How some sacred geometries are equivalent maps of all levels of reality

Last fm Leds Stephen M. Phillips (334 Article 26: How the seven musical scales relate to the disdyakis triacontahedron 42 p [939 Article 27: How the disdyakis triacontahedron embodies the structural parameter 1680 of the E8xE8 heterotic superstring 28 p L 83 q Article 28: Encoding ofthe roots of the superstring gauge symmetry group E8 in the inner Tree of Life and the disdyakis triacontahedron 22p. Article 29: The triakis tetrahedron and the disdyakis triacontahedron embody the fine- L 3 Lj g Structure constant and the structural parameter of the heterotic superstring ANN 15p Article 30: The equivalence of the triakis tetrahedron, disdyakis triacontahedron and Plato's ‘Lambda tetractys' 15 p Articlé 31 : The musical nature of the polyhedral Tree of Life Article 32: Derivation of the bone and classical acupuncture compositions of the human body and their relationship to the seven musical scales houd Lau Article 33: The human axial skeleton is the trunk of the Tree of Life 16 p Article 34: The seven layers of vertices in the disdyakis triacontahedron encode the 206 4 5 bones of the human skeleton, the superstring symmetry groups E8 and E8xE8 and the L Ô he boul L 34 superstring structural parameters 168, 336, 840 & 1680 Article 35: The Tree of Life nature of the Sri Yantra and some of its scientific meanings Arucle 36: The Sri Yanra-like pattern of the 15 layers of vertices in the disdyakis triacontahedron and its scientific meaning Avice 37: The seven octaves of the seven musical scales are a Tree of life pattern LS 4 3 mirrored in the disdyakis triacontahedron 36 p Article 38: The geometrization of the seven musical scales and its mathematical b34 a implications 15 p Article 39: The correspondence between the inner Tree of Life, the Sri Yantra & the I diagram and their realization in the seven musical scales 6350 Ching 17 p i Article 40 (Part 1): The unification of all sacred geometries and its implication for ‘ particle physics 40 p Article 40 (Part 2): The unification of all sacred geometries and its implication for particle physics 45 p ! Article 40 (Part 3): The unification of all sacred geometries and its implication for : particle physics 49 p | Article 40 (Part 4): The unification of all sacred geometries and its implication for i particle physics

41: The pyramidal form of the inner Tree of Life, its counterparts and its 18%4, Article encoding of the human skeleton 18 p Article 42: Comparison of the eight Church musical modes and the human skeleton as Lasi holistic systems 22 p ner Article 43: The Tree of Life nature of the {3,7} tessellation of the 168 automorphisms of the Klein quartic on the 3-torus 31 p Article 44: The polyhedral CTOL and its embedding of the 496 roots of the heterotic superstring gauge symmetry group 195° E8xE8 16 p Lost ey 354 Loke Article 45: The 1680 circularly polarized oscillations in the E8xE8 heterotic superstring as 1680 harmonics of the Pythagorean musical scale 12 p Article 46: How sacred geometries encode the 64 codons of mRNA and the 64 anticodons of tRNA 38 p Article 47 (Part 1): How sacred geometries embody structural/dynamical parameters of the E8xE8' heterotic superstring and the codon pattern of DNA Article 47 (Part 2): How sacred geometries embody structural/dynamical parameters of the E8xE8' heterotic superstring and the codon pattern of DNA 35p+31p Article 48: The holistic nature of the first (4+4) regular polygons of the inner Tree of Life 18 p Article 49: How some sacred geometries are equivalent maps of all levels of reality 32p Article 50 (Part 1): The Golden Ratio, Fibonacci & Lucas numbers in sacred geometries Article 50 (Part 2): The Golden Ratio, Fibonacci & Lucas numbers in sacred geometries 64p+66p Article 51: The connection between Fibonacci numbers and the Pythagorean musical scale 12p Article 52: How EHYEH, YAH & YAHWEH prescribe the root structure & dimension of E8 A breakthrough in relating sacred geometries to the superstring constituents of quarks 21p Article 53: The 10-fold division within five sacred geometries & its manifestation in the ten whorls of the UPA, the subquark state of the E8xE8 heterotic superstring Mathematical meaningsofthe Part 1 (PDF) 26p Part 2 (PDF) 53 p Names of God

ARTICLE 49 Hoow wS Soom mee S Saaccrreedd G Geeoom meettrrieess aarree E Eqquuiivvaalleenntt Maappss ooff A Allll LLeevveellss oof R Reeaaliityy "As above, so below." by Stephen M. Phillips Flat 3, 32 Surrey Road South. Bournemouth. Dorset BH4 9BP. England. E-mail: Stephen@smphillips.8m.com Website: http://smphillips.8m.com ABSTRACT The triangular array of ten points, or tetractys, at the heart of Pythagoras’ philosophy can be regarded as but the first member of an infinite sequence of higher order tetractyses. The mathematical realisation of the Hermetic axiom “as above, so below” is the Cosmic Tetractys in which the centre of a 2nd-order tetractys is another 2nd-order tetractys. Its 91 1st-order tetractyses represent the 91 subplanes of the seven cosmic planes of consciousness expounded by Theosophy. As the ten points of the tetractys denote the ten Sephiroth of the Tree of Life, the Cosmic Tetractys is equivalent to 91 overlapping Trees of Life (CTOL) with 550 Sephirothic emanations. As well as its outer form known to Kabbalists, the Tree of Life has an inner form consisting of two enfolded, similar sets of seven regular polygons that share one ‘root edge.’ Constructed from tetractyses, a unique set of twelve of these separate polygons, together with this root edge, have 91 corners and 550 yods. The Inner Tree of Life encodes the replication of its outer form to encompass all levels of reality. One set of seven polygons represents the cosmic physical plane and the five others represent the six cosmic superphysical planes. As independent evidence supporting this interpretation, the gematria number value of the Hebrew name of the Mundane Chakra of Malkuth (the physical level of the Tree of Life) determines the geometrical composition of the 49 Trees of Life that map the cosmic physical plane, whilst this map embodies the number value of Malkuth when constructed from tetractyses. The Sri Yantra is shown to be the counterpart of CTOL because the numbers 91 and 550 are embodied in its geometry. They are also embodied in the five Platonic solids because 550 points, lines & triangles are needed to construct their 50 faces, whilst the 90 interior triangles formed by their vertices and shared centre have 270 sectors with 91 corners. Constructed from tetractyses, the faces of the dodecahedron contain 550 points other than corners of tetractyses. The faces of the first four Platonic solids, which the ancient Greeks believed were the shapes of the elements Earth, Water, Air & Fire, are constructed from the 248 corners & sides of 120 triangles, showing how they embody the dimension 248 of the superstring gauge symmetry group E8. Previous work established that the outer form of the polyhedral version of the Tree of Life is the disdyakis triacontahedron. Its 60 vertices surrounding an axis passing through two opposite vertices of a certain type are corners of 15 polygons. Including these two vertices, they possess 550 vertices, sides & triangles when their sectors are divided into three triangles. The eight polygons in one half of the polyhedron are composed of 299 geometrical elements. This is the number of Sephirothic emanations in the 49 Trees of Life mapping the cosmic physical plane. The 251 elements in the remainder of the polyhedron correspond to the 251 emanations in the 42 Trees of Life that map the six cosmic superphysical planes. The interior geometry of the disdyakis triacontahedron therefore represents CTOL. This is confirmed by the fact that the number of yods in the 15 polygons surrounding the centre of the polyhedron is the number of yods on the central pillar of CTOL between its apex and nadir when its triangles are tetractyses.

