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 2ndorder 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 nexthigher version — the 2ndorder 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 2ndorder tetractys denotes Malkuth of the central tetractys, which itself corresponds to this Sephirah. It is surrounded by 84 yods. The 2ndorder 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 1storder tetractyses that are not at the corners of the 2ndorder 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 2ndorder 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 1storder tetractyses. ELOHA (אלה), the Godname of Geburah with number value 36, prescribes both the 36 yods lining the sides of the 2ndorder 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:
The Tetrad determines the number of yods surrounding the centre of a 2ndorder tetractys. These yods include 15 corners of 1storder 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 higherorder tetractys above the 1storder 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 4dimensional “tetrahedral numbers”: 1, 5, 15, 35, 70, etc. The fourth, nontrivial example of these numbers is 70. This is the number of hexagonal yods in the 2ndorder 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 1storder tetractyses in the 2ndorder tetractys is
15 = 2⁰ + 2¹ + 2² + 2³ = 1 + 2 + 4 + 8
whilst the number of yods in the 2ndorder 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 2ndorder tetractys. In mathematics, there are only four orders of normed division algebras**: the 1dimensional scalar numbers, the 2dimensional complex numbers, the 4dimensional quaternions and the 8dimensional 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 1storder tetractyses in the 2ndorder 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 2ndorder tetractys in terms of a basic, triangular array of 15 points, namely, the corners of 10 1storder tetractyses. Its third raised edge generates the complete “body” of the 2ndorder 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 1tree when its 19 triangles are Type A (see here). It is embodied in the UPA as the number of spacetime coordinates of points on the 10 whorls as 10 closed curves in 26dimensional spacetime: 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 2ndorder tetractys.
The correspondences between the 2ndorder tetractys, the 1tree 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 (“thricegreat 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 1storder tetractys symbolizing Malkuth (the “below”) at the centre of the 2ndorder tetractys by a 2ndorder 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 2ndorder 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 2ndorder 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 sevenfold 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 spacetime 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 (spacetime)  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 psychospiritual 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).
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 ladderlike structure is the basis of what is referred in the Old Testament as “Jacob’s ladder” (see Book of Genesis 28: 1019).
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 16 (see diagram). As
n² = 1 + 3 + 5 + … + 2n1
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:
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 sacredgeometrical 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 Ngons nested inside one another with n dots equally spaced along each side of the outermost Ngon:
Pⁿ₁ = 1. Writing P³ₙ ≡ Tₙ, P⁴ₙ ≡ Sₙ, P⁵ₙ ≡ Pₙ, P⁶ₙ ≡ Hₙ, P⁸ₙ ≡ Oₙ, P¹⁰ₙ ≡ Dₙ & P¹²ₙ ≡ dₙ, the sum of the first seven nontrivial 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 nsided 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 nsided 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 ntree, 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 49tree. 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 26tree 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 ntree, namely, the first 13 polygons, which are the counterpart of the 95tree. However, this is of no interest, because only a number of overlapping Trees of Life has significance, an ntree 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 49tree 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 86tree. 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:
dodecagontriangletriangledodecagon
dodecagontriangledodecagontriangle
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 nonmaterial 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 49tree mapping the cosmic physical plane. The remaining five polygons have 251 yods symbolising the 251 SLs in CTOL above the 49tree.
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 1tree 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 1tree, 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 MalkuthYesodHodNetzachTiphareth representing the human soul connects at Tiphareth to the Upper Face of TipharethBinahChokmahKether — 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 1tree. 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, centretocircumference, 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:
 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:
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 ChesedGeburah 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:
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 26dimensional spacetime 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 spacetime continuum to nontemporal 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 rearranging 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 ngon whose sectors are 2ndorder 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 2ndorder 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 2ndorder 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 10tree
ADONAI (Hebrew: אדני, ADNI), the Godname of Malkuth with number value 65 , prescribes the 10tree 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 10tree (see diagram). The 10tree 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 10tree contains 127 triangles, where 127 is the 31st prime number. This shows how EL, the Godname of Chesed with number value 31, prescribes the 10tree. 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 10tree are two numbers, 192 and 550, that are characteristic parameters of holistic systems (see The holistic pattern). The 10tree 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 11dimensional spacetime. It is an example of a sacredgeometrical 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 centuryold 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 10tree 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 10tree 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 2ndorder tetractyses as their sectors.
