there are (810+100+810=1720) sides. The number of corners, sides & triangles in the five Platonic solids = 550 + 1720 + 1350 = 3620 = 362×10, where 362 is the number of yods in the two Type B dodecagons present in the inner form of the Tree of Life:

Each dodecagon embodies (apart from the Pythagorean factor of 10) the number of geometrical elements in each half of the five Platonic solids. Of these geometrical elements, 20 corners & sides make up their axes, which are therefore surrounded by (3620−20=3600) geometrical elements, 1800 being in the five upper halves and 1800 being in their five lower halves. As 1800 = 50×36, where 50 is the number of ELOHIM, the Godname of Binah, and 36 is the number of ELOHA, the Godname of Geburah, the Sephirah below Binah in the Tree of Life, we see how these Sephiroth on the Pillar of Judgement determine the geometrical composition of the regular polyhedral representation of the Tree of Life. On average, the number of geometrical elements surrounding the axis of a Platonic solid constructed from Type A triangles = 3600/5 = 720. There are 360 (=36×10) such geometrical elements on average in each half of a Platonic solid. They correspond to the 360 yods surrounding the centres of the two Type B dodecagons. The number 720 = 72×10, where 72 is the number of Chesed, the Sephirah opposite Geburah in the outer Tree of Life. The centre of a decagon whose sectors are 2nd-order tetractyses is surrounded by 720 yods,* whilst the centres of the seven separate polygons of the inner Tree of Life are surrounded by 720 yods when all the polygons are Type B.**

Their possession of this parameter of the inner Tree of Life demonstrates the holistic character of the five Platonic solids — how they embody archetypal numbers that parameterise any holistic system. The number 720 appears also in the first four Platonic solids, as well as in the disdyakis triacontahedron (see Fig. 7 here) and as the 720 edges of the 600-cell, which is the polychoron counterpart of each half of the inner form of 10 Trees of Life (see section entitled “The holistic 120:720 division in the two 600-cells and in sacred geometries” here).

The Sri Yantra

Correspondence between the Cosmic Tree of Life and the 3-dimensional Sri Yantra.

The seven cosmic planes of consciousness.

There are 504 SLs in the 91 Trees of Life in CTOL down to the top of the 7-tree that maps the physical plane (26-dimensional space-time). The 84 Trees above it comprise the 42 Trees mapping the 42 subplanes of the six cosmic superphysical planes and the 42 analogous Trees that map the 42 subplanes of the six superphysical planes. The 252 SLs down to the top of the 49-tree mapping the 49 subplanes of the cosmic physical plane consist of three sets of 84 SLs. The 84 light red SLs span the Pillar of Mercy, the 84 light blue SLs span the Pillar of Judgement and the 84 light green SLs span the central Pillar of Equilibrium. There are three more similar sets of dark red, dark blue & dark green SLs down to the top of the 7-tree. Hence, 168 light & dark green SLs line the central pillar down to this point. This is how the number value 168 of Cholem Yesodoth, the Mundane Chakra of Malkuth, specifies that section of CTOL that is its Malkuth level, namely, the 7-tree, or physical plane. Forty-six more black SLs in the 7-tree extend to the lowest point of CTOL. Each of the seven cosmic planes expresses one of the seven Sephiroth of Construction, as do the seven subplanes comprising each cosmic plane, the seven planes of the cosmic physical plane and the seven subplanes of each of these planes.

Compare these properties with the 3-dimensional Sri Yantra when the 42 triangles are turned into Type A triangles and the central one is turned into a Type B triangle (for the definition of these two types of triangles, see Sacred Geometry/Tree of Life). The seven planes of consciousness have a one-to-one correspondence with the seven Sephiroth of Construction (physical↔Malkuth, astral↔Yesod, mental↔Hod, etc). The six superphysical planes are associated with the six Sephiroth of Construction other than Malkuth. In the tetractys, the latter are symbolized by the six hexagonal yods that lie on its edges, the central hexagonal yod denoting Malkuth. The 504 hexagonal yods that line the sides of the 126 tetractyses in the 42 triangles of the Sri Yantra are the counterpart of the 504 SLs belonging to the 84 Trees of Life mapping all superphysical realms of consciousness. The counterpart of the 46 SLs below the top of the 7-tree are the 46 black yods in the central triangle. The presence of the 504 hexagonal yods on sides of tetractyses would, of course, continue if it were the 2-dimensional version of the Sri Yantra that was being considered. So either version can be considered as equivalent to CTOL. However, the point at the centre of the central triangle is regarded in Tantra as the source of creation and this does not fit its symbolizing the bottom of CTOL — the opposite to this, being the final emanation of God. But then the central triangle symbolizes, according to Tantra, the trimûrti of Shiva, Brahma & Vishnu, and this is inconsistent with its identification with the physical universe mapped by the 7-tree. In order to retain the traditional meaning of the bindu point as standing outside all realms of consciousness, being their source, it does make a difference which version of the Sri Yantra is the correct counterpart of CTOL. Only its 3-dimensional form, which has this point hovering above the central triangle and the four layers of 42 triangles stacked on top of one another, is the right counterpart.

This encoding of CTOL in the Sri Yantra has its counterpart in a Type C dodecagon. The section Power of the polygons/dodecagon discusses the properties of this polygon. It is the single, polygonal representation of the archetypal pattern embodied in sacred geometries of western and eastern religions. The centre of the Type C dodecagon is surrounded by 504 yods:

There are 168 red SLs above the 7-tree on the left-hand Pillar of Judgement of the 91 Trees of Life representing CTOL. There are 168 blue SLs above the 7-tree on the right-hand Pillar of Mercy. 168 green SLs span the central Pillar of Equilibrium down to the top of the 7-tree. There are 14 green yods per sector of the Type C dodecagon that are either corners of tetractyses or hexagonal yods at their centres. The 9 tetractyses per sector have 14 sides, a red and a blue hexagonal yod lying on each one. Hence, the 504 yods surrounding the centre of the dodecagon comprise (12×14=168) yods that are either red, green or blue. They symbolise the three sets of 168 SLs lining the three pillars of CTOL down to the top of the 7-tree.

They form three sets of 168 yods (coloured red, green & blue) because each sector of the Type C dodecagon has nine tetractyses with 14 sides, each having a pair of red & blue hexagonal yods, and 14 green yods that are either corners of tetractyses or their centres, so that the (12×9=108) tetractyses contain (12×14=168) red hexagonal yods, 168 blue hexagonal yods and 168 green yods. These sets correspond to the 168 SLs on the three pillars of CTOL down to the top of the 7-tree mapping the physical plane (space-time). Hence, the Type C dodecagon is equivalent to the 3-dimensional Sri Yantra. Its centre corresponds to the central Type B triangle, whose 46 yods denote the 46 SLs below the top of the 7-tree. Its 504 yods correspond to the 504 hexagonal yods on the 252 sides of the 126 tetractyses making up the 42 Type A triangles of the Sri Yantra; they symbolise the 504 SLs in CTOL down to the top of the 7-tree.

The 91 overlapping Trees of Life in CTOL are composed of 3108* corners, sides & triangles, where

3108 = 1⁴ + 3⁴ + 5⁴ + 7⁴,

i.e., this number is the sum of the fourth powers of the first four odd integers. The geometrical composition of CTOL is determined by the Tetrad:

Notice that 3108 = 444×7, where 7 is the fourth odd integer. As 3108 = 37×84, where 37 is the number of yods in the Type A hexagon, the fourth type of regular polygon, and 84 = 1² + 3² + 5² + 7², we see that the Tetrad determines the number quantifying the geometrical composition of CTOL, as well as its number (91) of Trees of Life, because the Type B hexagon has 91 yods:

The Tetrad also expresses the geometrical composition of the 2-dimensional Sri Yantra because 240 corners, sides & triangles surround its centre, where 240 = 10×24 = 1×2×3×4×(1+2+3+4). The physical plane (the space-time continuum) is mapped in CTOL by its lowest seven Trees of Life and in the Sri Yantra by its central triangle. As space-time, it is the Malkuth level of the 49-tree mapping the cosmic physical plane, the gematria number value of this Sephirah being 496, where

t is no coincidence that this number is that found in 1984 by physicists Michael Green and Gary Schwarz to be the crucial dimension of a Yang-Mills gauge symmetry group governing the interactions of 10-dimensional superstrings that are free of quantum anomalies. It was then that research in theoretical physics encountered for the first time a mathematical aspect of the universe that is revealed by Kabbalah. But it is also revealed by the geometry of the Tree of Life, for this superstring parameter is embodied in the cosmic physical plane as the 496 yods either lying on or aligned with the Pillar of Equilibrium of 49 overlapping Trees of Life mapping its 49 subplanes when their triangles are tetractyses.** Making use of the formula proved in the first footnote, these 49 Trees are composed of (34×49 + 14 = 1680) points, lines & triangles. Here is a spectacular conjunction of the two most fundamental parameters of E₈×E₈ heterotic superstrings — one (496) referring to their forces described by the symmetries of E₈×E₈ and the other (1680), yet to be discovered by physicists, that refers to their 4-dimensional space-time structure in their subquark ground state, namely, the 1680 circular turns of each helical whorl of the UPA counted by C.W. Leadbeater when he observed it with micro-psi over a century ago. Here in the conjunction between two numbers referring to the same context is an undeniable meeting point of the mystical, the scientific and the paranormal. Provided he is honest, even the most die-hard sceptic towards the paranormal would admit that it is too unlikely that the simultaneous appearance of these two numbers (one scientifically based, the other paranormally obtained) in the Tree of Life representation of the cosmic physical plane might be due to chance. Yet the sceptic is forced to embrace the miraculously improbable in order to avoid believing what is anathema to him, namely, that remote viewing of the subatomic world is possible and was successfully achieved by Besant & Leadbeater. No one should be fooled by the sceptic’s ideological disbelief in the paranormal forcing him to reject all common sense concerning what can be reasonably regarded as coincidence. When prima facie evidence such as this is rejected for highly implausible reasons, one cannot fail to conclude that an unscientific attitude of bias is at work.

Heptagon with Type A triangles as sectors
A symbol of the seven-fold nature of Man and the seven Sephiroth of the spiritual cosmos, the heptagon has 91 hexagonal yods when its sectors are Type A triangles. Each denotes one of the subplanes of the seven cosmic planes of consciousness and one of the Trees of Life in CTOL. The 49 coloured, hexagonal yods either at the centres of tetractyses or lining their sides denote the 49 Trees of the 49-tree that maps the cosmic physical plane; the 42 white hexagonal yods on sides of tetractyses inside each sector of the heptagon denote the 42 Trees of Life above the 49-tree that map the 42 subplanes of the six cosmic superphysical planes. This is the single, polygonal representation of CTOL.

Heptagon with 2nd-order tetractyses as sectors
When its sectors become 2nd-order tetractyses, the 504 yods surrounding the centre of the heptagon denote the 504 SLs down to the top of the 7-tree, which is represented by its central yod. They are the counterpart of the 504 hexagonal yods that line the 126 tetractyses making up the 42 triangles of the Sri Yantra. Notice that there are 84 yods on the boundary of the heptagon. As they shape the polygons, it makes sense, intuitively speaking, to regard them as corresponding to the 84 dark green SLs on the central pillar between the top of the 49-tree and the top of the 7-tree, for they belong to the Tree of Life map of the cosmic physical plane. The 7-tree is the ‘Malkuth’ level of the 49-tree, which in turn is the ‘Malkuth’ level of CTOL — the most general sense of this word. The top of the 7-tree is the 168th SL on the Pillar of Equilibrium from the top of CTOL, where 168 is the number of Cholem Yesodoth, the Mundane Chakra of Malkuth (see the diagram on the previous page depicting the correspondence between CTOL and the Sri Yantra). This is amazing evidence of how the gematria number values of the Sephiroth in the four Kabbalistic Worlds of Atziluth, Beriah, Yetzirah & Assiyah quantify their properties as manifested in each World. The superstring structural parameter 168 discovered by C.W. Leadbeater over a hundred years ago in his micro-psi examination of the UPA actually marks out the physical universe (7-tree) from all superphysical levels of reality (84 higher Trees). That cannot be due to chance! Here, therefore, are two undeniable pieces of mathematical evidence that confirm his paranormal discovery: firstly, the gematria number value of the Mundane Chakra of Malkuth is 168, and, secondly, there are 168 SLs on the Pillar of Equilibrium down to the Malkuth level of CTOL, namely, space-time.

The ancients believed that the Earth was the centre of the universe — or so we now interpret their beliefs. A deeper idea lies behind their regarding our planet as the centre of physical reality. It is that Earth symbolized the plane of physical awareness that is the centre, or fulcrum, of the spiritual cosmos (CTOL), which is encoded in the inner Tree of Life (see here) and represented by the Sri Yantra. In the Kabbalistic, astrological correspondence between Sephiroth and astronomical bodies, the planet Earth is assigned to Malkuth at the bottom of the Tree. It represents the Element Earth — the substance of the physical universe, which includes both ordinary matter and yet to be detected dark matter.

We saw on the previous page that the 3-dimensional Sri Yantra is equivalent to CTOL in that the 550 yods on the boundaries of the 126 tetractyses in its 42 Type A triangles and in its central, Type B triangle denote the 550 SLs in CTOL. The Type A heptagon has 42 yods surrounding its central yod. They correspond to the 42 triangles surround the central triangle in the Sri Yantra:

Correspondence between heptagon & Sri Yantra

The number of yods in the Type C n-gon = 42n + 1. The Type C heptagon (n=7) has 295 yods. This is the number of yods in the seven separate Type A polygons of the inner Tree of Life:

This is also the number of SLs up to Chesed of the 49th Tree of Life in CTOL. In other words, this number measures the number of SLs in CTOL up to the first Sephirah of Construction of the Tree of Life that maps the highest of the 49 subplanes of the cosmic physical plane and which represents the same Sephirah, the first subplane of the Adi plane corresponding to Chesed. The seven centres of the polygons, or the seven corners of the heptagon, are the counterpart of the lowest seven SLs of the 1-tree and the 288 remaining yods in either case are the counterpart of the next 288 SLs up to Chesed of the 49th Tree, where

288 = 1¹ + 2² + 3³ + 4⁴

				10¹
			2¹		2¹
=		3²		3²		3²
	4³		4³		4³		4³

As 248 + 47 = 295, there are 248 SLs up to Chesed of the 49th Tree beyond the 47th SL, which is both the top of the 7-tree and the 168th SL on the Pillar of Equilibrium from the top of CTOL. This connects the superstring structural parameter 168 at the core of Leadbeater’s micro-psi observations to the dimension 248 of E8, the rank-8, exceptional Lie group describing E8×E8 heterotic superstrings. The significance of this remarkable relationship generated by the geometry of CTOL hardly needs to be emphasized. It mathematically connects the number (1680) of circular turns in each helical whorl of the UPA/subquark superstring remote-viewed by Besant & Leadbeater to the dimension of the very Lie group predicted by superstring theory to govern the forces between one of the five types of superstrings, just as we found on the previous page that the same number is connected to the dimension 496 of the two types of symmetry groups describing superstring forces that are free of quantum anomalies! The Type C heptagon is a representation of the 295 SLs up to Chesed of the 49th Tree, just as the Type A 49-gon is, because both polygons have 295 yods. As the Type C heptagon is the fourth in the sequence of successive types of heptagons:

heptagon → Type A heptagon → Type B heptagon → Type C heptagon →

this illustrates the Tetrad Principle formulated in Article 1, according to which the fourth member of a class of mathematical object is (or embodies) a characteristic parameter of holistic systems (in this case, the number 295). It is exemplified par excellence by the Type C hexagon (the fourth class of the fourth type of regular polygon), which contains 248 yods outside its root edge that surround its centre:

The two white yods denote the two simple roots of E8 that are not simple roots (denoted by the six yellow yods) of its exceptional subgroup E6, which has 72 roots denoted by the 72 red yods, the remaining 168 roots being denoted by the 168 blue yods. Also illustrated here is the amazing power of the tetractys to reveal in sacred geometries numbers of prime significance to theoretical physics (in this case, the dimension 496 of E8×E8′, one of the two possible symmetry groups describing the anomaly-free forces between heterotic superstrings). See also here.

c0mmando · June 24, 2024, 7:44pm

The 64 hexagrams of the I Ching table

Suppose that one tosses a coin three times. The result of tossing each time is either “heads” or “tails”. Two tosses leads to either heads-head, heads-tails, tails-head or tail-tail, that is, (2×2=4) possible outcomes. Three tosses leads to (2×2×2=8) outcomes. Now suppose that the coin is tosses another three times. Again, there are eight different combinations of head and tails. Each outcome from the first three throws may be followed by any one of the eight results of the second set of throws. This means that there are (8×8=64) possible pairs of outcomes of throwing three coins and then repeating them. Eight of these are when the outcome of the second set of throws is identical to that of the first not only in numbers of heads and tails but also in the order in which the faces of the coins turn up. The number of different pairing of eight objects without regard for their ordering is 28. The 64 pairs of outcomes consist therefore of eight pairs when the eight possible results of tossing the three coins merely repeat themselves, 28 pairs when the outcome for the second set of throws changes and 28 pairs when the outcome of the second set of throws is the reverse of these, e.g., instead of (say) head-tail-head being followed by tail-tail-head, the outcome is tail-tail-head followed by head-tail-head.