Table 1. Gematria number values of the ten Sephiroth in the four Worlds. SEPHIRAH Kether (Crown) GODNAME 2 Chokmah (Wisdom) 21 YAHVEH, YAH (The Lord) 73 3 Metatron (Angel of the Presence) EHYEH (I am) 620 Binah (Understanding) ARCHANGEL 314 Raziel (Herald of the Deity) 26, 15 248 ELOHIM (God in multiplicity) Tzaphkiel (Contemplation of God) 67 50 311 ORDER OF ANGELS MUNDANE CHAKRA Chaioth ha Qadesh (Holy Living Creatures) Rashith ha Gilgalim First Swirlings. (Primum Mobile) 833 Auphanim (Wheels) 187 Aralim (Thrones) 282 636 Masloth (The Sphere of the Zodiac) 140 Shabathai Rest. (Saturn) 317 Daath (Knowledge) 474 4 Chesed (Mercy) 72 5 Geburah (Severity) 216 6 Tiphareth (Beauty) 1081 7 Netzach (Victory) 148 8 9 Hod (Glory) 15 Yesod (Foundation) 80 10 Malkuth (Kingdom) 496 Tzadkiel (Benevolence of God) EL (God) 31 ELOHA (The Almighty) 36 YAHVEH ELOHIM (God the Creator) 76 YAHVEH SABAOTH (Lord of Hosts) 129 ELOHIM SABAOTH (God of Hosts) 153 62 Samael (Severity of God) 131 Michael (Like unto God) 101 Haniel (Grace of God) 97 Seraphim (Fiery Serpents) 630 Malachim (Kings) 140 Tarshishim or Elohim 1260 Beni Elohim (Sons of God) 311 112 Gabriel (Strong Man of God) 49, 363 246 65, 155 428 Raphael (Divine Physician) SHADDAI EL CHAI (Almighty Living God) ADONAI MELEKH (The Lord and King) Chasmalim (Shining Ones) Sandalphon (Manifest Messiah) 280 Cherubim (The Strong) Tzadekh Righteousness. (Jupiter) 194 Madim Vehement Strength. (Mars) 95 Shemesh The Solar Light. (Sun) 640 Nogah Glittering Splendour. (Venus) 64 Kokab The Stellar Light. (Mercury) 48 Levanah The Lunar Flame. (Moon) 272 Ashim (Souls of Fire) 351 87 Cholem Yesodeth The Breaker of the Foundations. The Elements. (Earth) 168 The Sephiroth exist in the four Worlds of Atziluth, Beriah, Yetzirah and Assiyah. Corresponding to them are the Godnames, Archangels, Order of Angels and Mundane Chakras (their physical manifestation, traditionally symbolised by celestial bodies). This table gives their number values obtained by the ancient practice of gematria, wherein a number is assigned to each letter of the alphabet, thereby giving to a word a number value that is the sum of the numbers of its letters. (All numbers in this table referred to in the article are written in boldface).

1. The Cosmic Tetractys At the heart of the ancient Pythagorean teachings was the triangular array of ten points known as the tetractys (Fig. 1). Figure 1. The tetractys. The modern academic view is that it represented what the Pythagoreans regarded as the perfect number 10 (Decad) as the fourth, so-called ‘polygonal number,’ the four rows of one, two, three & four dots symbolising, respectively, a mathematical point, the two points at the ends of a straight line, the corners of a triangle and the four vertices of a tetrahedron. However, it was far more than that, as the oath made by members of Pythagoras’ mystery school testifies: “I swear by the discoverer of the tetractys, Which is the spring of all our wisdom, The perennial fount and root of Nature.” The seminal importance of the tetractys to Pythagoreans was due to this figure being not merely symbolic, as most mathematicians have wrongly believed, but the actual template of sacred geometry that transforms the latter into numbers of fundamental significance to the mathematical design of the universe and therefore having scientific meaning. The notion of design, with its implication of a Designer, is anathema, of course, to many scientists who hold to the materialistic paradigm that the universe sprung Figure 2 . The one-to-one correspondence between the ten Sephiroth of the Tree of Life and the ten points (yods) of the tetractys. into existence without the assistance of God. Hence, they would reject the claim of the Pythagoreans that the tetractys has a profound, metaphysical meaning that reveals the mysteries of the universe. The author’s previous 48 research articles have demonstrated beyond doubt that this view is wrong. The tetractys is a representation of the Kabbalistic Tree of Life, each of its ten points (or ‘yods,’ from the Figure 3. The seven coloured triangles in the 1st-order tetractys differentiate into the 49 coloured triangles of the 2nd-order tetractys. 1st-order tetractys 2nd-order tetractys Hebrew name ‘yod’ of the tenth letter of the Hebrew alphabet) corresponding to one of the ten Sephiroth (Fig. 2). The yods either at the corners of a hexagon or at its centre will be called ‘hexagonal yods.’ They symbolize the seven Sephiroth of Construction. The yods at the corners of the tetractys denote the Supernal Triad of Kether, Chokmah and Binah and the central hexagonal yod denotes Malkuth. Suppose that we replace each point of a tetractys by a triangle. The motive behind this is to express the idea that each point of the tetractys (‘1st-order tetractys’) is a potential tetractys in itself and a member of a higher-order tetractys, or ‘2nd-order tetractys’ (Fig. 3). Each triangle is then turned into a copy of the whole of which it was a constituent. All the triangles in the 1st-order triangles at the corners of the 2ndorder tetractys shown in Fig. 3 have been left white in order to conform to the white triangles in the 2 original 1st-order tetractys. The seven coloured triangles in the 1st-order tetractys become (7 =49) coloured triangles in the 2nd-order tetractys. There are 17 points along each side of the 2nd-order tetractys that are corners of 16 triangles. The number of corners of the triangles on its boundary is 48.