A 2ndorder 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 ngon when it is a 2ndorder tetractys. The number of yods in an ngon constructed from 2ndorder tetractyses ≡ N(n) = 72n + 1, where “1” denotes its centre. The number of corners of 1storder 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) 1storder 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, 4dimensional, tetrahedral number* after 1, whilst 35 (number of corners of 1storder tetractyses in each square outside the root edge) is the fourth tetrahedral number after 1. It determines their 80 2ndorder 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 1storder 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 10tree 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 10tree have 550 sides (see #12);
 the 70 separate Type A polygons making up the inner form of the 10tree have 480 sectors with 550 corners.
The complete inner form of the 49tree 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 49tree. 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 49tree mapping the cosmic physical plane). The sum of the gematria number values of Daath and ADONAI aptly measures the inner geometry of the 49tree as the “cosmic physical” expression of the seven Sephiroth of Construction.
The number 539 is the number of SLs in CTOL above the 1tree, 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 shapedetermining, 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  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 10100) 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 7tree, there are 252 SLs up to the top of the 49tree. 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 ngon = 15n + 1 (see Section 2 in Power of the polygons/General view). 30n yods surround the centres of two separate, Type B ngons. 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 7tree 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 7tree in the 49tree mapping the cosmic physical plane and to the 42 coloured hexagonal yods in the six smaller 1storder tetractys in the Cosmic Tetractys; the other set of 42 corners corresponds to the 42 Trees above the 49tree and to the 42 coloured hexagonal yods in the six larger 1storder tetractyses in the Cosmic Tetractys. Remarkably, we see that the tetrahedron corresponds to the 7tree 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 3dimensional, 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:
Each half of the dodecahedron has:
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 7tree 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:
 The 248 hexagonal yods correspond to the 248 SLs above the 7tree 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 rank8, 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 7tree 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.
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 7tree mapping the physical plane;
 84 dark red SLs on the Pillar of Judgement up to the top of the 49tree;
 84 dark green SLs on the Pillar of Equilibrium up to (but not including) the top of the 49tree;
 84 dark blue SLs on the Pillar of Mercy up to the top of the 49tree;
 84 light red SLs on the Pillar of Judgement above the 49tree up to the top of CTOL;
 84 light green SLs on the Pillar of Equilibrium from the top of the 49tree 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 7tree 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 spacetime 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 7tree, 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 49tree 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 & facecentres 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 2ndorder 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 600cell, 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 600cells and in sacred geometries” here).
The Sri Yantra
Correspondence between the Cosmic Tree of Life and the 3dimensional 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 7tree that maps the physical plane (26dimensional spacetime). 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 49tree 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 7tree. 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 7tree, or physical plane. Fortysix more black SLs in the 7tree 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 3dimensional 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 onetoone 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 7tree 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 2dimensional 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 7tree. 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 3dimensional 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 7tree on the lefthand Pillar of Judgement of the 91 Trees of Life representing CTOL. There are 168 blue SLs above the 7tree on the righthand Pillar of Mercy. 168 green SLs span the central Pillar of Equilibrium down to the top of the 7tree. 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 7tree.
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 7tree mapping the physical plane (spacetime). Hence, the Type C dodecagon is equivalent to the 3dimensional Sri Yantra. Its centre corresponds to the central Type B triangle, whose 46 yods denote the 46 SLs below the top of the 7tree. 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 7tree.
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 2dimensional 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 spacetime continuum) is mapped in CTOL by its lowest seven Trees of Life and in the Sri Yantra by its central triangle. As spacetime, it is the Malkuth level of the 49tree 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 YangMills gauge symmetry group governing the interactions of 10dimensional 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 4dimensional spacetime 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 micropsi 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 diehard 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 sevenfold 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 49tree 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 49tree that map the 42 subplanes of the six cosmic superphysical planes. This is the single, polygonal representation of CTOL.
Heptagon with 2ndorder tetractyses as sectors
When its sectors become 2ndorder tetractyses, the 504 yods surrounding the centre of the heptagon denote the 504 SLs down to the top of the 7tree, 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 49tree and the top of the 7tree, for they belong to the Tree of Life map of the cosmic physical plane. The 7tree is the ‘Malkuth’ level of the 49tree, which in turn is the ‘Malkuth’ level of CTOL — the most general sense of this word. The top of the 7tree 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 micropsi examination of the UPA actually marks out the physical universe (7tree) 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, spacetime.
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 3dimensional 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:
The number of yods in the Type C ngon = 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 1tree 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 7tree 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 micropsi observations to the dimension 248 of E8, the rank8, 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 remoteviewed 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 49gon 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 anomalyfree forces between heterotic superstrings). See also here.