This, essentially, is how the ancient Chinese system of divination known as I Ching is used to answer questions. Originally, yarrow stalks were used to generate a set of six results indicating the answer to a question, but this method is a biased random generator, so that the possible outcomes are not equally probable. The yarrow stalk method was gradually replaced during the Han Dynasty by the three coins method. Many methods have evolved over the years to generate the results of the six tosses, which are called “hexagrams.” Some people prefer using dice to coins because they avoid the problems of whether the coin is tumbling when cupped in the hand and of it bouncing and scattering when flipped. In this case, if an odd number of pips appears in the die, it counts as “heads;” if an even number of pips appears, it counts as “tails.” Outcomes of a randomly generated event like tossing a coin or die are represented by either unbroken lines (—) representing the Yang aspect of the event or broken lines (− −) representing the Yin aspect. Either type of line is characterized as either stable (“young”) or changing (“old”), so that there are four possibilities for each line. The set of three tosses of the coin generates three lines called a “trigram.” The eight possible outcomes of throwing three coins are represented by eight trigrams. The hexagram is a pair of trigrams representing the outcome of the two sets of three throws. Once the hexagram has been generated, one consults the oracle in the form of commentaries in the I Ching (“Book of Changes”) for each of the 64 hexagrams in order to determine the answer to one’s question.

For the purpose of this website, no attempt needs to be made to answer the issue of whether I Ching “works” and (if so) how or why it does. That is irrelevant in the present context. What is important here is 1. the mathematical nature and meaning(s) of the 8×8 matrix of hexagrams that grew out of Taoism, and 2. their connection to the sacred geometries of other religions. This array is an abstract representation of the Tao. Literally translated as the “path” or “way,” although more accurately as the “right way,” this ancient word has no single meaning and is interpreted in many different ways even within Taoist sects. The Tao Te Ching is the ancient Chinese classic text written according to tradition by the Chinese sage Lao Tzu in about the 6th century BCE. But even then, the word “Tao” had no well-defined, single meaning. Indeed, the opening of the Tao Te Ching illustrates its indefinable aspect: “The Tao that can be spoken of is not the real Tao.” The Tao is the driving principle behind the natural order, yet it not Nature. It signifies “the way things are,” yet it is not a force or object existing in the world of duality. It is the causeless Cause, the ultimate source of all phenomena. In symbolically representing something as mundane as all possible outcomes of two sets of three tosses of a coin, the table of hexagrams is expressing something far more profound and fundamental about the mathematical nature of the Tao. At the most elementary level of space-time — the subatomic world of the superstring — we shall discover in the section Superstrings as sacred geometry how this nature manifests.

Stroke order of Tao
The stroke order for the Chinese character for “Tao”.

For an online introduction to I Ching by Richard Wilhelm, one of its greatest exponents, see here.

The 64 hexagrams consist of 384 lines & broken lines. To every one of the 192 unbroken lines expressing Yang, there corresponds its polar opposite, unbroken line expressing Yin. For convenience of expression, they will be referred to as Yang and Yin lines.

The red line passes through the eight hexagrams in the diagonal of the 8×8 array. Their 48 Yang & Yin lines comprise 24 Yang lines and 24 Yin lines. The eight trigrams in the upper halves of these hexagrams comprise 12 Yang lines and 12 Yin lines. As each diagonal hexagram consists of two similar trigrams, the eight trigrams in the lower halves of the diagonal hexagrams also comprise 12 Yang lines and 12 Yin lines.

The 28 hexagrams on each side of the diagonal line of eight hexagrams contain 168 Yang/Yin lines (84 Yang, 84 Yin). Hence, there are 168 off-diagonal Yang lines and 168 off-diagonal Yin lines, that is, 336 Yang/Yin lines. The number 168 is the number value of Cholem Yesodoth, the Mundane Chakra of Malkuth (see here). This is the second sign that the table of hexagrams has a connection with the Kabbalistic representation of the divine paradigm — the Tao, the first being the fact that the number 64 itself is the number value of Nogah, the Mundane Chakra of another Sephirah — Netzach, whilst the third sign is that the number 48 (the number of Yang/Yin lines in the eight diagonal hexagrams) is the number value of Kokab, the Mundane Chakra of a third Sephirah — Hod.

The array of 64 hexagrams displays two patterns that prove to be common to all sacred geometrical representations of holistic systems:

384 = 192 + 192. This expresses the Yang/Yin polarity of the two halves of a holistic system because the 64 hexagrams are composed of 192 Yang lines and 192 Yin lines. Alternatively, it may be thought as expressing the two diagonal halves of the 8×8 array of hexagrams because each half has 192 lines & broken lines (96 lines & 96 broken lines);
192 = 24 + 168. This expresses the 24 non-formative and 168 formative degrees of freedom needed to express each half of a holistic system. It manifests as the 24 lines & broken lines in the set of eight different trigrams in the upper or lower halves of the eight diagonal hexagrams and the 168 lines & broken lines in the 56 off-diagonal trigrams in each half that constitute seven copies of this basic set of trigrams.

The third pattern found in these systems:

384 = 48 + 336,

where

48 = 24 + 24

336 = 168 + 168

is a corollary of the first two patterns. Just as there are 12 Yang lines and 12 Yin lines in the eight diagonal trigrams:

8 trigrams
(The numbers indicate the order of generation of the trigrams)

so, too, the number 24 as a parameter of holistic systems divides into two sets of 12 degrees of freedom that are opposite, complementary or “dual,” in some sense (in the case of the three tossed coins, there are 12 heads and 12 tails in the eight possible outcomes). The division:

168 = 84 + 84

of the holistic parameter 168 is also characteristic of such systems (see here). It manifests in the 168 off-diagonal lines & broken lines in each half of the 8×8 array of hexagrams as the 84 lines expressing Yang and the 84 broken lines expressing Yin:

24-168 pattern in 64 hexagrams

It implies that

336 = 168 + 168 = 84 + 84 + 84 + 84.

This division is discussed for various sacred geometries in Article 64 and here.

The holistic pattern

Holistic systems embody the divine ideals. The nature of God can be said to be stamped on them. Whether one calls it the “image of God,” the “cosmic blueprint,” etc, there exists a universal, mathematical pattern or structure that appears in the sacred geometries of religions, although it is subtle enough not even to have been recognized up till now, let alone proved to exist in these geometries. Because this claim has profound spiritual, philosophical and scientific implications, it is not made lightly. As astronomer, free-thinker and skeptic Carl Sagan remarked in his Cosmos TV series, “extraordinary claims require extraordinary evidence.” So the purpose of this website is to convince its visitors by presenting ample, irrefutable evidence that such a universal design of divine origin exists in sacred geometries because they possess analogous features the existence of which cannot be explained in any other way and which are quantified by certain numbers far too many times to be due to chance. Its aim is not to prove that God exists. That is an inference which the visitor is invited to draw as the only possible reason for both the amazing correlations between these geometries and their beautiful properties, which are revealed for the first time in this website.

The gematria number values of the ten Sephiroth in the four Kabbalistic Worlds are given here for reference:

(The listed numbers are written in bold wherever they appear in the text).

The inner form of the Tree of Life, the I Ching table of 64 hexagrams, the five Platonic solids, the Sri Yantra and the disdyakis triacontahedron are equivalent representations of holistic systems that express the divine paradigm. This is demonstrated in this website, firstly, by showing how the Godname numbers of the ten Sephiroth mathematically determine aspects of their structure, thereby revealing their holistic nature, and, secondly, by proving that their properties are quantified in analogous ways by the same set of parameters, which display the following divisions:

The structure of holistic systems

Holistic systems always exhibit these patterns of divisions (apart, perhaps, from a factor of 10). Explicitly, they are:

384 =	192 + 192
=	[24+168] + [24+168]
=	[3+21+84+84] + [3+21+84+84]
=	(3+3) + (21+21) + (84+84) + (84+84)
=	6 + 42 + (168+168)
=	48 + 336,

where

48 = 24 + 24,
24 = 3 + 21,
336 = 168 + 168,
168 = 84 + 84,

and

192 = 24 + 168.

Each division will now be illustrated in the context of the various sacred geometries that are discussed in this website. The author’s research articles and latest book analyze in more detail how various sacred geometries display these parameters and divisions.

384 = 192 + 192

The primary split of the holistic parameter 384 into two 192’s represents a bifurcation or — more accurately —a polarisation of the system into two complementary but polar-opposite sets of elements. For the case of sacred geometries, it divides into two halves that are either: 1. mirror images of each other, if the sacred geometry has inversion symmetry; 2. related by a 1:1 isomorphic mapping between pairs of structural elements belonging to each half; 3. related to each other as metaphysical opposites. For example:

1. 7-tree
Counting from its base, there are 384 yods up to the top of the 7-tree mapping the Theosophical “physical plane” (the physical universe) when its triangles are tetractyses. There are 192 yods below Chesed of the fourth Tree and 192 yods from the top of the 7-tree down to the Path connecting Chesed and Geburah of this Tree (see here).
2. Inner Tree of Life

Each set of the seven separate polygons consists of 192 points, lines & triangles surrounding their centres (see here, here & here). As one set is the mirror image of the other, whether separate or enfolded, corresponding geometrical elements that are mirror images of each other form complementary pairs;
the two sets of the first six regular polygons belonging to the inner form of successive, overlapping Trees of Life comprise 384 yods that are intrinsic to each set, each of the 192 yods in one set having a position that makes it the mirror image of its counterpart in the other set (see here & here).

3. Outer & inner Tree of Life
When combined, they comprise 384 points, lines & triangles outside the root edge. Because each is symmetric with respect to reflection in a plane containing the central Pillar of Equilibrium, 192 such geometrical elements can be associated with the left-hand side of the combined pair of objects and 192 geometrical elements can be associated with their right-hand side (see here). Also, the 384 geometrical elements comprise 192 points & triangles and 192 sides (see previous link).

4. The I Ching table of 64 hexagrams

The 384 lines & broken lines making up the 64 hexagrams of the I Ching table comprise 192 (Yang) lines and 192 (Yin) broken lines (see here & here).

5. Sri Yantra
When its 43 triangles are turned into tetractyses, 192 yods are in one half of the Sri Yantra and 192 yods are in its other half (see here & here). However, because these halves are not mirror images of each other, the corresponding members of each pair of yods are not reflections of each other across the horizontal plane. There is mirror symmetry across the vertical plane, but the pairing disappears for those yods lying along the vertical axis. What is crucial here is not the lack of mirror symmetry but the inversion symmetry through the centre of the Sri Yantra. This means that each hexagonal yod on the sides of the central tetractys has its counterpart in the other triangular array whose overlap forms a Star of David, so that the 384 yods really do divide into 192 pairs.

6. Disdyakis triacontahedron

Surrounding an axis passing through two diametrically opposite A vertices are 384 geometrical elements in its faces (60 vertices, 180 edges & 120 triangles) and in the central 12-gon (12 triangular sectors & their 12 internal sides) that is perpendicular to this axis. They divide into the 192 elements making up one half of the polyhedron & 12-gon and the 192 inverted images of these elements in their other half (see here).

7. {3,7} tiling of 3-torus

When the 56 triangles of the {3,7} tiling of the 3-torus are turned into tetractyses, 192 yods line their sides. Similarly, 192 yods line the sides of the 56 tetractyses in the turned inside-out version of the 3-torus (see here).

8. F₄ as an exceptional subgroup of E₈
The dimension of the rank-8, exceptional Lie group E₈ is 248. It has 8 simple roots and 240 roots. Its exceptional subgroup F₄ has 48 roots. There are (240−48=192) roots of E₈ that are not roots of F₄. In the case of E₈×E₈, one of the two anomaly-free symmetry groups describing superstrings, it has (192+192=384) roots that are not roots of the two similar F₄ groups it contains. The 48 roots of F₄ divide into two sets of 24, each defined by an 8-d root vector. Each set has (8×24=192) coordinates in its 24 root vectors. For more details see (7) here.

In the case of the seven musical scales, the primary 192:192 division in this holistic system manifests as the 192 rising intervals between their notes and their 192 falling counterparts, when the tonic, unit interval and the octave are included (see here).

Shiva Shakti
Shiva and Shakti

Holistic systems, therefore, display a fundamental two-foldness, or duality, being composed of 192 pairs of elements that are either mirror-images or polar opposites of each other in some sense, just as members of a biological species are divided into males and females. In the case of the notes of the seven musical scales, rising and falling intervals are the musical counterpart of reflected or inverted geometrical elements. The 192:192 division of the global parameter 384 is a manifestation of a fundamental duality present in all holistic systems. In Taoism it is embodied in the cosmic duality of Yang and Yin. In the Kabbalistic Tree of Life it is represented by the right-hand Pillar of Mercy and the left-hand Pillar of Judgement. In Tantric Hinduism it is personified by the divine masculine power invested in the God Shiva and the divine feminine power of the Goddess Shakti, sometimes referred to in Hinduism as “The Great, Divine Mother.” One of the ways this cosmic polarity manifests in physics is as the basic distinction between matter and antimatter. Another way is the difference between the positive and negative UPAs — the two chiral forms of the basic units of physical matter that Annie Besant & C.W. Leadbeater claimed over a century ago make up the subatomic particles that they described with micro-psi vision (see here) and which the author has proved are the constituents of the quarks making up protons and neutrons inside atomic nuclei (see here). The two types of UPAs carry “colour magnetic charges” of opposite sign.