This is the number value of Kokab, the Mundane Chakra of Hod. The number 48 is always a basic measure of the global nature of a holistic system, as has been demonstrated in many previous articles and as will be illustrated in subsequent sections. A triangular array of 153 points (Fig. 4), 17 points along each side, comprises 300 sides of 100 triangles that form a 2nd-order tetractys. Each of its ten 1st-order tetractyses is made up of the 15 corners & 30 sides of ten triangles, that is, 55 geometrical elements, where 1 2 3 55 = 4 5 6 7 8 9 10. This demonstrates how the Decad determines the geometrical composition of a 1st-order tetractys whose yods are triangles. As 150 = 15×10, YAH, the Godname of Chokmah with number value 15 and ELOHIM SABAOTH, the Godname of Hod with number value 153, prescribe the geometrical array of points Figure 4. ELOHIM SABAOTH, the Godname of Hod with number value 153, prescribes the triangular array of 153 points underlying the 10 1st-order tetractyses of the 2nd-order tetractys. underlying the 2nd-order tetractys, the ten 1st-order tetractyses each having 15 corners. The ten separate 1st-order tetractyses have (10×55=550) geometrical elements that become 535 geometrical elements in the 2nd-order tetractys when their corners coincide, 15 corners disappearing in the process. Including the 18 points in the six triangular gaps between 1st-order tetractyses, the triangular array consists of (18+535=553) geometrical elements. Starting with three points as the three corners of a triangle, 550 more elements (150 corners, 300 sides & 100 triangles) are needed to create the triangular array, where 55 55 55 550 = 55 55 55 55 55 55 55 = 10 20 30 40 50 60 70 80 90 100. This demonstrates how the Decad determines the geometrical composition of a 2nd-order tetractys as well as a 1st-order tetractys. The number 550 will manifest in various sacred geometries discussed later. Let us next take this transformation a step further but with the difference that only the central 1st-order tetractys in the 2nd-order tetractys is replaced by the 2nd-order tetractys (Fig. 5). This generates what will be called the ‘Cosmic Tetractys.’ Its central 2nd-order tetractys, which denotes the Sephirah Malkuth, contains (7×7=49) coloured triangles symbolizing the seven Sephiroth of Construction. 49 is the number Figure 5. Replacing the central 1st-order tetractys of a 2nd-order tetractys by a 2nd-order tetractys generates the Cosmic Tetractys.

value of EL ChAI, the Godname of Yesod. The six 1st-order tetractyses surrounding it refer to the six Sephiroth of Construction above Malkuth and contain (6×7=42) coloured triangles. The seven Sephiroth of Construction are expressed in the Cosmic Tetractys by (49+42=91) coloured triangles, where 2 2 2 2 2 2 91 = 1 + 2 + 3 + 4 + 5 + 6 . Because the square of any integer n is the sum of the first n odd integers, 91 is the sum of (1+2+3+4+5+6=21) odd integers. This is the number value of EHYEH, the Godname of Kether. Explicitly, 91 = 1 1+3 1+3+5 1+3+5+7 1+3+5+7+9 1 + 3 + 5 + 7 + 9 + 11 = 65 + 26 15 integers form the boundary of the triangular array, showing how YAH prescribes this number. YAH also prescribes it because a 15-sided polygon with its sectors turned into 1st-order tetractyses has 91 yods. The sum of the integers is 65, the remaining six integers inside the boundary adding to 26. 65 is the Figure 6. The Cosmic Tetractys. The 91 coloured triangles represent 7-fold differentiations of the 7 Sephiroth of Construction denoted by the central 2nd-order tetractys and by the six 1st-order tetractyses that surround it. They are expressed by the compound Divine Name ADONAI TETRAGRAMMATON. number value of ADONAI (“My Lord”) and 26 is the number value of YAHWEH. Out of reverence for the holy nature of YHVH (Tetragrammaton) and because most Jewish denominations teach that it is forbidden to utter the Name of God except by the High Priest in the Temple, Jews substitute it in prayers with ADONAI. 91 is the number value of ADONAI TETRAGRAMMATON. The fact that the centre of the Cosmic Tetractys is another 2nd-order tetractys like itself can be regarded as the geometrical realisation of the Hermetic principle “As above, so below.” As a map of all levels of consciousness (as we shall shortly see) — the Cosmic Tetractys is somewhat analogous to a hologram. This reproduces a three-dimensional image of a scene, each point in the hologram containing light from the whole of the original scene. In principle, the latter can be reproduced from any part of the hologram, however small, although in practice, the image becomes fuzzier because diffraction causes a loss of resolution. All information about the whole holographic image is contained in any part. The purist may object that this analogy between a hologram and the Cosmic Tetractys breaks down because only the central 2nd-order tetractys is identical to the parent one — its nine other 1st-order tetractyses remain such in the Cosmic Tetractys. The criticism is misplaced. Every triangle representing a Sephirah is potentially a 1st-order tetractys and therefore also a 2nd-order tetractys. It is this fundamental property that is analogous to a hologram, not the geometry of the Cosmic Tetractys per se. 2. The Cosmic Tree of Life The 91 coloured triangles in the Cosmic Tetractys symbolize all possible differentiations of the seven

Sephiroth of Construction subject to the requirement of consistency with the Hermetic principle of correspondences that all their 49 differentiations should appear as well in the last one, Malkuth. As a triangle is a potential 1st-order tetractys, which is simply a Pythagorean representation of the Tree of Life, it is deduced that the seven Sephiroth of Construction require 91 Trees of Life represent their cosmic manifestation. The Tree of Life counterpart of the Cosmic Tetractys therefore consists of 91 Trees of Life that overlap one another (Fig. 7). This scheme will be called the ‘Cosmic Tree of Life,’ or CTOL. As the yod at the centre of a tetractys corresponds to Malkuth in the Tree of Life, the cosmic physical level of CTOL corresponding to Malkuth consists of the lowest 49 overlapping Trees of Life, each one denoted by one of the 49 coloured triangles (potential tetractyses) that form the central yod (Malkuth) of the Cosmic Tetractys. The cosmic superphysical region of CTOL is defined by the set of 42 Trees of Life above the lowest 49 Trees of Life. They correspond to the 42 larger, coloured triangles (potential tetractyses) in Fig. 6 symbolizing the six Sephiroth of Construction above Malkuth. A THEOSOPHICAL INTERPRETATION H.P. Blavatsky, one of the founders of the Theosophical Society, stated in her book The Secret Doctrine that Creation is divided into seven great planes of consciousness. Many of her ideas, which her book proved are found in the creation mythologies of ancient religions like Judaism and Hinduism, were developed by later Theosophical writers, such as Annie Besant and Charles W. Leadbeater. In particular, Alice Bailey gave a more precise and detailed formulation of the Theosophical classification of the planes. We shall follow her terminology, which is now commonly used in mainstream, Theosophical literature. The seven cosmic planes of consciousness are: 1) Adi, or Divine, plane; 2) Anupadaka, or Monadic, plane; 3) Atmic plane; 4) Buddhic plane; 5) Mental plane; 6) Astral plane; 7) Physical plane. Each cosmic plane is divided into seven cosmic subplanes. The seven subplanes of the cosmic physical plane are called ‘solar planes.’ The word solar was used not to imply any connection with the Sun but to convey a loose analogy (that is, not to be regarded in any literal sense) between the difference between the Solar System and the rest of the universe and the distinction between the cosmic physical plane and the six cosmic superphysical planes of consciousness. The solar planes, too, have the names listed above, e.g., Theosophists speak of the ‘solar physical plane,’ the solar astral plane,’ etc. Each solar plane is further divided into seven solar subplanes. The solar physical plane consists of seven solar physical subplanes; the solar astral plane consists of seven solar astral subplanes, etc. The seven solar planes of the cosmic physical plane have (7×7=49) solar subplanes and the six cosmic superphysical planes (astral→adi) have (6×7=42) subplanes. All seven cosmic planes have (49+42=91) subplanes. The seven planes (cosmic or solar) denote primary modalities of consciousness that correspond in a oneto-one way to the seven Sephiroth of Construction: Adi Anupadaka Atmic Buddhic Mental Astral Physical → → → → → → → Chesed, Geburah, Tiphareth, Netzach, Hod, Yesod, Malkuth. Only the four lowest Sephiroth of the Tree of Life: Malkuth, Yesod, Hod and Netzach, are at present components of normal consciousness. They correspond in the human personality to the four psychological functions of, respectively, sensing, feeling, thinking and intuiting, identified by the great psychoanalyst, C. J. Jung. Atmic, or mystical, consciousness corresponds formally to what Jung called the ‘transcendent function of the psyche.’ It is the expression of Tiphareth. A one-to-one association similarly exists between the seven subplanes of a cosmic or solar plane and the seven Sephiroth of Construction. It must be emphasized that Theosophical planes are not synonymous with the latter. The Kabbalistic notion of the Sephiroth is more fundamental than the Theosophical concept of planes. The latter are modalities of consciousness that express the particular quality of their corresponding Sephirah. Comparison of the Theosophical scheme of planes of consciousness with the Cosmic Tetractys shown in Fig. 6 establishes the following set of correspondences (▲ = coloured triangle):

91-617 À 9 1}-/ —9 J]-| Y

obviously not the 91-tree but 91 overlapping TOLs. (3) A ‘tree level’ is any one of the seven horizontal divisions of a TOL in CTOL marked by SLs that are shared by adjoining TOLs (Fig. 8). Tree levels are thus defined by the five SLs on the central Pillar of Equilibrium of the TOL, the Path linking its Chokmah and Binah and the Path joining its Netzach and Hod, which are Chokmah and 7 Binah of the next lower TOL. Each TOL has seven tree levels shared by 6 adjacent, overlapping TOLs. The horizontal Path linking Chesed and Geburah 5 of a TOL does not define a tree level because these Sephiroth are not shared by adjacent TOLs. Daath defines a tree level provided that it is Yesod of the 4 next higher TOL. No tree level is therefore defined for Daath of the 91st TOL 3 2 because that is the last one in CTOL. The scientific meaning of the 25 tree levels of the 7-tree representing the physical plane (the physical universe) is 1 that they denote the 25 spatial dimensions of the 26-dimensional space-time continuum predicted by quantum mechanics for so-called ‘spinless bosonic Figure 8. Tree levels on overlapping Trees of Life. strings.’ Tree levels may be thought of as the rungs of the ‘Jacob’s ladder’ referred to in the bible when it is interpreted as a set of overlapping TOLs that map all levels of reality. The structure of CTOL is described by the following formulae: 1. Kether of the nth TOL is number S(n) = 6n + 5 (1≤n≤90; S(91) = 550) 2. Kether of the nth TOL is the (2n+3)th SL on the central Pillar of Equilibrium (1≤n≤90); Kether of the 91st TOL is the 184th SL on the central pillar. 3. The nth SL on the central pillar is number C(n) = 3n – 4 (3≤n≤183; C(1) = 1, C(2) = 2, C(184) = 550) 5. Kether of the nth TOL is tree level number L(n) = 3n + 4 (1≤n≤90; L(91) = 276) 6. Number of tree levels up to the nth SL on the central Pillar is T(n) = (3n–2)/2 (n even) = (3n–1)/2 (n odd) 7. The n-tree has (n+1) Upper and Lower Faces. 8. Number of Paths in the n-tree is P(n) = 16n + 6 (n overlapping TOLs) = 16n + 9 (n-tree) The 91 overlapping TOLs in CTOL has 550 SLs. Compare this with the facts established earlier that the ten 1st-order tetractyses that form the 2nd-order tetractys have 550 geometrical elements when they are separate and that 550 geometrical elements are needed to construct its underling triangular array of 153 points, starting from the three corners of a triangle. According to formula 1, the 49-tree mapping the cosmic physical plane has S(49) = 299 SLs. Hence, there are 251 SLs above it. The 7-tree has 47 SLs, so that its apex is the 504th SL from the top of CTOL. The lowest TOL is the 1-tree with 11 SLs, so that its top is the 540th from the top of CTOL. The Decad determines the number 91 because a decagonal array of the integers 1, 2, 3 & 4 symbolised by the tetractys add up to 91: As the 13th triangular number, 91 is the number of points in the triangular array underlying a 1st-order tetractys in which each triangle is a tetractys: The odd integers in the sum of the first 13 integers: 91 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13