The Decad expresses the beautiful, mathematical properties of holistic systems. An example of this is that 385 yods other than the three corners of the central triangle (symbolizing the trimûrti, or Hindu trinity, of Shiva, Brahma & Vishnu) surround the central bindu point of the 3-dimensional Sri Yantra when it is constructed from tetractyses, where

		1²
		2²	3²
385 =	4²	5²	6²
	7²	8²	9²	10²

see here). The Tree of Life counterpart of the 384 yods in the Sri Yantra that surround the centre of the central triangle are the 384 yods up to the top of the 7-tree that maps the Theosophical physical plane, namely, the physical universe with 25 spatial dimensions (see here). They divide into 192 yods below the Chesed-Geburah Path of the fourth Tree of Life and 192 yods down to this point. As another example of the influence of the Decad, there are 385 tetractyses below the top of the 10-tree prescribed by ADONAI, the Godname of Malkuth, that contain 1680 yods — the Tree of Life basis of the 1680 turns of each ten-dimensional whorl making up the UPA as the subquark state of the E8×E8 heterotic superstring (see here). Their counterparts in the disdyakis triacontahedron are the 385 extra yods needed to transform into Type A triangles the sectors of the seven polygons perpendicular to an axis passing through two opposite C vertices that are formed by its vertices and which lie either above or below its central plane [1]. The centres of these seven polygons correspond in the Sri Yantra to the lowest corner of the central triangle and the six hexagonal yods on its three sides.

192 = 24 + 168

Each half of a holistic system embodying the holistic parameter 192 divides into a basic set of 24 elements and a set of 168 elements that fill out the form delineated by the primary elements. For example:

nner Tree of Life

Their counterparts in each half of the inner Tree of Life are the 24 geometrical elements surrounding the centre of the hexagon and the 168 geometrical elements surrounding the centres of the six other polygons when they are all divided into their sectors. Their counterparts in the first (6+6) enfolded polygons are the 24 corners of each set of the first six enfolded polygons outside the root edge and the 168 other yods associated with them when their sectors become tetractyses.

The I Ching table of 64 hexagrams

In each diagonal half of the I Ching table, they are the 24 Yang & Yin lines making up the eight upper or lower trigrams in its eight diagonal hexagrams and the 168 Yang & Yin lines in the 28 off-diagonal hexagrams.

Sri Yantra

There are 24 hexagonal yods either at the centres of the 21 triangles in each half of the Sri Yantra or at the corners of each overlapping triangle forming a Sign of Vishnu/Star of David within its central triangle. There are also 168 yods lying on the 63 edges of these triangles.

Disdyakis triacontahedron

24 geometrical elements make up each half of the central, 12-sided polygon perpendicular to an axis passing through two opposite A vertices; 168 geometrical elements (24 vertices, 84 edges & 60 triangles) are above or below it.

Seven musical scales

There are 21 notes above the tonic comprising second, third & fourth notes, as well as the tonic, unit interval & octave interval, totalling 24 intervals, and 168 remaining intervals between the notes of the seven musical scales (including notes below the octave that are the inversions* of these 21 notes).

Roots of E₈

The 192 roots in E₈ that are not roots of F₄ consist of the 168 roots that are not roots of E₆ and the 24 roots in E₆ that are not roots of F₄.

In this division, the number 24 denotes the primary/basic elements that determine the global nature of a holistic system and the number 168 measures those elements that build up the complete form of the system delineated in skeletal outline by these 24 building blocks.

The number 4 (tetrad) generates the division 192 = 24 + 168 in the following way: four different objects have:

(4!/2!=12) permutations of 2 objects (a total of 24 objects, each type appearing 6 times);
(4!/1!=24) permutations of 3 objects (a total of 72 objects, each type appearing 18 times);
(4!=24) permutations of 4 objects (a total of 96 objects, each type appearing 24 times).

Therefore, (24+72+96=24+168=192) objects make up the (12+24+24=60) permutations of 2, 3 & 4 objects, each object appearing 48 times. The 192 objects consist of the 24 objects making up the 12 permutations of 2 objects and the 168 objects that make up the 48 permutations of 3 & 4 objects. The 24:168 distinction in holistic systems has its arithmetic counterpart in the 192 objects making up the 60 permutations of 2, 3 & 4 objects, which consist of 12 permutations of 2 objects, i.e., 24 objects, and 48 permutations of 3 & 4 objects, i.e., 168 objects.

24 = 3 + 21

The three hexagonal yods at the corners of each triangle that form a Star of David in the central triangle of the Sri Yantra correspond in the inner Tree of Life to the corners of the triangle and hexagon that coincide with Sephiroth on each side pillar of the outer Tree of Life. In the I Ching table of 64 hexagrams, they correspond to the three Yang lines of the Heaven trigram and to the three Yin lines of the Earth trigram. In the 14 separate polygons of the inner Tree of Life, they correspond to three corners of each hexagon that form an equilateral triangle, the remainder of the hexagon having 21 geometrical elements surrounding its centre. The 21 corners in each set of the first six enfolded polygons that do not coincide with Sephiroth correspond to the hexagonal yods at the centres of the 21 triangles in each half of the Sri Yantra when they are regarded as tetractyses. They correspond to the 21 Yang & Yin lines of the seven upper or seven lower trigrams in the diagonal hexagrams other than the Heaven trigram. The counterparts in the seven musical scales of the two sets of 21 elements are the 21 notes (2nd, 3rd & 4th notes of each scale) and their 21 inversions. The two sets of three hexagonal yods in the central tetractys/triangle of the Sri Yantra, the two triads of Sephiroth in the side pillars of the Tree of Life and the Heaven and Earth trigrams have their musical counterparts in, respectively, the tonic, the octave & rising octave interval and the unit interval, the subharmonic octave with tone ratio ½ and the falling octave interval of ½.

The 24 geometrical elements in each half of the central, 12-sided polygon of the disdyakis triacontahedron:

A, B, C vertices in disdyakis triacontahedron

The central 12-gon

comprise two copies of three sets of four elements: three types of vertices (A, B, C), three types of outer edges (AB, BC, AC), three types of inner edges (AO, BO, CO) & three types of triangular sectors (Tₐ = OAB, T₋ = OBC & Tᵇ = OCA). The three kinds of vertices, the outer edges & inner sides and the triangular sectors are denoted by the three rows of Yang/Yin lines of a trigram:

A	AB	AO	Tₐ	A’	A’B’	A’O	Tₐ’
A	AB	AO	Tₐ	A’	A’B’	A’O	Tₐ’
B	BC	BO	T₋	B’	B’C’	B’O	T₋’
C	CA	CO	Tₑ	C’	C’A’	C’O	Tₑ’

(a prime indicates not the mirror image of its unprimed counterpart but its second version or copy, rotated through 90° around an axis perpendicular to the plane containing the 12-gon; for clarity, the prime notation is omitted from the diagram shown above).

The polyhedral counterpart of the Heaven trigram, etc is (A, B, C); the counterpart of the Earth trigram, etc is (Tₐ, T₋, Tₑ).

The two upper sets of four trigrams in the eight diagonal hexagrams of the I Ching table correspond to the four types of geometrical elements in the 12-gon, each being of three kinds, and their repetition, these making up half the 12-gon.

The lower group of eight trigrams in the diagonal hexagrams correspond to the mirror images of these geometrical elements in the other half of the 12-gon.

(a prime indicates not the mirror image of its unprimed counterpart but its second version or copy, rotated through 90° around an axis perpendicular to the plane containing the 12-gon; for clarity, the prime notation is omitted from the diagram shown above).

The polyhedral counterpart of the Heaven trigram, etc is (A, B, C); the counterpart of the Earth trigram, etc is (Tₐ, T₋, Tₑ).

The two upper sets of four trigrams in the eight diagonal hexagrams of the I Ching table correspond to the four types of geometrical elements in the 12-gon, each being of three kinds, and their repetition, these making up half the 12-gon.

The lower group of eight trigrams in the diagonal hexagrams correspond to the mirror images of these geometrical elements in the other half of the 12-gon.

Just as the three types of vertices generate 189 more geometrical elements in each half of the disdyakis triacontahedron, so the seven types of musical scales contain 189 intervals below the octave, there being 27 such intervals for each eight-note scale, and 7×27 = 189. The three types of vertices can be said to play the defining role of the tonic, octave & octave interval. The 84 edges above or below the central 12-gon form 28 sets of three edges (AB, BC, CA). The 84 vertices & triangles above or below the polygon form 28 sets of vertices & triangles: (8 A vertices, 20 triangles), (8 C vertices, 20 triangles) and (4 A, 4 B vertices, 20 triangles). Hence, above or below the 12-gon are 28 sets of three edges and 28 sets of three geometrical elements that are either vertices or triangles. They correspond in the I Ching table to the 28 pairs of trigrams either above or below the eight hexagrams in its diagonal. The eight sets of three types of geometrical elements in half the 12-gon correspond to the eight upper trigrams in the diagonal hexagrams; the eight sets of three types of elements in the other half of the 12-gon correspond to the eight lower trigrams in these hexagrams. The 84 vertices & triangles and the 84 edges in an upper or lower half of the disdyakis triacontahedron correspond to the 84 lines and 84 broken lines in the upper or lower diagonal halves of the I Ching table. In other words, the basic Yang/Yin duality of holistic systems manifests in the disdyakis triacontahedron as the distinction between the two types of geometrical elements that are, mathematically speaking, scalar in nature (points and the areas bounded by triangles) and the third type (straight line edges) that is a vector because it has direction. In a metaphysical sense, the geometrical correspondence that emerges here illustrates the Pythagorean principles of apeiron (Unlimited) and peras (Limit, or Boundary, such as that created in the polyhedra by their edges). The former is a masculine quality because it is the inchoate pre-cursor to fully developed structure, the latter is a feminine principle because it bestows form by organizing and imposing limitation). The 12 points & triangles and the 12 straight lines in each half of the 12-gon are the geometrical counterpart of the 12 Yang lines and the 12 Yin lines in the eight upper or lower trigrams making up the hexagrams in the diagonal of the I Ching table of 64 hexagrams.

An alternative way of understanding how the (192+192) elements in a holistic system manifest in the disdyakis triacontahedron (and one that is clearer because it provides a more obvious geometrical counterpart to the Yang/Yin distinction between lines and broken lines) is to regard each of the 192 polyhedral edges & internal sides of the central, 12-sided polygon as orientated lines, i.e., as arrows that can point in opposite directions. Then the 12 Yang lines and 12 Yin lines of the eight trigrams making up the upper and lower halves of the diagonal hexagrams would correspond to the 12 external edges and internal sides in each half of the 12-gon, each straight line generating two oppositely pointing arrows, whilst the 84 Yang and 84 Yin lines in the 28 hexagrams on either side of the diagonal in the I Ching table would correspond to the 84 pairs of arrows created by the 84 edges in the upper or lower halves of the polyhedron. This time, the three rows of a trigram would represent just the three types of edges or internal sides, and the eight trigrams would signify triplets of arrows of each type, the Yang or Yin polarity denoting whether the arrow pointed, respectively, towards, say, the right or left. As before, the two diagonal halves of the I Ching table would denote a half of the polyhedron and its mirror-image half. But the 192 Yang lines and the 192 Yin lines in the table would symbolize, respectively, the 192 arrows that point to the right and the 192 arrows that point to the left. Suppose in the diagram above that the centre of the polygon is labelled “O”. Then we can write the arrow pointing from B to A as –Eₐ, the arrow pointing from C to B as –E₋, & the arrow pointing from A to C as –Eₑ. Similarly, the arrow AO pointing from A to O is eₐ and the arrow OA pointing from O to A is –eₐ (and similarly for vertices B & C). Using primes to indicate the repeated set of vertices & edges, there are four triplets of arrows:


AB = Eₐ	AO = eₐ	A’B’ = Eₐ’	A’O = eₐ’
BC = E₋	BO = e₋	B’C’ = E₋’	B’O = e₋’
CA = Eₑ	CO = eₑ	C’A’ = Eₑ’	C’O = eₑ’

and four triplets of their counterparts pointing in the opposite direction. The 12 positively pointing and 12 negatively pointing arrows are symbolized, respectively, by the 12 Yang and 12 Yin lines in the eight trigrams, which can be identified in terms of the following sets of arrows:

24 lines & broken lines in eight trigrams as sides of sectors of 12-gon

The pairing of similar trigrams in the eight diagonal hexagrams of the table of 64 hexagrams expresses the fact that every set of three arrows belonging to one half of the central polygon has its mirror counterpart in the other half. We saw a similar interpretation for the (24+24=48) geometrical elements in the pair of hexagons when we considered the I Ching table as a representation of the (192+192) geometrical elements surrounding the centres of the two sets of seven separate polygons making up the inner Tree of Life (see here).

The 84 Yang lines and the 84 Yin lines in the 28 pairs of trigrams on either side of the diagonal of the table denote the 84 positive and 84 negative arrows defined by the 84 edges of the disdyakis triacontahedron above or below the central plane containing its central 12-gon. As with the edges of the latter, they consist of three types of edges, so that there are 28 edges of each type, i.e., 28 (±Eₐ), 28 (±E₋) & 28 (±Eₑ) arrows that form two groups of 28 sets of three arrows. Each hexagram symbolizes a pair of sets of three arrows. The inversion of trigrams in a hexagram on the other side of the diagonal corresponds to the spatially inverted counterparts of these arrows in the other half of the polyhedron. What demonstrates the holistic nature of the disdyakis triacontahedron is that it is the polyhedral representation of all 64 possible pairs of eight trigrams when Yang and Yin lines are identified as the orientated edges of its faces and sectors of its central polygon.

A musical manifestation of the holistic pattern

The divisions:

384 = 48 + 168 + 168

and

48 = 24 + 24

embodied in the first (6+6) enfolded polygons, the I Ching table of 64 hexagrams, the Sri Yantra & the disdyakis triacontahedron have a remarkable, musical interpretation. According to Table 1, G₅, the perfect fifth of the fifth octave, the tenth overtone, the 33rd note and the 11th Pythagorean harmonic, has a tone ratio of 24 and G₆, the perfect fifth of the sixth octave, the 40th note and the 15th Pythagorean harmonic, has a tone ratio of 48. The Tetrad expresses both tone ratios because 10 = 1 + 2 + 3 + 4, 33 = 1 + 1×2 + 1×2×3 + 1×2×3×4, 24 = 1×2×3×4, and 40 = 4 + 8 + 12 + 16, whilst 15 is the number of combinations of four objects, taken 1, 2, 3 & 4 at a time.

Table 1. Table of tone ratios of the first 11 octaves of the Pythagorean musical scale.
(overtones are in pink cells)

The 26ˢᵗ Pythagorean harmonic is A₈ with tone ratio 216 (the number value of Geburah). Therefore, the Godname YAH prescribes G₆, up to which there are 48 harmonics (15 Pythagorean) and the Godname YAHWEH prescribes A₈, up to which there are 216 harmonics, i.e., 168 extra harmonics. The note G₉, the perfect fifth of the ninth octave, has the tone ratio 384; it is the 30ᵗʰ overtone. The Godname EL (אל) with number value 31 prescribes this note as the 31ˢᵗ Pythagorean harmonic and as the 61ˢᵗ note, Decagonal representation of 61 notes up to G9 where 61 is the 31ˢᵗ odd integer. The value 1 of the letter E (א) denotes the tonic and the value 30 of L (ל) denotes the 30 overtones up to G₉.