add up to 49 and the even integers sum to 42. This arithmetic division created by the even and odd integers mirrors the division of the seven cosmic planes into the 49 subplanes of the cosmic physical plane and the 42 subplanes of the six cosmic superphysical planes. The Godname ELOHA with number value 36 prescribes CTOL because 91 is the sum of the 36 integers (15 even, 21 odd) in the array: 6 5 5 4 4 4 3 3 3 3 2 2 2 2 2 1 1 1 1 1 1 5 4 4 3 3 3 2 2 2 2 1 1 1 1 1. ELOHIM, the Godname of Binah with number value 50, prescribes the 550 SLs in CTOL because 550 = 10×55, where 2 2 2 2 2 55 = 1 + 2 + 3 + 4 + 5 , i.e., 550 is the sum of (10×5=50) squares of the integers 1-5. Therefore, 10 10 + 30 10 + 30 + 50 10 + 30 + 50 + 70 10 + 30 + 50 + 70 + 90. 550 = The Godname YAH with number value 15 prescribes the number 550 as the sum of 15 numbers. We saw earlier how the Decad determines the number 550 as 10×55, where 55 is the sum of the first ten integers. 55 is also the tenth, so-called ‘Fibonacci number.’ Named after the 13th century Italian mathematician Leonardo of Pisa, also called Fibonacci, these are numbers appearing in the infinite sequence of integers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 … They have the property that each is the sum of the two previous numbers. They are known to be present in the philotaxis of many flowers and plants. The ratio of successive Fibonacci numbers converges towards the Golden Ratio Φ = (1+√5)/2 ≈1.6180339887… This is the ratio of adjacent sides of the Golden Rectangle ABCD: A F B a b/a = Φ D E C b It has the remarkable property that, when a square ADEF is cut away from it with a side equal to the shorter side of the Golden Rectangle, the sides of the remaining rectangle BCEF has the same proportion as those of the original rectangle and therefore it is another Golden Rectangle. 3. Encoding of CTOL in the Inner Tree of Life Polygonal representation of the n-tree When the n sectors of an n-sided polygon (not necessarily regular) are each turned into tetractyses, six yods are added per sector, so that (6n+1) yods are generated, the number “1” referring to its centre. According to formula 1 given above, the number of SLs in the n-tree is S(n) = 6n + 5. There are four SLs beyond Chesed of the nth TOL up to its top. Therefore, the number of SLs up to Chesed of the nth TOL = S(n) – 4 = 6n + 1, where the number ‘1’ denotes the lowest SL and six SLs are added for each overlapping TOL: S(n+1) – S(n) = 6. But (6n+1) is the number of yods in an n-sided polygon. Such a tetractys-constructed polygon is therefore a representation of n overlapping TOLs as far as its highest Chesed. The central yod corresponds to the lowest SL, the pair of hexagonal yods on each side of the polygon, being unshared by adjacent sectors, correspond to Chesed and Geburah of each TOL because these are unshared with adjacent TOLs. The yods at the corners of the polygon correspond to Yesod of each TOL and the hexagonal yods at the centres of the sectors correspond to each Malkuth, except for the lowest one, which is denoted by the centre of the polygon. Enclosing the n-sided polygon in

Y4aw2oo©NhadLef©afA(()5Nomeo a AN if \ N, s DaL"vU4wS-|“/xyeo QiY iSSNiaN, +LNàV/\INON JNAdC,á\€v4aAAoaStUT7yAD)NnteaT;e>-wleAyNaIe3CNà\XS\o\A\,NAoyA\oar yw 7= < / veA pr y / a x? MATAN \ LI gr >Py \\ N Pa Nan, fr / a a

from the bottom of the table. They are separated by the root edge, one set of seven polygons being arranged in the reverse sequence of the other set (the issue of other possible arrangements will be addressed later). Reading upwards from the base of the table, we find: 1. the first five polygons have 26 vertices or tetractyses and 161 yods. Compare this with the 161 SLs of the 26-tree; 2. the first seven polygons have 295 yods. Chesed of the 49th TOL is the 295th SL; 3. the first seven polygons & the root edge have 299 yods. Kether of the 49th TOL is the 299th SL; 4. the first 12 polygons & the root edge have 550 yods and 91 vertices (from a formal, mathematical point of view, the endpoints of the root edge count as vertices). CTOL has 550 SLs and 91 TOLs. Any set of five polygons with N corners will have a yod population obeying the formula S (N) = 6N + 5 for the number of SLs in the N-tree because Σ(6n+1) = 6 Σn + 5 = 6N + 5, set set where N ≡Σn set is the number of corners of the five polygons. Therefore, it is not surprising that the yod population of the first five polygons obeys this formula. However, what is interesting is that this set of polygons has 26 vertices, where 26 is the number value of YAHWEH, and 161 yods, which is the number of SLs in the 26-tree. What is even more interesting (in fact, remarkable) is that the seven polygons and the root edge have the same number of yods as there are SLs in the 49-tree, which maps the 49 subplanes of the seven planes of the cosmic physical plane! The four yods of the root edge therefore denote Kether, Chokmah, Binah and Daath of the 49th TOL. What is most interesting of all is that there exists a subset of the 14 polygons that — together with the root edge — have not only 91 vertices symbolizing the 91 TOLs of CTOL but also 550 yods that denote their 550 SLs. That subset is the set of the first 12 polygons and the root edge. This is amazing, for even if it were coincidence that a set of polygons can be found that contains 550 yods when the root edge is included, it is highly implausible that they would by chance possess 91 vertices as well! Nor is it plausible that it could be just coincidence that the number of SLs in the 49-tree is equal to the number of yods in the root edge and all seven polygons of the Inner TOL. Twelve of the 14 polygons collectively encode the self-replication of a single TOL into CTOL as a map of all levels of reality. These 12 enfolded polygons have 67 corners (65 outside the root edge) and 76 corners and centres of 88 tetractyses. This is how the respective numbers of Binah, ADONAI and YAHWEH ELOHIM prescribe the polygonal encoding of CTOL (the complete prescription by all Godnames is given 1 in Article 4 ). Moreover, only this particular combination of polygons has such a property, because only the triangle and square have the necessary seven vertices that must be excluded from Figure 11. Alternative, possible orderings of polygons belonging to the Inner Tree of Life. Allowed numbers of overlapping TOLs in the last column are shown ticked. The thick lines separate one set of seven polygons & the root edge from the other set. The running total number of yods should be read upwards, starting from the base in each case.