As the Pythagorean measure of wholeness and perfection, the Decad determines all the 61 notes up to G₉ because, constructed from tetractyses, the decagon has 61 yods (see here). The central yod denotes the tonic, the 30 internal yods surrounding it denote the 30 partials up to G₉ and the 30 boundary yods denote the 30 overtones, the ten corners denoting the ten overtones up to G₅. The Decad also determines the 26ᵗʰ musical harmonic A₈ with tone ratio 216 as the 55ᵗʰ note, where

55 = 1 + 2 + 3 + … + 10.

This geometrical correspondence is evidence that the 60 note intervals above the tonic up to G9 constitute a holistic set symbolized in the tetractys-constructed TOL by its 60 hexagonal yods symbolizing Sephiroth of Construction, in the dodecagon by its 60 hexagonal yods and in the disdyakis triacontahedron by the 60 vertices surrounding an axis joining two opposite vertices. The Tetrad also defines the number 384 because the 61 notes can be assigned to the 61 yods in a Type B square (see here).

The 61 notes up to G9 are the counterparts of the 61 sounds created by playing the 15 notes of the Hypatôn, Mesôn, Diezeugmenôn and Hyperbolaiôn tetrachords making up the Greater Perfect System of ancient Greek music — another holistic system, discussed here. The 31 sounds (notes, harmonic intervals & chords) that can be made by playing the eight notes of the first octave correspond to the 31 harmonics up to G9; the 30 sounds created by playing the second octave correspond to the 30 partials.

Table 2 displays the 31 harmonics of the Pythagorean scale up to G₉ and their tone ratios. YAH (YH) separates the 48 harmonics up to G₆ from the remaining 336 harmonics up to G₉, which is prescribed by EL. The last 168 harmonics with five notes are separated by the Godname YAHWEH (YHVH). Amazingly, the number value 10 of yod (Y) and the value 5 of heh (H) divide the 48 harmonics into two sets of 24, which is characteristic of this number when it is a parameter of sacred geometries. The 48 harmonics contain 15 musical notes prescribed by YAH. The first set of 168 harmonics up to A₈ contains 11 notes — the sum of the remaining values of the Hebrew letters vav (V)and heh (H) of YAHWEH — and 157 non-musical harmonics. The second set of 168 harmonics up to G₉ contains five notes and 163 non-musical harmonics. Of the 336 harmonics, 16 are notes and 320 are non-musical harmonics. Of the 384 harmonics, 31 are notes and 353 are non-musical harmonics. Of the former, 22 are notes other than octaves. As 22 is the 21ˢᵗ odd integer after 1, EHYEH with number value 21 prescribes the overtones that are not octaves. It also prescribes G₉ because this is the 21ˢᵗ note above G₆. The 61 notes have 52 notes other than octaves. 52 is the 26ᵗʰ even integer, showing how YAHWEH prescribes this archetypal set of notes.

What confirms that these 48:336, 168:168 and 24:24 divisions in the Pythagorean scale have significance is that they are prescribed by the Godnames YAH, YAHWEH and EL through their number values. This cannot, plausibly, be attributed to chance. The 384 harmonics up to G₉ consist of the fundamental frequency 1 and 383 higher harmonics. Amazingly, 383 is the 76th prime number, showing how the Godname YAHWEH ELOHIM with number value 76 prescribes the archetypal set of the first 61 notes of the Pythagorean scale that span 384 harmonics. There are 383 hexagonal yods at the centres of the tetractyses whose yods make up the 1680 yods below the top of the tenth Tree of Life (65th SL) that symbolize the 1680 turns of a helical whorl of the heterotic superstring (see here). In other words, these 1680 yods comprise 383 yods of the type that symbolize Malkuth (see here the equivalence between the Tree of Life and the tetractys). This shows how YAHWEH ELOHIM prescribes this superstring structural parameter. ELOHIM with number value 50 prescribes the 50 notes other than the 11 musical harmonics up to G₅, the perfect fifth with tone ratio 24. The first ten overtones and the 22 partials up to G₅ were shown in The seven musical scales to constitute a Tree of Life pattern because they are analogous to its ten Sephiroth and 22 Paths (see #19).

The 11:50 division of the 61 notes corresponds in the tetractys-constructed 1-tree to the 11 corners of its 19 triangles and to the 50 hexagonal yods lying on their 25 sides. The number 61 creates the shape of the 1-tree as the number of yods lining the sides (Paths) of its tetractyses. Its counterparts in the tetractys-constructed dodecagon are the 61 yods on the sides of its sectors:

Its counterparts in the disdyakis triacontahedron are its centre, the ten B vertices and the 50 A & C vertices that surround an axis passing through two opposite B vertices. The 20 C vertices correspond to the 20 hexagonal yods lying on the ten sides of triangles in the trunk of the 1-tree and the 30 A vertices correspond to the 30 hexagonal yods lying on the 15 sides of triangles making up its branches. The musical counterparts of the 20 C vertices, the ten B vertices and the 30 A vertices are, respectively, the 20 partials in the first four octaves, the ten remaining partials and the 30 overtones, with the centre of the polyhedron denoting the fundamental, C1. The disdyakis triacontahedron is the geometrical representation of the 60 notes above the tonic up to G₉, the 384th harmonic and the 31st musical harmonic.

The division:

24 = 3 + 21

found in sacred geometries manifests in the Pythagorean scale, firstly, as the three harmonics (all notes) up to G₅ (see Table 1) and the 21 higher harmonics (eight notes & 13 non-musical harmonics) up to G₅ and, secondly, as the three harmonics (one note & two non-musical harmonics) beyond G₅ up to the next note A₅ and the 21 harmonics (three notes & 18 non-musical harmonics) beyond that up to G₆. The (3+3=6) harmonics comprise four notes and two non-musical harmonics; the (21+21=42) harmonics comprise 11 notes and 31 non-musical harmonics. The composition of the 384 harmonics is:

Composition of 1st 384 harmonics in Pythagorean scale

Its 31 musical harmonics prescribed by the Godname EL comprise every type of note except perfect fourths.

Disdyakistriacontahedron
Disdyakis triacontahedron

Central 12-gon

Surrounding an axis passing through any two opposite A vertices of the disdyakis triacontahedron are 24 vertices, 84 edges & 60 triangles, i.e., 168 geometrical elements, either above or below the central, 12-sided polygon perpendicular to this axis (see here). The 12-gon has 48 geometrical elements surrounding its centre made up of 12 corners, 24 sides & 12 triangles. Compare these geometrical compositions with the composition of harmonics up to G₉:

Starting from G₆, the 15ᵗʰ musical harmonic with tone ratio 48, there are 60 more harmonics up to A₇, the 21ˢᵗ musical harmonic, then 84 more harmonics up to G₈ and 24 more harmonics up to A₈, the 26ᵗʰ musical harmonic. The composition of the 168 geometrical elements either above or below the central polygon of the disdyakis triacontahedron matches the 168 harmonics between G₆ and A₈, making it too unlikely that the correspondence between the geometry and the notes of the Pythagorean scale could be coincidental. As the 48 geometrical elements of the polygon correspond to the 48 harmonics up to G₆, this note prescribed by YAH corresponds to the central polygon, whilst the note A₈ prescribed by YAHWEH corresponds to half of the polyhedron with 216 geometrical elements. The 168 harmonics beyond A₈ up to G₈ correspond to the 168 geometrical elements in the other half of the polyhedron below the polygon. Each element is the geometrical counterpart of a harmonic, the 336 harmonics between G₆ and G₉ being the musical counterpart of the 336 geometrical elements above and below the central polygon. This is the musical manifestation of the division:

384 = 48 + 168 + 168

that is characteristic of holistic systems. It is powerful evidence of the universality of this pattern, as embodied in sacred geometries like the first (6+6) enfolded polygons of the inner Tree of Life, its seven separate Type B polygons, the I Ching table of 64 hexagrams, the Sri Yantra and the disdyakis triacontahedron.

The Tetrad Principle (see page 4 in Article 1) expresses the 384 harmonics (the musical manifestation of this holistic parameter) because 384 = 4²4!:

	4!	4!	4!	4!
384 =	4!	4!	4!	4!	(4! = 1×2×3×4)
	4!	4!	4!	4!
	4!	4!	4!	4!

They include (4²=16) octaves and perfect fifths. The 378 geometrical elements above and below the 48 elements of the central polygon are the geometrical counterpart of the 378 harmonics above G₆. They contain (4²=16) overtones. The 48 harmonics up to G₆ contain 15 Pythagorean harmonics, where 15 is the fourth triangular number after 1. The three harmonics up to G₂ and the three harmonics above G₅ to A₅ comprise four Pythagorean harmonics (1, 2, 3 & 27).

168 = 84 + 84

We pointed out in #19 at Superstrings as sacred geometry/Tree of Life that the first six polygons of the inner Tree of Life have 84 yods lying on their edges outside their shared root edge. Both sets have 168 yods:

First_6-6_enfolded_polygons_and_Type B_dodecagon_embody_168

This property demonstrates par excellence the shape-defining character of the number value 168 of Cholem Yesodoth, the Mundane Chakra of Malkuth. The same 84:84 pattern is displayed in the 168 extra yods needed to construct the Type B dodecagon the last of the regular polygons making up the inner Tree of Life. In each case, every yod has its mirror image. The 7-tree has 168 corners & sides of its 91 triangles [2]. They comprise 84 corners & complete sides below Chesed of the fourth Tree of Life and 84 corners & sides above it.

In the I Ching table, there are 84 Yang lines and 84 Yin lines in the 28 hexagrams on either side of the diagonal. We saw in Article 18 that the eight types of trigrams symbolize the binary numbers corresponding to the integers 0–7. We showed in Article 20 that they also express the eight sets of three faces of a cube whose intersection defines its corners. This means that these integers can be assigned to the corners, with lines that join pairs of corners representing hexagrams. Numbers can be assigned to lines whose values are the sum of the integers assigned to the two corners that they join. The 12 diagonals of the six square faces of the cube are the edges of two interpenetrating tetrahedra. The integers associated with them add up to 84, as do the integers associated with the 12 edges of the cube. The 84:84 division of Yang & Yin lines therefore manifests geometrically as the two tetrahedra whose vertices form a cube.

In Article 16 we found that there are 84 rising and 84 falling Pythagorean intervals between notes of the seven musical scales that are repetitions of the basic set of 12 notes between the tonic and octave belonging to these scales. They are the musical counterpart of the 84 Yang and 84 Yin lines in the off-diagonal hexagrams. They are composed of 78 rising and falling D & E intervals and 90 rising and falling F, G & A intervals — the gematria number values of the Hebrew words “Cholem” and “Yesodoth” in the Mundane Chakra of Malkuth. These are the numbers of lines in, respectively, two and three types of off-diagonal hexagrams (see Article 20). We saw in The power of the polygons/hexagon that 168 yods are needed to transform a pair of hexagons joined at one edge into Type B hexagons, 84 per hexagon. There are also 84 Yang lines and 84 Yin lines in the 28 hexagrams that line the sides of the square array of hexagrams in the I Ching table.

The 28 triangles in the first three layers of the three-dimensional Sri Yantra have 84 edges and 84 vertices & triangles. The four layers have 168 edges & triangles made up of 84 edges of triangles in the first three layers, 42 triangles in all four layers and 42 edges in the fourth layer. When the triangles become tetractyses, there are 168 hexagonal yods on the edges of the triangles in the first three layers (84 in each half). When the triangles are divided into their sectors, the fourth layer has 84 edges and 84 vertices & triangles, that is, 168 geometrical elements.

As already mentioned when comparing it with the 64 hexagrams in the I Ching table, the disdyakis triacontahedron has 168 edges above and below the central, 12-sided polygon, 84 on each side. Its 60 vertices surrounding an axis joining two opposite A vertices are the corners of seven polygons parallel to the central one. The polygons in the inner Tree of Life with corresponding numbers are its last four ones. They have 168 yods outside their shared edge lying on their boundaries. The uppermost two polygons in the polyhedron have 84 such yods, as do the next two. The shapes of the four types of polygons making up half the polyhedron are therefore defined by the number value 168 of Cholem Yesodoth. Moreover, they display the basic 84:84 division of this holistic parameter. It is the 206 yods associated with the last four polygons enfolded in the inner Tree of Life with corresponding numbers of corners that symbolize the 206 bones of the human skeleton. The 84:84 division of the boundary yods in the four types of polygons in half the disdyakis triacontahedron therefore reflects the distinction between the axial and appendicular skeletons. Here is a mathematical connection between the two parts of the human skeleton and the core and outer half of the subquark superstring, each with 8400 1st-order spirillae that are manifesting the basic 84:84 division of holistic systems. They are different manifestations of the same holistic pattern.

We saw earlier in the discussion of the division 192 = 24 + 168 that 168 objects make up the 48 permutations of 3 and 4 objects selected from a set of 4 objects. Every permutation has its mirror image, so that there are 24 permutations of 3 and 4 objects and 24 mirror images of them, each set of 24 permutations comprising 84 objects. The division 168 = 84 + 84 that manifests in holistic systems has its arithmetic counterpart in the 84 objects making up the 24 permutations of 3 and 4 objects and the 84 objects that make up their 24 mirror images. Each set of 84 objects comprises the 36 objects in 12 permutations of 3 objects taken 3 at a time and the 48 objects in 12 permutations of 4 objects taken 4 at a time. This 36:48 division manifests in the 2nd-order tetractys as the 36 yods lining its sides and the 48 internal yods that surround its centre (black yod):

36-48 division of yods in 2nd-order tetractys

Article 64 analyses in detail the 24:24, 168:168 & 84:84 divisions in seven sacred geometries and the seven diatonic musical scales.

3360 = 1680 + 1680 (or 336 = 168 + 168)

The ten whorls of the UPA wind five times around its spin axis, each revolution containing 3360 spirillae and each half-revolution containing 1680 spirillae. Sacred geometries embody this division in the following ways:

Outer Tree of Life

Below Binah of the 1-tree are 67 yods (see here). This is the gematria number value of Binah (see here). Below Binah of 67 overlapping Trees of Life are 3360 yods [3]. They divide into the 1680 yods making up the 33-tree, the highest Binah of which is the 67th SL on the Pillar of Judgement, and the 1680 yods above them. This is the remarkable way in which Binah, embodying the feminine principle in nature, encodes not only the superstring structural parameter 3360 but also its division into two 1680s that correspond to the 1680 1st-order spirillae in an outer or an inner half of one revolution of the ten whorls of the UPA/subquark superstring. Truly, Binah, “The great mother,” is the mother of all forms because it gives birth to the 3360 yods symbolizing the 3360 turns in one revolution of the ten whorls of the very fundamental particle that makes up the atomic nuclei of all the chemical elements.