the 96 corners of the 14 polygons to leave 91 corners, including the two endpoints of the root edge. As will be now proved, if the three other possible orderings of the 14 polygons are considered: 1. triangle→dodecagon→triangle→dodecagon; 2. dodecagon→triangle→dodecagon→triangle; 3. dodecagon→triangle→triangle→dodecagon, none of them has a yod population that, including the root edge, fits the formula 6N + 4 for N overlapping TOLs with N greater than 91. Proof of uniqueness of encoding of CTOL in the (7+5) polygons The two possible orderings: triangle→dodecagon and dodecagon→triangle create (2×2=4) possible combinations of the two sets of polygons. Those not considered earlier are: 1) Triangle→dodecagon The section: triangle→octagon encodes 76 overlapping TOLs (Fig. 11), which is allowable, although uninteresting, being less than the 91 TOLs of CTOL. The section: triangle→decagon encodes the 86-tree, which is disallowed because it only makes sense to look for a number of overlapping TOLs as a complete map of all levels of reality. The n-tree is always part of N overlapping TOLs, where N>n. 2) Dodecagon→triangle The section: dodecagon→hexagon encodes 36 overlapping TOLs, which is permissible, but uninteresting, being less than 91. The section: dodecagon→square encodes the 95-tree, which is disallowed for the reason given above. CTOL is encoded in the section: dodecagon→pentagon. 3) Dodecagon→dodecagon The two sections: dodecagon→hexagon & dodecagon→octagon encode, respectively, 36 & 76 overlapping TOLs, which are both uninteresting. The section: dodecagon→decagon encodes the 86-tree, which is disallowed. Although combination (2) is the only one that encodes a number of overlapping TOLs as large as that in CTOL, it also encodes 36 overlapping TOLs in one set of polygons. This is not permitted, because it is only meaningful for a set of polygons encoding as many TOLs as CTOL to have subsets encoding groups of TOLs that are part of CTOL, not separate, overlapping groups. In the 12-polygon section: triangle→pentagon considered earlier, it encodes not only a single number (91) of overlapping TOLs but also the 26-tree and the 49-tree, which are both parts of CTOL. It is concluded that, of the four possibilities, only this section is permitted. We conclude that the encoding of CTOL in this particular set and ordering of 12 polygons is unique. (En passim, it is worth noting that the triangle and square uninvolved in this encoding have 44 yods — a very ‘Pythagorean’ feature in view of the importance of the Tetrad to this philosophy! It is another example of 2 how the Tetrad Principle formulated in Article 1 governs properties of TOL). In Theosophical terms, the 26th TOL in CTOL maps the fifth subplane of the fourth (Buddhic) plane, counting in both cases from the lowest plane and subplane of each plane. Malkuth of the 26th TOL is Tiphareth of the 25th TOL, which is the centre of the seven TOLs mapping the fourth plane, the central one of the seven planes. Malkuth of the 26th TOL is the pivotal centre of the 49-tree — its ‘half-way point,’ so to speak. These 26 TOLs corresponding to the first five polygons denote the region of CTOL that is the counterpart of 26-dimensional space-time predicted by quantum mechanics for the one-dimensional 50 10 5 2 67 corners BINH = 2 + 10 + 50 + 5 = 67 Figure 12. The number value 67 of Binah is the number of yods below this Sephirah in the 1tree. The letter values of the Hebrew word BINH denote sets of yods. 67 is the number of corners of the (7+5) enfolded polygons belonging to the Inner Tree of Life that separately and together with the root edge have 550 yods and 91 corners when their 89 sectors are tetractyses.

© > «i

AAA \AA/V\AA/ \/ IV soo. 000 < Ne Le . a en . / ? # / + 7 Paves SAEZ a > / . / A ive Ne . . (i = su. . fe N . . / LI SS E 14VA 4 Da O NN © x / « e N e LI Doo N / 4 A Li e_\ . . “ \ s_\ = N"o. N . \ |

physical plane. Enfolded, they have five vertices. This is an example of a 5:7 pattern that is characteristic of seven-fold, holistic systems. An example is the fact that the seven tone intervals of each of the seven types of musical scales include five whole tone intervals. This pattern is the geometrical manifestation of the fundamental distinction between the five Sephiroth of Construction belonging to the Lower Face of the Tree of Life and the two Sephiroth of Construction (Chesed and Geburah) that are part of its Upper Face. Enfolded, the (7+5) polygons have 67 corners (Fig. 12). This is the number value of Binah, the Sephirah at the head of the Pillar of Judgement that embodies the archetype governing the basic form s of holistic systems. Remarkably, it is the number of yods below Binah of the 1-tree when its 19 triangles become tetractyses. ELOHIM, the Godname of Binah with number value 50, prescribes the (7+5) separate polygons because their 89 sectors have 101 corners, where 101 is the 50th odd integer after 1. They are prescribed by YAHWEH, Godname of Chokmah with number value 26, because 101 is the 26th prime number. It is also the number value of Michael, the Archangel of Tiphareth. The (7+5) enfolded polygons have 65 corners outside their shared root edge, showing how ADONAI, the Godname of Malkuth with number value 65, prescribes their outer form. The 87 sectors of the enfolded (7+5) polygons have 76 corners, where 87 is the number value of Levanah, the Mundane Chakra of Yesod, and 76 is the number value of YAHWEH ELOHIM, the Godname of Tiphareth (see Article 4 in ref. 1 for how the Godnames of the other Sephiroth prescribe the (7+5) polygons). What other evidence indicates that the 12 polygons encoding CTOL constitute a holistic system, as indicated by the way in which the number values of the ten Sephiroth in the four Worlds parameterize their properties? Converted into tetractyses, their 87 sectors contain 494 yods (Fig. 13). 247 yods lie on the sides of the 48 tetractys sectors of the seven separate polygons of the Inner TOL. This means that (2×247=494) yods line the 192 sides of the 96 tetractyses making up both sets of seven polygons. Of these, 110 yods are corners of the latter, leaving 384 hexagonal yods on their sides. Many previous articles have demonstrated that this number is a parameter of holistic systems. For example, it is the number of corners, lines & triangles surrounding the centres of both sets of seven separate polygons. It is also the number of lines and broken lines in the 64 hexagrams of the I Ching table, as well as the number of yods in CTOL up to the top of the seventh TOL, i.e., the number of yods needed to construct from tetractyses the map of the physical plane. The Pythagorean integers 1, 2, 3 & 4 express the number 494 as: 1 494 = 1 1 2 1 3 1 4 1 1 2 2 2 3 2 4 2 1 3 2 3 3 3 4 3 4 2 4 3 4 4 4 They also express the 30 yods in the missing triangle and square because 2 2 2 2 30 = 1 + 2 + 3 + 4 . EL ChAI, the Godname of Yesod with number value 49, prescribes the (7+5) enfolded polygons because they have 490 (=49×10) yods outside their root edge. Their 87 sectors have 165 sides, where 2 2 2 2 2 165 = 1 + 3 + 5 + 7 + 9 . Such simple, mathematical beauty is characteristic of holistic systems. The 295 yods in the seven separate polygons of the Inner TOL symbolize the 295 SLs up to Chesed of the 49th TOL. The four yods in the separate root edge denote Kether, Chokmah, Binah & Daath of the 49th TOL. The 251 yods in the five other polygons of the set of 12 symbolize the 251 SLs in CTOL above the top of the 49-tree (Fig. 14). There are 183 SLs on each side pillar of CTOL and 184 SLs on its central pillar. There are 99 SLs on each side pillar of the 49-tree and 101 SLs on its central pillar, where 101 is the 26th prime number and the number value of Michael, the Archangel of Tiphareth. Hence, there are (183–99=84) SLs on each side pillar above the 49-tree and (184–100=84) SLs on the central pillar down to its top. There are 15 SLs on each side pillar of the 7-tree, which is analogous to the 49-tree because both correspond on their own levels to the Sephirah Malkuth, as well as 17 SLs on the central pillar of the 7-tree. Hence, there are (99–15=84) SLs on each side pillar of the 49-tree above the 7-tree and (100– 16=84) SLs on its central pillar below the top of the 49-tree down to the top of the 7-tree. This SL, the 47th from the bottom of CTOL, is the 168th SL on the central pillar, counting from the top of CTOL. 168 is the number value of Cholem Yesodeth, the Mundane Chakra of Malkuth. This is remarkable evidence that the 7-tree does, indeed, represent Malkuth. Furthermore, there are (84+84=168) SLs on the side pillar above the 49-tree, which, too, represents this Sephirah on a greater cosmic level. The 504 SLs down to the top of the 7-tree consist of 168 SLs on each side pillar and 168 SLs on the central pillar. This 168:168:168 division will prove to be highly significant when CTOL is compared in the next section with