Inner Tree of Life

As discussed here, there are 3360 yods in the seven enfolded polygons of the inner Tree of Life when their 48 sectors are each transformed into a 2nd-order tetractys:

Inside the latter are 49 yods, nine of which (white yods in the diagram above) belong to the tetractyses at the corners of the 2nd-order tetractys corresponding to the Supernal Triad of Kether, Chokmah & Binah. This leaves 40 internal (black) yods (36 hexagonal yods surrounding its centre) that belong to tetractyses expressing the Sephiroth of Construction. In conformity with the pattern characteristic of holistic systems:

48 = 6 + 42 = 6 + 21 + 21,

the 48 sectors group into the six sectors of the hexagon and the 42 sectors of the six other polygons. The latter form two unique sets, each with 21 sectors: (triangle, octagon & decagon) and (square, pentagon & dodecagon). They contain (42×40=1680) internal black yods, 840 in each set. There are also 1680 yods either in the hexagon, in tetractyses at corners of 2nd-order tetractyses or on their sides.

Eight hexagonal yods line each of the 42 sides of the seven enfolded polygons, totalling 336 hexagonal yods. This property has its counterpart in the Sri Yantra when its 42 triangles are turned into tetractyses, namely, 336 yods line their sides (see here). It demonstrates the shape-determining character of the Tree of Life parameter 336, which manifests in the subquark state of the E8×E8 heterotic superstring as the 336 circularly-polarized waves in each revolution of a whorl around the spin axis of the particle. Eleven yods lie on each side of the seven enfolded polygons with 36 corners. The number of yods lining their sides = 42×11 + 36 = 498. Given that one corner of the triangle is the centre of the hexagon and that one corner of the pentagon is the centre of the decagon, there are 496 yods that belong to and line only the boundaries of the seven enfolded polygons, thereby shaping them. 496 is both the gematria number value of Malkuth (see here) and the number of spin-1 particles that transmit the unified superstring force (see here). This is a very remarkable example of how sacred geometries embody numbers discovered by advances in theoretical physics to have fundamental significance. We saw in #20 at Superstrings as sacred geometry/Tree of Life that the twenty Type B dodecagons enfolded in the 10-tree have 3360 yods other than their 220 corners. In other words, 1680 new yods are associated with each set of ten dodecagons. One revolution of all ten whorls of the UPA/superstring represents a whole in itself that manifests in all seven enfolded polygons as their 3360 yods, when their sectors are 2nd-order tetractyses.

We also found in #20 that 1680 yods other than corners are associated with each set of the first six types of polygons enfolded in ten overlapping Trees of Life. As the Tree of Life representation of each whorl, each set has 168 yods other than corners that denote the 168 spirillae in a half-revolution of the whorl, both sets representing one complete revolution of 336 spirillae. The division: 336 = 168 + 168 is characteristic of holistic systems.

Sri Yantra

When the 42 triangles surrounding the centre of the Sri Yantra are converted into tetractyses, their sides are lined by 336 yods, 168 yods in each half (see here). A yod is a potential tetractys of ten yods. If, therefore, the Decad (10) is assigned to these boundary yods shaping the Sri Yantra, its two halves generate the number 3360 as the sum of 1680 and 1680.

Disdyakis triacontahedron

The vertices of the disdyakis triacontahedron form 21 Platonic solids of the four types thought by the ancient Greeks to be the shapes of the particles of the four Elements Fire, Air, Earth & Water (see here). When their faces are constructed from tetractyses, they contain 1680 hexagonal yods. The polyhedral vertices also form a dodecahedron, five rhombic dodecahedra and a rhombic triacontahedron. Constructed from tetractyses, their faces, too, contain 1680 hexagonal yods.

1680 = 840 + 840

This division appears in the micro-psi description by Besant & Leadbeater of the 1680 1st-order spirillae of each whorl of the UPA, because there are 840 spirillae in each of its 2½ outer or inner revolutions around the axis about which this spin-½ superstring spins. It is hardly unexpected, therefore, that a fundamental feature of a holistic system such as a whorl — itself a whole within another whole — should turn out to be a property of sacred geometries.

Outer Tree of Life

The lowest ten Trees of Life in CTOL are the counterpart of the lowest ten dimensions in 26-dimensional space-time. When all triangles are turned into Type A triangles, there are 1680 yods below the top of the tenth Tree, which is prescribed by the Godname ADONAI with number value 65 because it is the 65th SL (see here).

Inner Tree of Life

We saw in #20 at Superstrings as sacred geometry/Tree of Life that, when the 700 sectors of the 120 polygons of the first six types enfolded in the lowest ten Trees of Life of CTOL are turned into tetractyses, 1680 yods line the boundaries of these polygons other than their shared edges, 840 yods being on either side of the Pillar of Equilibrium. They symbolize the 840 spirillae in the inner and outer halves of each whorl. 480 of these yods are corners of polygons and 1200 are hexagonal yods. As there are 120 yods on the boundaries of the seven enfolded polygons, which have 48 corners when separate, the number 480 is embodied in the ten sets of seven separate polygons as the number of their corners, whilst the number 1200 is embodied as the number of boundary yods in ten separate sets of seven enfolded polygons.

We also pointed out in #20 that the dodecagon in the inner Tree of Life has 168 yods other than corners of its sectors, that is, 168 extra yods. They comprise 84 yods and their 84 counterparts on the other side of its centre. The ten dodecagons enfolded in the 10-tree contain 840 extra yods and their 840 mirror images. They symbolize the 840 1st-order spirillae in a quarter-revolution of all ten whorls of the UPA and the 840 spirillae in the next quarter-revolution. Their counterparts in the ten dodecagons enfolded on the other side of the Pillar of Equilibrium denote the spirillae in the other half-revolution of the ten whorls. The 1680 yods in the ten dodecagons include 120 hexagonal yods at centres of tetractyses formally symbolizing Malkuth, leaving 1560 yods (the yods in 156 tetractyses) surrounding these centres. 156 is the 155th integer after 1, showing how ADONAI MELEKH, the complete Godname of Malkuth with number value 155, prescribes the superstring structural parameter 1680 determined by Charles W. Leadbeater about 1908, when he counted the spirillae in each helical coil of a whorl in the UPA (see p. 23 of Occult Chemistry, 3rd ed.).

Sri Yantra

The intersection of the nine primary triangles in the two-dimensional version generates 42 triangles, which, when Type B triangles, possess 1680 new yods other than corners of tetractyses inside each one (see here). They divide into 840 yods belonging to the set of 21 triangles in one half of the Sri Yantra and 840 yods belonging to its other half. Unlike in the inner Tree of Life, they are not mirror images of each other because the Sri Yantra is not exactly symmetric. However, they are still counterparts that form pairs. In the three-dimensional case, there are 1680 yods other than tips of triangles and internal vertices of tetractyses (see here). Again, each set of 21 triangles has 840 such yods.

Disdyakis triacontahedron

When the polyhedron is constructed from triangles, with each internal triangle formed by its centre and an edge divided into three sectors, 1680 corners, triangles and their sides surround an axis joining any two diametrically opposite vertices, whether they are A, B or C vertices (see here & here). The 1680 geometrical elements comprise 840 elements in one half of the polyhedron and their 840 diametrically opposite mirror images, reflected through the centre of the disdyakis triacontahedron, which possesses inversion symmetry.

Inspection of Table 1 reveals that there are 1680 harmonics between note G₆ with tone ratio 48 and note A₁₁ with tone ratio 1728. The latter is the 76ᵗʰ note. YAHWEH ELOHIM with number value 76 prescribes the very note in the Pythagorean scale that is separated by 1680 harmonics from G₆! This amazing fact demonstrates how YAHWEH ELOHIM prescribes the musical counterpart of the superstring structural parameter 1680 as well as the structural parameter 336. The Godname ELOHA with number value 36 prescribes A₁₁ as the 36ᵗʰ note above G₆, the 48ᵗʰ harmonic, whilst there are 151 notes and successive intervals up to A₁₁, where 151 is the 36ᵗʰ prime number. The Tetrad aptly expresses A₁₁ because it is the 44ᵗʰ musical harmonic. Moreover, there are ten (=1+2+3+4) octaves, ten perfect fifths and 24 (=1×2×3×4) other Pythagorean harmonics up to A₁₁. This note is the only overtone other than A₈ and G₉ that is separated from G₆ by an integer multiple of 168. It is readily verified that G₆ and A₁₁ are the only notes of the Pythagorean scale whose tone ratios differ by 1680. Clearly, A₁₁ is unique in the context of the 1680 turns of the helical whorl of the heterotic superstring, whose embodiment in the inner form of ten Trees of Life shows that they consist of ten sets of 168 turns. Its connection to superstring physics is further demonstrated by the fact that, as the 75ᵗʰ note above the fundamental, A₁₁ is the 65ᵗʰ note other than the ten octaves, i.e., it is prescribed by ADONAI, the Godname of Malkuth (denoting the physical universe), whose number value is 65. A₁₁ is the 33ʳᵈ overtone other than octaves, where 33 = 1! + 2! + 3! +4!. The 33ʳᵈ note and the tenth overtone is G₅ with tone ratio 24. Therefore, A₁₁ is the 33ʳᵈ non-octave overtone above the 33ʳᵈ note! Given that there are 1680 yods in the 33-tree, the role played by the number 33 in each context in determining this superstring structural parameter is remarkable. The 50ᵗʰ note prescribed by ELOHIM is the seventh octave C₈ with tone ratio 128. It is 80 harmonics above G₆, so that the next 26 notes prescribed by YAHWEH up to A₁₁ span 1600 harmonics. The two words in the Godname of Tiphareth define a 80:1600 division in the number 1680 that appears in the 33-tree as the 80 yods in the 1-tree and as the 1600 yods in the 32 Trees above it. This section of CTOL is prescribed by ADONAI because Malkuth of the 33ʳᵈ Tree of Life is the 65ᵗʰ SL on the central pillar. There are 1730 yods up to the top of the 34ᵗʰ Tree of Life, that is, 1728 yods below its top other than that at Daath of this Tree. Alternatively, there are 48 yods below the top of the 34th Tree of Life other than its Daath that are outside the 33-tree. This 48:1728 division is the Tree of Life counterpart of the 48 harmonics up to G₆ and the 1728 harmonics up to A₁₁. Each yod denotes a harmonic. The 48 harmonics from G₆ to G₇ are the counterpart of the 48 yods up to Chesed of the 1-tree. The 80 harmonics from G₆ to C₈ are the counterpart of the 80 yods of the 1-tree. The first of the 1680 harmonics above G₆ up to A₁₁ is the 49ᵗʰ harmonic, so that it is prescribed by the Godname EL CHAI of Yesod.

None of the 14 basic types of notes in the seven musical scales:

	256/243	9/8	32/27	81/64	4/3	1024/729	729/512	3/2	128/81	27/16	16/9	243/128	2

(see here) has a higher octave whose tone ratio is 1680. This number must be understood as referring, not to a note of any scale, but to the number of harmonics between G₆ and A₁₁. Inspection of Table 1 confirms that A₁₁, which is the first note to have a tone ratio larger than 1680, is also the only note differing by 1680 harmonics from a note of lower pitch! The note A₁₄ is the last note to differ from its predecessor by less than 1680 harmonics, whilst all subsequent, consecutive notes differ by more than 1680 harmonics. G₆ and A₁₁ are, therefore, the only notes in any octave of the Pythagorean scale whose tone ratios differ by 1680. The uniqueness of this pair of notes is highly significant because it eliminates the criticism that any discussion based upon a particular choice of pairs of notes would be ad hoc if other pairs existed that differ by 1680 harmonics.

The 1680 harmonics from G₆ to A₁₁ contain 29 overtones. 29 is the tenth prime number. The Decad, therefore, determines how many of these 1680 harmonics are overtones! Table 1 shows that they contain 24 overtones other than octaves, where 24 = 1×2×3×4 and 10 = 1 + 2 + 3 + 4. Counting from G₆, we see that the 1680th harmonic is the 24th Pythagorean harmonic other than octaves. The first 168 harmonics up to A₈ include nine such notes, the next 168 harmonics up to G₉ include four such notes and the remaining harmonics include 11 notes that are not octaves. The 24 non-octave overtones are distributed nine to the first 168 harmonics and 15 to the remainder. This 9:15 division reminds us of the distinction in string theory between the nine transverse dimensions existing in the 11-dimensional space-time of M-theory and the 15 higher transverse dimensions belonging to bosonic strings in 26-dimensional space-time. If we take the analogy seriously, it suggests that oscillations in the 24 transverse directions generate 24 particles as 24 vibration modes that correspond to the 24 overtones, the nine overtones up to A₈ corresponding to the nine particles representing Kaluza-Klein-type oscillations in each of the nine transverse dimensions and the 15 remaining overtones corresponding to the 15 vibration modes for the 15 higher dimensions. Are the 1680 helical turns of a closed whorl the manifestation of the charge sources of 24 gauge fields of E₈, the ten whorls of the UPA/superstring being the manifestation of the sources of the 240 gauge fields associated with its 240 roots? The musical analogy strongly implies such an interpretation.

When G₆ is regarded as the fundamental frequency, the tone ratios of the 24 higher overtones other than octaves are shown below (new overtones are written in red):

The overtones, starting with A₆ with tone ratio 54, span exactly five octaves, the last note A₁₁ having the new tone ratio 36 relative to G₆. The set of 24 overtones is prescribed by ELOHA, Godname of Geburah with number value 36. The note in the Pythagorean scale with tone ratio 36 is D₆. It is the 36th note above the tonic. It is the only note in the scale whose tone ratio is identical to its position number above the tonic — another reason why A₁₁ is special.

There are three overtones up to the note A₈ with tone ratio 216 that remain overtones when G₆ is regarded as the fundamental. The remaining 21 overtones comprise A₆, the first one, five partials up to A₈, followed by ten harmonics and five partials spanning the last three octaves. Amazingly, these are the letter values of EHYEH (AHIH) with number value 21, namely, A = 1, H = 5, I = 10 & H = 5! In terms of its geometrical counterpart in the disdyakis triacontahedron, A₈ represents the last of the 216 geometrical elements constituting exactly half of the polyhedron. It is the reason for the thick, vertical line dividing the 24 overtones at this note. The 24 overtones display the 3:21 division characteristic of holistic systems, as illustrated earlier.

Just as the 1680 helical turns of each whorl of the UPA/heterotic superstring revolve five times around its spin axis, so the 1680 harmonics between G₆ with tone ratio 54 and A₁₁ with tone ratio 1728 contain 24 overtones other than octaves that span five complete octaves, that is, five musical cycles. In both cases, the number 1680 quantifies an underlying five-fold cycle. The relative frequencies of the 24 overtones increase by a factor of 1728/54 = 2⁵ = 32). This is the number of components of the Dirac wave function describing spin-½ fermions in 10-dimensional space-time, e.g., the UPA. A remarkable analogy exists between the vibrating, string-like whorl and the harmonics of the Pythagorean scale because they are both holistic systems.

Table 3. The 1680 harmonics between G₆ and A₁₁.

Let us next compare the pattern of 1680 harmonics with the 1680 geometrical elements surrounding any axis of the disdyakis triacontahedron. If the former truly constitute a holistic set, their composition should correlate with the geometrical structure of this polyhedron, which many research articles posted to this website by the author have confirmed is the polyhedral form of the outer Tree of Life. When the axis passes through two diametrically opposite A vertices, the 1680 elements comprise 24 vertices above the 12-gon in the central plane of the polyhedron, 24 vertices below it, 12 vertices of this 12-gon and 180 vertices of the 540 internal triangles, i.e., 240 vertices. Their musical counterparts are the 24 harmonics up to D₇ (Table 3), then 24 harmonics up to G₇, 12 more harmonics to A₇ (the first of the five octaves and the 21st Pythagorean harmonic) and the next 180 harmonics up to D₉, which, as the perfect fourth of the third of these octaves, is the middle of the five octaves. Then, there are 180 edges, 60 internal sides ending on polyhedral vertices & 120 external triangles, i.e., 360 edges & triangles. Their Pythagorean musical counterpart is the 360 harmonics beyond D₉ up to E₁₀. Finally, there are 540 more internal edges of 540 internal triangles, i.e., 1080 internal edges & triangles. Their counterpart is the 1080 harmonics above E₁₀ up to A₁₁.

There is no overtone that is the 840th harmonic above G₆. The only two notes that differ by 840 harmonics are G₅ with tone ratio 24 and A₁₀ with tone ratio 864, but they do not all fall within the range of the 1680 harmonics. However, there are 840 odd harmonics from 49 to 1727 and 840 even harmonics from 50 to 1728. The musical counterpart of the 840 helical turns in the 2½ revolutions of an outer or inner half of a whorl in the UPA/heterotic superstring are the 840 even and 840 odd harmonics between G₆ and A₁₁. As the former note is the 15th Pythagorean harmonic and the 40th note, and the latter note is the 44th Pythagorean harmonic, the 76th note and the 36th note after G₆, this shows how YAH, YAHWEH ELOHIM and ELOHA prescribe these five octaves spanned by the 1680 harmonics. Their counterparts in the disdyakis triacontahedron are the 840 geometrical elements in one half of the polyhedron that surround an axis joining two opposite vertices and their 840 mirror images in its other half.

The 24 overtones other than octaves in the 1680 harmonics between G₆ and A₁₁ are notes of the first five octaves of the A scale (Hypodorian mode). In terms of this scale, they comprise 12 harmonics and 12 partials with the following tone ratios:

	A	B	D	E	G
1st octave:	1		4/3	3/2	16/9
2nd octave:	2		8/3	3	32/9
3rd octave:	4	9/2	16/3	6	64/9
4th octave:	8	9	32/3	12	128/9
5th octave:	16	18	64/3	24	256/9
6th octave:	32

22 of the 24 overtones have even tone ratios and two overtones (E₇ with tone ratio 81 and B₈ with tone ratio 243) have odd tone ratios.

Compare this with the prediction by string theory that bosonic strings exist in 26-dimensional space-time and vibrate along two transverse, large-scale dimensions of space and 22 transverse, microscopic dimensions, i.e., 24 transverse dimensions = 22 compactified dimensions + 2 large-scale dimensions.

One of the 22 curled-up dimensions is the dimensional segment separating the two space-time sheets occupied by E₈×E₈′ heterotic superstrings of ordinary matter and shadow matter, so that the compactification of 21 dimensions is instrumental in creating each type of superstring, where 21 is the number value of EHYEH, Godname of Kether.

They correspond to the 21 even overtones above A₆, which, as the lowest note of the five octaves of notes shown above, corresponds to the dimensional gap between the two space-time sheets.

A = 1: lowest octave with tone ratio 2.
H = 5: five octaves of partials (16/9, 32/9, 64/9, 128/9, 256/9).
I = 10: ten overtones (3, 4, 6, 8, 9, 12, 16, 18, 24, 32).
H = 5: five octaves of partials (4/3, 8/3, 16/3, 32/3, 64/3).

Notice that the letter values Y = 10 & H = 5 of the Godname YAH (Hebrew: YH, or יה) with number value 15 denote, respectively, the ten overtones and the five octaves of perfect fourths. Notice also that the first ten overtones in the A scale are the same as the first ten overtones in the Pythagorean scale (see Table 1). These and the 22 partials up to the tenth overtone were shown in #19 at The seven musical scales to constitute a Tree of Life pattern. Seven of them (notes B, D & E) belong also to the A scale.

Given this correlation between the letter values of EHYEH and the numbers of overtones and partials spanning the first five octaves of the A scale — a pattern of such detail that it could not, plausibly, arise by chance — the following question arises: what compactified dimensions correspond to the numbers? Six of them are predicted by superstring theory and 15 are dimensions of bosonic strings. This division is indicated by the letters A and H of AHIH, whose gematria sum is 6, and by the letters I and H, which sum to 15.

It means that the ten overtones up to that with relative tone ratio 32 (the 1680th harmonic after G₆) and the five octaves of perfect fourths in the A scale have their counterparts in the 15 higher, bosonic dimensions. What is the string counterpart of this 10:5 distinction between overtones and partials? It is the role played by ten of the bosonic dimensions when a membrane wraps itself around each of them to generate the ten whorls of the UPA/superstring.

The dimensional counterparts of the three overtones with tone ratios 3, 4 & 6 create the three major whorls corresponding to the Supernal Triad of the Tree of Life. The counterparts of the remaining seven overtones 8, 9, 12, 16, 18, 24 & 32 create the seven minor whorls corresponding to the seven Sephiroth of Construction.

A₁₁, the 1680th harmonic beyond G₆ in the Pythagorean scale, is the fifth octave of the A scale with tone ratio 32. It is its 36th note, prescribed by ELOHA, the Godname of Geburah with number value 36. Relative to one another, the first three overtones are the tonic, perfect fourth and octave of the Pythagorean scale and that the last seven overtones form two successive octaves of the tonic, major second and perfect fifth, ending with the third octave. This 3:3:1 pattern is analogous in the Tree of Life to the two triads of Sephiroth of Construction (Chesed-Geburah-Tiphareth & Netzach-Hod-Yesod) and Malkuth, to which the seven minor whorls correspond.

The compound of two 600-cells

The 600-cell is the 4-dimensional counterpart of the icosahedron. It is well-known to mathematicians that the compound of two concentric 600-cells, one of different size inside the other, is the 4-dimensional, Coxeter plane projection of the 8-dimensional polytope called the “4₂₁ polytope” whose 240 vertices define 240 vectors that are identical to the 240 root vectors of the Lie group E₈ appearing in heterotic superstring theory.

The 600-cell has 120 vertices and 720 edges, i.e., 840 vertices & edges. The two 600-cells have (840+840=1680) vertices & edges. This provides a connection to the 1680 turns in a helical whorl of the UPA described by Besant & Leadbeater (and therefore irrefutable evidence that the latter is an E₈×E₈′ heterotic superstring) that cannot, plausibly, be dismissed as due to chance.

Basic parameters of holistic systems

Holistic systems are hierarchical, that is, they display different levels of differentiation, each a whole in itself. For example, starting with the Tree of Life, we can generate its inner form, then analyze the inner form of ten Trees of Life or a subset of its 14 polygons that itself constitutes a whole. Similarly, we can start with the nine primary triangles of the Sri Yantra, then analyze the properties of its two-dimensional and three-dimensional forms when constructed from tetractyses and their higher-order differentiations. What is remarkable (although not surprising from a theoretical perspective) about holistic systems is that they should display the same set of parameters. Those discussed up till now refer to higher levels of differentiation of holistic systems. Their more basic parameters and their characteristic divisions are discussed below.

70 = 10 + 60

Tree of Life

Transformed into tetractyses, its 16 triangles contain 70 yods. They comprise ten yods at corners and 60 hexagonal yods (see here);
The 14 enfolded polygons of the inner Tree of Life have 70 corners (see here). They comprise either the ten corners of the pair of hexagons and the 60 corners of the 12 other polygons or, alternatively, the ten corners that either are endpoints of the root edge, coincide with Sephiroth or are centres of polygons and their 60 remaining corners;
The 70 polygons enfolded in the 10-tree comprise ten dodecagons and 60 other polygons;
The first (6+6) enfolded polygons have 70 sectors, ten of which belong to the pair of pentagons;
When its 16 triangles are Type A, the outer Tree of Life consists of 48 triangles with 70 sides (see a):

a

b

Of these, 10 blue sides (see b) are solid, straight lines making up the boundary of the “trunk” of the Tree of Life, namely, the Chokmah-Binah Path (one side), the Chesed-Geburah-Tiphareth triangle (three sides), and the Netzach-Hod-Yesod-Malkuth tetrahedron (six sides), whilst 60 red sides are either dashed sides of the 48 sectors of the 16 triangles or solid sides of the 11 triangles making up the “branches” of the Tree of Life.

Platonic solids

Constructed from tetractyses, the tetrahedron has 70 yods surrounding its centre (see here). Ten yods are either vertices or centres of internal tetractyses. The simplest Platonic solid is a polyhedral representation of the Tree of Life that has its apotheosis in the disdyakis triacontahedron, which is the polyhedral representation of ten overlapping Trees of Life. The five Platonic solids have 50 vertices and 90 edges. (25+45=70) vertices & edges make up each set of five halves of the regular polyhedra:

70 vertices & edges in 5 half-Platonic solids

2nd-order tetractys

This Pythagorean representation of holistic systems is an array of ten 1st-order tetractyses with 70 hexagonal yods (see here), ten of which are at their centres and 60 of which are at corners of hexagons.

Sri Yantra

The 43 triangles of the two-dimensional Sri Yantra have 70 vertices if we include the central bindu point formally as an isolated vertex (see here). Ten vertices lie on the central, vertical axis passing through it, so that the 70 vertices divide into sets of 10 & 60.

Disdyakis triacontahedron

There are 28 vertices and seven centres of the polygons either above or below the central, 6-sided polygon that is perpendicular to an axis passing through two opposite C vertices (see here). The (28+7+28+7=70) points that are either polyhedral vertices or corners of the 54 sectors of these 14 polygons include the centres of 10 triangles.

70 = 35 + 35

Outer Tree of Life

There are 70 yods in the Tree of Life when its 16 triangles are tetractyses. Its trunk (see here) contains 35 red yods, leaving 35 blue yods in its branches.

Inner Tree of Life
The (7+7) enfolded regular polygons of the inner Tree of Life have 70 corners. Associated with one set of seven polygons are 35 red corners and associated with the other set are 35 blue corners. The former corresponds to the trunk and the latter corresponds to the branches of its outer form.

2-d Sri Yantra

The 2-dimensional Sri Yantra is made up of 70 points (the bindu point at its centre & the 69 corners of its 43 triangles). Two points belong exclusively to the central triangle, its uppermost two corners coinciding with corners of the innermost violet triangles. One of the latter is associated with the upper half of the Sri Yantra and is coloured red; the other corner is coloured blue because it is associated with the lower half of the Sri Yantra. This means that the bindu and the lowest corner of the central triangle must have different colours. But which? These two points cannot correspond in the outer Tree of Life to Kether and Malkuth because both Sephirothic points belong to its trunk, whereas one has to belong to its branches, which are composed of blue yods. They correspond in the inner Tree of Life to the two endpoints of the root edge, which represent the start and end of the generation of this geometrical structure. They are the projection onto the plane containing the left-hand and right-hand pillars of the Tree of Life of, respectively, Daath, which is a blue yod because it belongs to the branches, and Tiphareth, which is a red yod, belonging to the trunk. The bindu symbolizes the source of all levels of reality and corresponds to the upper (blue) endpoint of the root edge in the inner Tree of Life because Daath projects onto this point. Hence, it must be coloured blue and the lowest corner of the central triangle must be coloured red, as it corresponds to the other endpoint. The two halves of the Sri Yantra correspond to the trunk and branches of the outer Tree of Life and to the two sets of seven enfolded polygons of its inner form.

Disdyakis triacontahedron
Sixty vertices of the disdyakis triacontahedron surround an axis that passes through two diametrically opposite C vertices. They form the corners of 15 polygons, the central one of which is a hexagon (see here & here). The 14 polygons in the upper & lower halves of the polyhedron (shown red & blue) have (60−6=54) corners, so that 27 belong to each half. Including the isolated C vertex (denoted by a red or blue dot), there are (1+27+7=35) points in the upper half of the polyhedron that are either vertices or corners of sectors of polygons and 35 points in the lower half that are likewise. The 35:35 division that is characteristic of holistic systems manifests in the disdyakis triacontahedron as its upper and lower halves and the internal polygons that its vertices form. The two C vertices correspond to:

the two endpoints of the root edge of the (7+7) enfolded polygons of the inner Tree of Life;
the bindu & lower corner of the central triangle of the Sri Yantra;
Daath and Tiphareth of the Tree of Life.

The icosahedron
When the sectors of the faces of the icosahedron are turned into tetractyses, there are (30×2=60) hexagonal yods on its 30 edges. Ten vertices surround an axis that passes through two opposite vertices. Hence, (10+60=70) yods surround this axis, lining its edges to create its form. This pattern is another example of the 10:60 division discussed earlier in the case of the tetrahedron which, constructed from tetractyses, has 70 yods, 35 yods being distributed in each half of it. 35 red yods line the 15 edges of one half of the icosahedron and 35 blue yods line the 15 edges of its other half (for clarity, the diagram above displays only an illustrative set of four yods of each colour).

The 35:35 pattern in sacred geometries is also discussed here.

120 & 144

Tree of Life

The outer Tree of Life consist of 10 corners and 22 sides of 16 triangles, i.e., 48 corners, sides & triangles (see here); its inner form has 48 corners (see here). When its 16 triangles are Type A, it has (3×16=48) triangles with (10+16=26) corners and (22+48=70) sides, i.e., (26+70+48=144) corners, sides & triangles. The counterparts of this in the inner Tree of Life are the 144 yods along the boundaries of the seven separate, regular polygons, 48 of which are corners. Enfolded, the seven polygons have 120 boundary yods and 144 internal yods (see here). This is how the inner Tree of Life encodes the Polyhedral Tree of Life. This is a pair of polyhedra consisting of one with 74 vertices & 144 faces and the disdyakis triacontahedron with 62 vertices & 120 faces (see here). The 48 sectors of the seven separate polygons have 144 corners & sides surrounding their centres. There are 120 hexagonal yods in the pair of Type A dodecagons, which constitute a Tree of Life structure in themselves. A single, Type B dodecagon has 144 yods surrounding its centre other than hexagonal yods at centres of its 36 tetractyses. It also has 120 geometrical elements surrounding its centre. There are 120 hexagonal yods on the 60 edges of the first (6+6) enfolded polygons outside their shared root edge, indicating that they are in themselves a holistic structure within the inner form of the Tree of Life (see Article 4 for how they are prescribed by the Godnames).

Sri Yantra

There are 240 geometrical elements in the 43 triangles of the two-dimensional Sri Yantra that surround its central bindu point (see here). 120 elements belong to each half. Unlike the 42 triangles surrounding it, the central triangle with three sides and with its lowest corner unshared with the surrounding triangles does not divide into two identical halves. However, these four elements do form two sets of two elements: (side, side) & (corner, side). This makes it possible to divide the 240 geometrical elements into two same-sized sets. There are 144 yods on the 54 sides of the 18 triangles in the first and second layers.

Disdyakis triacontahedron

The central 12-gon formed by the 12 vertices in its equatorial plane:

is perpendicular to an axis passing through two opposite A vertices. Surrounding its centre are 12 sectors composed of 48 geometrical elements. The polyhedron has 120 faces (60 in each half). The 84 edges on each side of the central plane containing the 12-gon are sides of 84 internal triangles that meet at the centre of the polyhedron. (60+84=144) internal and external triangles exist on each side of the central polygon.