550 Sephirothic emanations

given by Plato in his Timaeus. They have 68 corners (30 polyhedral vertices, where 30 = 1 + 2 + 3 + 2 4 ), and 180 sides (60 polyhedral edges). The holistic parameter 180, which was earlier encountered as the 180 sectors of the five Platonic solids, reappears in the first four Platonic solids as the number of sides of their 120 triangular sectors. E8 has eight simple (or zero) roots and 240 non-zero roots. Comparing this with the 248 corners & sides in the 38 faces of the first four Platonic solids, Table 5 indicates that the number 8 appears three times, namely, as the eight corners of the 12 sectors in the tetrahedron, as the eight corners at the centres of the faces of the octahedron and as the eight corners of the cube. Rather than try to answer the question of which polyhedron embodies the corners corresponding to the simple roots, let us just note that the 120 sectors in the 38 faces have 60 corners and 180 sides, i.e., 240 corners & sides. They correspond to the 240 non-zero roots of E8 . The 180 sides comprise 12 sides that belong either to the tetrahedron, the octahedron or the cube and 168 other sides, where 168 is the number value of Cholem Yesodeth, the Mundane Chakra of Malkuth. The latter comprise 78 sides of the 60 sectors of the first three solids and 90 sides of the 60 sectors of the icosahedron. This is remarkable because it reproduces the number values of Cholem and Yesodeth: ‫ות‬ ‫ד‬ ‫ו‬ ‫ס‬ ‫םי‬ ‫ל‬ ‫ח‬ 168 = T U D US Y 4←400 6 4 6 6010 M L Ch 40 30 8 90 78 The gematria number value of Cholem Yesodeth. The number value 400 of T (tau) is contracted to 4. However, it is unsurprising, as this division always manifests in holistic systems, as has been demonstrated in many previous articles. The counterpart of the 240 non-zero roots of E8 in the 38 faces of the first four Platonic solids are the 60 corners, the 12 unspecified sides and the 168 other sides, i.e., 72 corners & sides and 168 sides, totalling 240 corners & sides. The former correspond to the 72 roots of E6 , the rank-6 exceptional subgroup of E8 and the latter correspond to the 168 roots of E8 that are not also roots of E6 . 4) When the 60 sectors of the 12 pentagonal faces of the dodecahedron are turned into tetractyses and the 90 sectors of the 30 internal triangles formed by its centre and edges are turned into tetractyses, 8 550 hexagonal yods are generated. Plato regarded this regular solid as representing the celestial sphere because it resembled the perfect shape of a sphere more than any other one. We now see that not only do the five Platonic solids represent CTOL but so, too, does the fifth one by itself! Each hexagonal yod in the 150 tetractyses needed to construct the dodecahedron symbolizes one of its 550 SLs. The holistic parameter 550 embodied in all five Platonic solid manifests in the last one as well. 6. The disdyakis triacontahedron represents CTOL The faces of the Platonic solids are regular polygons with the same shape and size. The 13 Archimedean solids have faces with the shapes of two or more regular polygons. Interchanging vertices and faces generates their duals — the 13 Catalan solids. The Catalan solid with the most 9 vertices is called the disdyakis triacontahedron. Article 23 proved that this semi-regular polygon is the outer form of the polyhedral version of the Tree of Life and all its equivalent sacred geometries (its a C B C A B B C B b C Figure 24. The disdyakis triacontahedron and its three types of vertices. a/b = Φ= 1.618…. Figure 25. The 30 golden rhombic faces of the rhombic triacontahedron underlying the disdyakis triacontahedron are the bases of 30 pyramids (outlined in yellow) with 120 faces. inner form — the polyhedral counterpart of the polygonal Inner Tree of Life — is a polyhedron with 74 vertices, 216 edges & 144 faces). The disdyakis triacontahedron (Fig. 24) has 62 vertices, 180 edges and 120 triangular faces, that is, 362 geometrical elements make up its faces. Including its centre, there are 363 elements, where 363 is the number value of SHADDAI EL ChAI, the complete Godname