350 = 90 + 260,

where

90 = 36 + 54

Lambda Tetractys

This is the division generated by the Tetrahedral Lambda, the arithmetic version of the universal pattern displayed by sacred geometries (see here). The sum of the ten integers in the Lambda Tetractys is 90 and the sum of the remaining ten integers is 260. The Tetrad defines this division of numbers because, in a square whose sectors are tetractyses, the 12 red integers 2–13 assigned to the 12 yods on its edges add up to 90, whilst the next 13 black integers 14–26 that can be assigned to its 13 internal yods add up to 260. In other words, the sum of the 25 integers 2–26 is 350, which is the sum of the 20 integers making up the Tetrahedral Lambda:

Square representation of 350

Notice that not only do the integers 2, 5, 8 & 11 at the four corners add up to 26, the number value of YHVH, but also the three smallest ones add up to 15, the number value of YH. The 11:15 division that this generates is characteristic of holistic systems. For example, the trunk of the Tree of Life consists of the geometrical sequence: point, line, triangle & tetrahedron (see here and Figure 4 here). The first three comprise 11 points, lines & triangles and the fourth consists of 15 points, lines, triangles & tetrahedra.

Tree of Life

350 corners are intrinsic to the 70 polygons enfolded in the inner form of the 10-tree (see here). Ninety corners outside their root edges belong to the first four types of polygons and 260 corners belong to the last three types. Every ten overlapping Trees consist of 350 vertices, edges, triangles & tetrahedra. Divided into their sectors, the seven enfolded polygons of the inner Tree of Life have 176 vertices, edges & triangles (see here). The topmost vertex of the hexagon coincides with the lowest vertex of the hexagon enfolded in the next higher Tree of Life, leaving 175 geometrical elements intrinsic to that set of polygons. The two separate sets of enfolded polygons have 350 intrinsic geometrical elements (90 in the first three types of polygons and 260 in the last four types). 350 hexagonal yods line the 175 edges of the 94 sectors of the (7+7) enfolded polygons (see here). Ninety of them are in the pairs of squares & octagons.

Sri Yantra

Regarded as tetractyses, the nine basic triangles generating the Sri Yantra have 90 yods. 36 yods are at their corners or centres and 54 yods are hexagonal yods (see here).

The Platonic solids

The 90 edges of the five Platonic solids consist of the 36 edges of the tetrahedron & dodecahedron and the 54 edges of the octahedron, cube & icosahedron. Table 4 shows the number of geometrical elements in their faces and interiors when internal sectors are Type A triangles (“1” denotes the centre of a regular polyhedron):

	Corners	Sides	Triangles	Total
	Interior	Exterior	Interior	Exterior
Tetrahedron	6 + 1 = 7	4 + 4 = 8	4 + 6×3 = 22	6 + 4×3 = 18
Octahedron	12 + 1 = 13	6 + 8 = 14	6 + 12×3 = 42	12 + 8×3 = 36
Cube	12 + 1 = 13	8 + 6 = 14	8 + 12×3 = 44	12 + 6×4 = 36
Icosahedron	30 + 1 = 31	12 + 20 = 32	12 + 30×3 = 102	30 + 20×3 = 90
Dodecahedron	30 + 1 = 31	20 + 12 = 32	20 + 30×3 = 110	30 + 12×5 = 90
Subtotal =	60 + 4 = 64	30 + 38 = 68	210	60 + 120 = 180
Total =	90 + 5 = 95	50 + 50 = 100	320	90 + 180 = 270

The 150 triangles in the dodecahedron have 62 corners surrounding its centre (the same number as the number of vertices surrounding the centre of the disdyakis triacontahedron). YAH prescribes the dodecahedron because 150 = 15×10. EL prescribes it because 62 is the 31st even integer, 31 being the number value of this Godname of Chesed. The fifth Platonic solid has 90 sides of triangles in its faces, 110 internal sides and 150 triangles, i.e., 260 triangles & internal sides, making a total of 350 sides & triangles. Its 90 external sides correspond to the sum of the ten integers in the Lambda Tetractys and the 260 triangles & internal sides correspond to the sum of the remaining ten integers in the Tetrahedral Lambda. The dodecahedron has 200 sides and 150 triangles. This 200:150 division corresponds to the sum (200) of the 12 integers at the corners of the six-sided polygons in each face and to the sum (150) of the eight integers at the vertices and centres of faces of the tetrahedral array of 20 integers (see Figure 3 here). These proportions do not occur in the icosahedron. Although, mathematically speaking, the dodecahedron is merely its dual, it can be said to be more perfect than the icosahedron because its geometrical composition is characterized by various parameters of the Tetrahedral Lambda. Moreover, it is the only Platonic solid that embodies these numbers. Perhaps, however, “perfect” is not quite the right word. Rather, it is more correct to say that the complete realisation of the divine archetype is most explicit in the dodecahedron.

The table shows that the five Platonic solids have 1230 geometrical elements surrounding their centres. The average number is 246, which is the number value of Gabriel, the Archangel of Yesod. The average number of triangles is 90 (54 internal, 36 external). This 36:54 division is displayed by the Lambda Tetractys. 550 geometrical elements make up their faces. This is the number of SLs in CTOL. Notice that the Decad measures not only this number:

550 = 10×(1+2+3+4+5+6+7+8+9+10)

but also the 190 corners of triangles surrounding the axes of the five Platonic solids. This is because 190 = 10×19, where 19 is the tenth odd integer, there being 100 (=10²) comers of triangles in their faces. Including their centres, the five solids have 95 internal vertices. This is the number value of Madim, the Mundane Chakra of Geburah. The 450 triangles in the five solids have 500 (=50×10) sides that are not polyhedral edges. The Godname ELOHIM with number value 50 prescribes not only their 50 faces but also the number of extra sides of triangles needed to construct them from the latter. The number of extra corners = 90 + 50 = 140, which is the number value of Malachim, the Order of Angels assigned to Tiphareth.

Disdyakis triacontahedron

When its internal triangles are divided into their sectors, the polyhedron has (180+120+60=360=36×10) corners & external triangles and (180×3=540=54×10) internal triangles that surround an axis joining two opposite vertices, i.e., 900 (=90×10) vertices & triangles. The polygons formed by the 60 vertices that surround this axis contain 900 yods distributed around it when their sectors are Type A triangles because such an n-gon has 15n yods surrounding its centre, so that the yod population of the polygons with 60 corners = ∑15n = 15 ∑n = 15×60 = 900. They comprise (60×6=360) yods either on their edges or at centres of tetractyses and (60×9=540) yods on internal sides of tetractyses.

Each half of this Catalan solid has 90 edges above and below the central polygon perpendicular to an axis passing through two opposite C vertices. This is a hexagon, none of whose sides are polyhedral edges because its six corners are all A vertices (see here). They form an edge with B and C vertices both above and below the central hexagon. As there are 30 AB, 30 AC and 30 BC edges in each half, there are 24 AC edges and 30 BC edges that do not touch this hexagon and which contain a C vertex. This leaves 24 AB edges that do not touch the polygon and six AB edges and six AC edges that do touch it. The 90 edges in each half of the polyhedron divide into 54 edges (30 BC & 24 AC) containing C vertices that do not touch the polygon and 36 edges (30 AB & six AC) that either do touch it or do not contain C vertices. The 90 edges in each half of the polyhedron therefore display the 36:54 division of the Lambda Tetractys.

When internal and external triangles are Type A, the disdyakis triacontahedron is composed of 900 triangles [360 (=36×10) external, 540 (=54×10) internal] (see here).

The ten-fold sequence of generation of the disdyakis triacontahedron (point–line–triangle–tetrahedron, etc) contains 351 vertices and faces, where 351 is both the number value of Ashim, the Order of Angels assigned to Malkuth, and the 26th triangular number, where 26 is the number value of YAHWEH, the Godname of Chokmah. (By “vertices” we mean not only the polyhedral vertices but also the starting point, the two endpoints of the single, straight line & the three corners of the triangle in the sequence — that is, mathematical points). Therefore, 350 vertices & faces are needed to generate the sequence of objects, given this point (the term “faces” includes the triangle). Their composition is shown below:

Vertices	Faces
line:	2
triangle:	3
tetrahedron:	4
octahedron:	6
cube:	8
icosahedron:	12
dodecahedron:	20
rhombic triacontahedron:	32
disdyakis triacontahedron:	62
Total =	149

= 149 201

The seven polyhedra in the sequence have 200 faces. The 150 (=15×10) other geometrical elements in the sequence consist of the 50 vertices of the five Platonic solids, the triangle and 99 other vertices, i.e., 100 elements. Compare these numbers with the arithmetic properties of the Tetrahedral Lambda:

Sum of integers at centres of faces = 50.

Sum of integers at corners = 100.

Sum of remaining integers = 200.

Correspondence also exists between the 6:36:48 division in the Lambda Tetractys, which is the first face of the Tetrahedral Lambda:

     1

   2   3

 4   6   9

8  12  18  27

The integer 6 at the centre of the Lambda Tetractys denotes the triangle and the five vertices of the line & triangle, the sum (36) of the integers 1, 8 & 27 at its corners denotes the four vertices of the tetrahedron and the 32 vertices of the rhombic triacontahedron and the sum (48) of the six other integers denotes the four faces of the tetrahedron (4), the 14 vertices & faces of the octahedron (2+3+9=14) and the 30 faces of the rhombic triacontahedron (12+18). This leaves the 100 vertices of the cube, icosahedron, dodecahedron and disdyakis triacontahedron and their 160 faces, i.e., 260 vertices & faces. This is the sum of the remaining 16 integers in the Tetrahedral Lambda. We conclude that the sequence of ten geometrical objects that lead to the disdyakis triacontahedron is archetypal because their vertices and faces conform to the 6:36:48 and 90:260 proportions of the Tetrahedral Lambda. Notice that the sequence: line→rhombic triacontahedron contains 87 vertices and 81 faces, i.e., 168 vertices & faces. 87 is the number value of Levanah, the Mundane Chakra of Yesod, which is the penultimate Sephirah, and 168 is the number value of Cholem Yesodoth, the Mundane Chakra of Malkuth, which is the last Sephirah. Hence, the sequence leading to the rhombic triacontahedron comprises the number values of the Mundane Chakras of the last two Sephiroth.

137

The number that determines the fine-structure constant (see here) is always embodied in holistic systems expressing the divine archetypes. Because matter itself is the manifestation of their designing influence, there is no mystery why this parameter of such systems plays such a central, mathematical role in determining the physics of the atomic world.

Tree of Life

Below the top of the 7-tree mapping space-time (see Article 2) are 137 vertices & triangles [4]. The 14 enfolded polygons of the inner Tree of Life whose sectors are Type A triangles contain 1370 (=137×10) yods (see here). Divided into their sectors, the last four enfolded polygons have 137 sides of 72 triangles (see here).

Tetrahedral Lambda

The sum of the integers 2, 3 & 4 and their squares & cubes on the three inclined edges of the Tetrahedral Lambda is 137 (see here):

	2¹	3¹	4¹
137 =	2²	3²	4²
	2³	3³	4³

These are the primary numbers of the archetypal set of 20 integers on these edges, including the number 1, the Pythagorean Monad, at the apex of the Tetrahedral Lambda. As:

2¹ + 2² + 2³ = 2(1+2¹+2²) = 2×7,

3¹ + 3² + 3³ = 3(1+3¹+3²) = 3×13,

and

4¹ + 4² + 4³ = 4(1+4¹+4²) = 4×21,

the 10 powers of 1, 2, 3 & 4 on the inclined edges can be written as a tetractys array of the integers 1, 7, 13 & 21:

Their sum is the number of yods outside the root edge in the two Type A dodecagons present in the inner form of the Tree of Life:

= 138

(For the sake of clarity, the four yods on the root edge are omitted). The number 137 is the sum of the nine odd integers below the apex 1 of this tetractys array. The sum of the four integers 21 (gematria number of EHYEH, the Godname of Kether) at the base of the tetractys is 84. This number is the sum of the nine integers in the Lambda Tetractys that surround its central integer 6 (see here). It is the number of circular turns in one-quarter of a revolution of each whorl of the UPA. The number 84 is expressed by the Tetrad because

1² + 3² + 5² + 7² = 84 =
Gnomon representation of 84

This 4×4 array of the first four odd integers is also significant for the group mathematics underlying superstrings because, when the squares of these integers are present, it generates the dimension 496 of both E8×E8 and SO(32), which are the two possible symmetries of their unified interactions that are free of quantum anomalies (see here).

The Platonic solids

The average number of yods needed to construct the faces of the four Platonic solids representing the four Elements is 137 (see here). The average number of corners & triangles making up half a Platonic solid is 137 (see here). Including their centres, the total number of such geometrical elements needed to construct the five halves of the Platonic solids = 137×5 = 685. As the inner Tree of Life has 1370 yods when the sectors of its 14 regular polygons are Type A triangles (see here), this, remarkably, is the number of yods associated with each half of the inner Tree of Life! It demonstrates the holistic nature of the mathematically complete set of five regular polyhedra. As further confirmation of its archetypal character, the number 685 is also the number of geometrical elements present in the sequence of development in 10 steps of the disdyakis triacontahedron from a point:

	Vertices	Edges	Faces	Total
1. Point	1	0	0	1
2. Line	2	1	0	3
3. Triangle	3	3	1	7
4. Tetrahedron	4	6	4	14
5. Octahedron	6	12	8	26
6. Cube	8	12	6	26
7. Icosahedron	12	30	20	62
8. Dodecahedron	20	30	12	62
9. Rhombic triacontahedron	32	60	30	122
10. Disdyakis triacontahedron	62	180	120	362
Total	150	334	201	685

The cube has 137 internal yods when it is constructed from Type A triangles (see #10 at The Platonic solids and the table in #16).

Seven diatonic musical scales

Including their tonics, or starting notes, the seven diatonic scales have 137 notes and intervals that belong to the Pythagorean scale (see table on p. 13 of Article 16).

Disdyakis triacontahedron

The Type A sectors of the 15 polygons perpendicular to an axis joining two opposite C vertices that are formed by 60 of the 62 polyhedral vertices have (15+60+60=135) corners. The triangles making up the polyhedron and its polygons have (2+135=137) vertices (see here).

206 = 80 + 126

This is the embodiment by sacred geometry of the 80 bones of the human axial skeleton and the 126 bones of the human appendicular skeleton.

Outer Tree of Life

The 1-tree whose 19 triangles are Type A triangles has 80 yods in its “trunk” (see here for its definition). 126 yods are added to its branches when these triangles become Type A triangles (see Article 33).

Inner Tree of Life

There are 206 yods associated with the last four enfolded polygons, whose 35 sectors have 138 vertices, edges & triangles (see here). Associated with the hexagon and octagon are 80 yods symbolizing the 80 bones of the human axial skeleton. Outside the shared edge of the decagon and dodecagon are 126 yods that symbolize the 126 bones of the human appendicular skeleton.

Disdyakis triacontahedron

In the case of its seven layers of vertices perpendicular to an axis passing through two opposite A vertices, layers 1 and 3 consist of 6- & 8-sided polygons, and layers 2 and 4 consist of 10- & 12-sided polygons. Their enfolded counterparts in the inner Tree of Life contain, respectively, 80 and 126 yods.

240 = 72 + 168

This expresses the fact that 72 of the 240 roots of the superstring symmetry group E₈ are roots of its exceptional subgroup E₆, leaving 168 roots. See here for how five sacred geometries embody this division.

1-tree

Constructed from Type A triangles, the 1-tree contains 240 yods other than SLs (see here). Noting that the Tree of Life with 22 Paths and 16 triangles is converted to the 1-tree with 25 Paths and 19 triangles by adding one corner and three sides of three triangles, the composition of the 240 yods is:

	Hexagonal yods on Paths	Hexagonal yods at centres of tetractyses	Vertices	Hexagonal yods on internal edges
Tree of Life:	22×2 = 44	16×3 = 48	16	16×3×2 = 96
Addition:	3×2 = 6	3×3 = 9	3	3×3×2 = 18
Total:	50	57	19	114

126 yods comprise 50 hexagonal yods on Paths and (57+19=76) yods that are either corners or centres of tetractyses:

126 =	44 + 6 + 48 + 9 + 16 + 3
=	6 + 48 + (44+9+16+3)
=	6 + 48 + 72 .

The 126 yods include 72 yods that are either the 44 hexagonal yods on the Paths of the Tree of Life, the 19 corners of tetractyses inside the triangles of the 1-tree and the nine centres of the three triangles added in the conversion of the Tree of Life to the 1-tree. The 126 yods symbolize the 126 roots of E₇, the rank-7 exceptional subgroup of E₈, and the 72 yods symbolize the 72 roots of E₆. The remaining 114 hexagonal yods on internal sides symbolize the 114 roots of E₈ that are not roots of E₇. The trunk of the Tree of Life has five Type A triangles on whose 15 internal sides of their tetractyses there are 30 hexagonal yods. 84 hexagonal yods lie on the 42 sides of the other 14 triangles making up the branches of the 1-tree, i.e., 114 = 30 + 84 and

168 = 6 + 48 + 114 = 6 + 48 + 30 + 84 = (48+30) + (6+84) = 78 + 90.

There are 78 hexagonal yods that are either centres of tetractyses in the Tree of Life or on internal sides of tetractyses in its trunk. There are 90 yods that are either hexagonal yods on Paths added by the conversion or hexagonal yods on internal sides of tetractyses in the branches of the 1-tree. These two numbers are the number values of, respectively, the words Cholem and Yesodoth in the Kabbalistic name of the Mundane Chakra of Malkuth:

Number value of Cholem Yesodeth

Notice also that

168 = (6+48+30) + 84 = 84 + 84.

This 84:84 division of the number 168 is characteristic of holistic systems, as we have seen previously. See also Article 64.

Inner Tree of Life

The 240 hexagonal yods of the seven separate polygons consist of 72 hexagonal yods lining the sides of the first six polygons and 168 other hexagonal yods. Of the latter, 78 belong to the hexagon and to the dodecagon and 90 belong to the other polygons. Once again, we see that the number values of Cholem and Yesodoth are reproduced in the inner Tree of Life as well as in the 1-tree. As the first six polygons are a holistic system, this 240:72 division signals a symmetry-breakdown from E₈ with 240 roots to E₆ with 72 roots.

Dodecagon

Surrounding the centres of two separate dodecagons with Type A sectors are 240 geometrical elements. They comprise 72 triangles and 168 corners & sides (84 in each dodecagon).

Sri Yantra

Surrounding the central bindu point of the two-dimensional Sri Yantra are 240 geometrical elements (see here). They comprise the 126 sides of the 42 triangles surrounding the central one, i.e., 168 sides & triangles (84 in each half of the Sri Yantra), as well as 69 corners and the three sides of the central triangle, i.e., 72 corners & sides.

When they are Type B triangles, the 42 triangles have 1680 yods other than their 168 internal corners and the 26 tips of the nine primary triangles that belong to them (see here). Noting that the nine tetractyses of a Type B triangle have four corners inside it, the 240 corners in the (1+42) triangles that surround the bindu are made up of the (4×42=168) corners of tetractyses making up the 42 triangles, the 68 corners of the latter and the four corners of the nine tetractyses inside the central triangle that are unshared with surrounding triangles. Therefore, the 240 corners display a 72:168 division.

Disdyakis triacontahedron

2400 (=240×10) geometrical elements surround its axis when its faces and internal triangles are Type A triangles (see here). 1680 (=168×10) elements surround it when only internal triangles are Type A. 720 (=72×10) elements are added by the further division of its faces into their sectors.

Surrounding an axis passing through two opposite A vertices are 180 edges (12 edges of its central polygon, 168 other edges). Its 180 internal triangles have 60 internal edges surrounding the axis. Its (120+180=300) triangles have (12+60+168=72+168=240) edges surrounding the axis. 72 edges are either internal or line the middle of the polyhedron. 168 edges line its 120 faces, 84 above the middle, 84 below it.

Divided into their sectors, the seven polygons formed by the 60 vertices of the disdyakis triacontahedron that surround an axis passing through two opposite A vertices are composed of 240 geometrical elements surrounding their centres (see here). They comprise 72 elements in the two topmost polygons with 18 vertices and 168 elements in the five other polygons with 42 vertices. There are 240 geometrical elements in the faces and internal polygons in each half of the polyhedron. Of these, 168 elements are in these faces and 72 elements are in the polygons.

In the case of the 15 polygons perpendicular to an axis through two opposite C vertices that are formed by vertices of the disdyakis triacontahedron, there are 72 corners, sides & triangles in the uppermost and lowest three polygons with 18 vertices and 168 vertices, edges & triangles in the remainder of the polyhedron with 42 vertices (see here).

496 = 248 + 248

Outer Tree of Life

It was shown here that, when ten overlapping Trees of Life are interpreted as representing the ten dimensions of the space-time of superstrings, there are 496 yods up to (but not including) Chesed of the tenth Tree when their triangles are turned into tetractyses. There are 248 yods up to Chesed of the fifth Tree, each yod symbolizing one of the 248 gauge fields of E8. For E8×E8 heterotic superstrings, the division of the 496 gauge fields of the superstring symmetry group E8×E8 into two sets of 248 fields is the manifestation of the 5:5 division of Sephiroth in the Tree of Life. It results in the existence of an as yet little-understood shadow matter universe co-existing with and parallel to the familiar universe of ordinary matter. The superstrings in each universe have unified forces described by the same symmetry group E8. In terms of the four overlapping Trees of Life that represent in Kabbalah the four Worlds of Atziluth, Beriah, Yetzirah & Assiyah in which Adam Kadmon, or “Heavenly Man” exists, the Upper Face of the Assiyatic Tree is the subtle energy body comprising the matter of the etheric body, prana/chi, acupuncture meridians, etc. Called in Kabbalah the “Zelim”, or image, and in Vedantic philosophy the “pranamaya kosha,” it exists not in exactly the same universe as ordinary matter (in particular, the physical body, or “annamaya kosha”) but in the shadow matter universe, a narrow gap that extends along the seventh of the curled-up dimensions predicted by M-theory separating the two 10-dimensional space-time sheets that contain the two universes, one visible, the other invisible.

Inner Tree of Life

The 248 roots of E8 are symbolized in the seven separate polygons of the inner Tree of Life by the endpoint of the root edge, the centres of the seven polygons and their 240 hexagonal yods (see here). The mirror symmetry of the two sets of polygons is responsible for the direct product E8×E8 of two similar Lie groups E8.

When the 70 yods of the outer Tree of Life are projected onto the plane containing the two sets of seven enfolded polygons of the inner Tree of Life, there are 248 yods in each set that are intrinsic to them in the sense that none of them coincide with projections onto this plane of yods on Paths. As evidence that this is neither coincidental nor the result of an arbitrary choice made merely in order to generate the superstring number 248, there are 78 yods associated with the first four polygons, where 78 is the dimension of E₆, a subgroup of E₈. The 114 yods in the octagon and dodecagon symbolize the 114 roots of E₈ that do not belong to E₇. The 56 yods in the decagon outside the root edge symbolize the two simple roots of E₈ not belonging to E₆ and the 54 roots of E₇ not belonging to E₆.

Sri Yantra

According to Table 8 in Article 35, when its 43 triangles are transformed into Type A triangles, the two-dimensional Sri Yantra contains 757 yods. 756 yods surround the bindu point. Of these, 16 green yods belong solely to the central triangle (its two upper corners are also corners of the innermost set of eight triangles, so that green yods are not assigned to them). According to this table, 504 hexagonal yods line the 252 sides of the 126 tetractyses in the 42 triangles surrounding the central one. The tips of some of these triangles touch twelve of their sides. From a strict, mathematical point of view, these particular sides are not single, straight lines but pairs of lines joined at the tip of another triangle. This leaves (252–12=240) true sides (i.e., single, straight lines) with 480 hexagonal yods. These yods cannot be divided into two sets of 240 that are mirror images of one another, if the mirror is horizontal, because three of the four sides excluded from the layer of ten blue triangles in Figure 1 of Article 35 (or see here) belong solely to its upper half, creating an imbalance. However, a mirror symmetry persists after excluding the 12 sides if the mirror is vertical, for, then, 240 red hexagonal yods and 240 blue hexagonal yods lie either on or, respectively, to the right and left of the vertical axis passing through the centre of the central triangle, which is surrounded by (240+240+16=496) yods on true sides:

The 24 hexagonal yods lining sides that are touched by other true triangles are coloured black. The pair of hexagonal yods in the side of the triangle touched by the lowest corner of the central triangle are not black because, although it has the appearance of a triangle, the central triangle is, mathematically speaking not a true, triangular area, the bindu (a separate point) coinciding with its centre. In the 3-dimensional Sri Yantra, the central triangle really is a triangle because the bindu hovers above it instead of being embedded in its surface. If it were traversed by various lines, it would be obvious that this “triangle” could not be regarded as a pure triangle. To the mathematical novice, it is not so obvious that it loses this status when merely the bindu is embedded in its surface. But the principle remains the same even if the “triangle” is divided into its sectors. The 16 green yods in the central triangular perimeter that surround the bindu are intrinsic to this triangle because they are unshared with any of the 42 triangles around it. They symbolize the 16 simple roots of E₈×E₈. The (240+240=480) hexagonal yods in the 240 single lines making up the true sides of the 42 triangles denote its (240+240=480) roots. Without this axial symmetry, the 496 yods could not be divided into two similar sets of 248 yods. Instead, the Sri Yantra would embody the 496 roots of SO(32), the second Lie group with the dimension 496 that governs Type 1 open and closed superstrings and one of the two types of closed heterotic superstrings. It is as if the two possible gauge symmetry groups governing superstring forces allowed by superstring theory correspond to the two orthogonal orientations of the Sri Yantra, its traditional one (E₈×E₈) displaying a left/right mirror symmetry and the other one rotated by 90° (SO(32)) not doing so.

The 72:168 division characteristic of holistic systems manifests in the 2-d Sri Yantra composed of Type A triangles as the (6×12=72) hexagonal yods in the six red triangles on either side of its central axis and as the 168 hexagonal yods in the remaining 19 triangles.

As supporting evidence that this is the correct way of seeing how the number 496 at the heart of superstring theory is embodied in the two-dimensional Sri Yantra, it was pointed out here that this sacred geometrical object is composed of 240 geometrical elements surrounding its centre when all triangles are not divided into their sectors. In other words, including this centre, there are (241–12=229) true, geometrical elements when the 12 sides touched by triangles are left out of consideration. 229 is the 50th prime number, showing how ELOHIM with number value 50 prescribes the true, geometrical composition of the two-dimensional Sri Yantra. There are 70 points and 117 true sides, i.e., 187 true points & lines, where 187 is the number value of Auphanim, the Order of Angels assigned to Chokmah. There are (70+42=112) points & triangles, where 112 is the number value of Beni Elohim, the Order of Angels assigned to Hod.

The Platonic solids

When the vertices and centres of faces of the five Platonic solids are connected to their centres, there are on average 496 geometrical elements other than vertices that surround their axes passing through two vertices and their centres. 248 elements other than vertices on average surround these axes in each half of the five Platonic solids (see here). They are the regular polyhedral counterpart of the 248 roots of the rank-8 Lie group E8. This remarkable analogy allows us to say that the direct product nature of the symmetry group E8×E8 governing the unified interactions between one of the two types of heterotic superstrings arises from the fact that every geometrical element surrounding these axes that make up the five Platonic solids has a mirror-image counterpart.

Disdyakis triacontahedron

The geometrical composition of the polyhedron and the seven polygons formed by the 60 vertices surrounding an axis that passes through two diametrically opposite A vertices is analyzed here. We found that 248 vertices, sides & triangles surround the axis in each half of the polyhedron, i.e., a total of 496 geometrical elements surround this axis. The 248:248 division of this number — the polyhedral counterpart of the (248+248) roots of E8×E8 — is the result of the mirror symmetry of the disdyakis triacontahedron whereby every geometrical element or yod generated by dividing its triangles into their sectors or by turning them into tetractyses has a mirror-image counterpart.

References

Proof: The 15 polygons have 60 corners surrounding the axis passing through two opposite C vertices. The central polygon has six corners, so that the seven polygons above or below it have 27 corners. A Type B n-gon has (15n+1) yods when its n sectors are Type A triangles, so that, given just its n corners (vertices of the disdyakis triacontahedron), (14n+1) extra yods are needed to transform it into these sectors. Each set of seven polygons requires (14×27 + 7 = 385) more yods to turn them into Type B polygons.
Proof: the n-tree is composed of (12n+7) triangles with (6n+5) corners and (16n+9) sides, i.e., (22n+14) corners & sides. The 7-tree has (12×7 + 7 = 91) triangles with (22×7 + 14 = 168) corners & sides.
Proof: n overlapping Trees of Life have (50n+20) yods. 67 overlapping Trees have 3370 yods. Their uppermost triangle with Kether, Chokmah & Binah at its corners has ten yods when it is a tetractys. Below Binah of the 67th Tree are (3370−10=3360) yods.
Proof: the n-tree is composed of (12n+7) triangles with (6n+5) corners and (16n+9) sides, i.e., (18n+12) corners & triangles. The 7-tree has (18×7 + 12 = 138) corners & triangles. Below its apex are 137 corners & triangles. Compare this with the number 1 at the apex of the Tetrahedral Lambda and the numbers on its inclined edges that add up to 137 (see Fig. 3 here). The apex of the 7-tree is the 168th Sephirothic level (SL) on the central Pillar of Equilibrium from the top of CTOL.* This connects the superstring structural parameter 168 to the mysterious number 137 that determines the fine-structure constant α = e2/ħc well-known to physicists (see previous discussion of this parameter of holistic systems). Below the 168th SL on the central pillar from the top of CTOL are the 137 corners & triangles of the 7-tree. Its 91 triangles have (6×7 + 5 = 47) corners and (16×7 + 9 = 121) sides, i.e., 168 corners & sides, further demonstrating how the gematria number 168 of Cholem Yesodoth, the Mundane Chakra of Malkuth, not only marks out the commencement in CTOL of the 7-tree mapping the physical plane/universe (the plane formally corresponding to Malkuth) but also measures the number of points & lines needed to construct this map! Here is an amazing conjunction of two fundamental parameters of particle physics (one yet to be discovered) that is too implausible to be attributed to chance. Indeed, what is the likelihood that it could be mere coincidence that there are 168 corners & sides of 91 triangles below the 168th SL on the central pillar of CTOL from the top of its 91 Trees of Life?