| height à x 4 N x 4 \ \ i74 [€ Î ASPit<P

of Yesod. The vertices are of three kinds: the 12 B vertices are vertices of a icosahedron, the 20 C vertices are vertices of a dodecahedron and the 30 A vertices are vertices of rhombic pyramids stuck on the 30 faces of a rhombic triacontahedron (Fig. 25). They are in fact golden rhombic faces because the ratio of their two diagonals (the edges of the underlying icosahedron and dodecahedron) is the Golden Ratio Φ. This means that the bases of all the diamond-shaped facets of the disdyakis triacontahedron have this aesthetically pleasing proportion used for centuries by artists and architects. Φ is present in this polyhedron in a more subtle way. Imagine a straight line passing through two opposite C vertices. It will be referred to as the C-C axis. As there are 20 C vertices, there are ten different C-C axes. For present purposes, however, only one need be considered. The 60 vertices (30 A, 12 B & 18 C) surrounding a C axis are arranged in 15 sheets or layers. They are the corners of seven polygons either above or below the central plane, which have six vertices arranged at the corners of a hexagon. Table 6 lists the rectangular coordinates of all 62 vertices. Each is a function of Φ, the distances between adjacent polygons being an integer power of Φ. Figure 26 displays the positions of polyhedral vertices plotted in the XY plane with the C axis along the Z axis. It shows that each of the 15 polygons has corners that are polyhedral vertices of the same type. This means that none of their 60 sides are edges of the disdyakis triacontahedron. Moreover, if the sectors of the polygons are Type A triangles, i.e., themselves divided into three triangles, only the centre of the central hexagon and the internal sides of its six sectors (shown red in Fig. 27) are shared with the 180 ordinary triangles inside the polyhedron formed by its centre and its 180 edges. The centres of the 14 polygons above and below the central polygon, the edges of their sectors and the sides of the three triangles in each sector of a polygon are unshared with the polyhedron, as are the 24 internal sides of the 18 triangles in the sectors of the central Type B hexagon. The two opposite C vertices on the Z axis are unshared with the polygons. Including these, the number of unshared geometrical elements is shown below: vertices 2 14 60 Total = 76 sides triangles 5×60=300 -6 (shared) + 294 60×3=180 + 180 = 550. There are 550 such elements, shown in detail in Fig. 27. They consist of 76 vertices, 294 sides & 180 triangles, i.e., 474 sides & triangles, where 474 is the number value of Daath (“knowledge) and 76 is the number value of YAHWEH ELOHIM, the Godname of Tiphareth. There are 256 vertices & triangles, 4 where 256 = 4 , showing how the Tetrad expresses this number. Remarkably, the 180 triangles and the 370 vertices & sides correspond to the 180 triangles and the 370 vertices & sides making up the 50 faces of the five Platonic solids (see Table 1). Indeed, the same 180:370 division appears in the tetractys representation of the number 550 discussed in Section 5: 100 370 vertices & sides 80 50 10 90 60 20 30 70 180 triangles 40 The 548 unshared geometrical elements other than the two opposite C vertices are the counterpart of the 548 SLs between the apex and nadir of CTOL, the C vertices playing the role of the latter. The disdyakis triacontahedron is the polyhedral representation of CTOL, the unshared geometrical elements of its 15 polygons corresponding to SLs. The central hexagon has (6×3=18) unshared triangles with six unshared corners and (6×4=24) unshared sides surrounding its centre. The 14 polygons above and below the central hexagon have 14 unshared centres and (60-6=54) unshared corners of (180-18=162) unshared triangles with (294-24=270) unshared sides, i.e., 500 (=50×10) geometrical elements unshared with the internal triangles of the disdyakis triacontahedron. This shows how ELOHIM, the Godname of Binah with number value 50, prescribes the 14 polygons. Each set of seven polygons has 250 unshared elements. Including the two C vertices, 251 elements are either above or below the central hexagon, which has (6×8=48) unshared elements (one corner, four sides & three triangles per sector). 48 is the number value of Kokab, the Mundane Chakra of Hod. Hence, the eight polygons making up the lower half of the disdyakis triacontahedron have (251+48=299) unshared geometrical elements, 251 unshared elements being above the central hexagon.

Number of yods 45 + 1 = 46 90 + 1 = 91 Figure 28. The 504 yods in the lowest eight Type B polygons (C vertex included) comprise 496 yods that surround their eight centres. 45 + 1 = 46 90 + 1 = 91 45 + 1 = 46 45 + 1 = 46 1 45 + 1 = 46 C vertex Total = 496 + 8 = 504 them. Including the C vertex, there are 504 yods in half the polyhedron (Table 7 & Fig. 28). There are 496 yods other than centres of its eight polygons. This is the number value of Malkuth, the last Sephirah of the Tree of Life denoting its physical form. The very half of the polyhedron whose 299 geometrical elements are analogous to the 299 SLs of the 49-tree mapping the cosmic physical plane embodies the number value of the Kabbalistic name of the Sephirah that is associated with this section of CTOL! Such an amazing property refutes the suggestion that it might be merely coincidence that there are 550 geometrical elements in the 15 Type B polygons and the two opposite C vertices, quite apart, of course, from the equally remarkable 299:251 division in elements that compose the polygons in one half of the polyhedron and its remainder. Another piece of evidence that discounts the possibility of coincidence is the fact that 66 corners of tetractyses surround the centres of the eight polygons in half the polyhedron. As 66 is the 65th integer after 1, this property is aptly prescribed by ADONAI, the Godname of Malkuth with number value 65. According to T able 7, including a C vertex, the number of yods in the seven polygons either above or below the central hexagon with 91 yods is 413. Including the two C vertices, the number of yods in the 15 polygons = 2×413 + 91 = 917. 916 yods surround the centre of the disdyakis triacontahedron. The number of yods on the central Pillar of Equilibrium of n overlapping T OLs constructed from tetractyses is given by 10 N(n) = 10n + 6. Hence, N(91) = 916. The number of yods surrounding the centre of the disdyakis triacontahedron is exactly equal to the number of yods that lie on the central pillar of CTOL when its 1096 triangles are tetractyses! Here is further, spectacular evidence that this polyhedron is the polyhedral counterpart of CTOL. The geometrical elements composing the 15 polygons perpendicular to a C-C axis correspond to SLs between the apex and nadir of CTOL, whilst their yods correspond to yods lying on its central pillar. The number of yods on the central Pillar of Equilibrium of the n-tree (the lowest n TOLs in CTOL) is given by Ň(n) = 10n + 10. 11 Hence, Ň(49) = 500. There are four more yods up to, but not including, Daath of the 50th TOL. The 504 yods in the eight polygons (including the C vertex) correspond to the 504 yods on the central pillar of CTOL up to this SL. Its significance is that it is the next lower SL after Binah of the 50th TOL, where 50 is

the inner form of four overlapping TOLs = 4×229 = 916. This is the number of yods lying on the Pillar of Equilibrium of CTOL. Different representations of the same holistic system are always parameterized by the same set of numbers. The counterpart in the Sri Yantra of the 504 SLs in CTOL down to the top of the 7-tree mapping the physical universe are the 504 hexagonal yods lying on the sides of tetractyses in the 42 triangles. They consist of 252 hexagonal yods on the outer sides of these triangles (three sets of 84) and 252 hexagonal yods (three sets of 84) inside them. They symbolized the 84 SLs in each of the three pillars of CTOL down to the top of the 49-tree and the 84 SLs in each of the three pillars below the top of the 49-tree down to the top of the 7-tree. What is the counterpart of this 3×84:3×84 pattern in the 504 yods (including Figure 34. The Type A dodecagon has as many hexagonal yods as the Tree of Life because this polygon is its polygonal counterpart. = 60 ( ) = the upper C vertex) in the eight polygons in the upper half of the disdyakis triacontahedron? Figure 32 shows that 84 yods other than corners surround the centre of each of the three six-sided polygons. 84 yods other than corners surround the centres of two triangles, 84 yods other than corners surround the centres of two more triangles, leaving 84 yods made up of the C vertex, the yods in the last triangle and the corners & centres of all eight polygons. That particular triangle has been chosen only for illustration, for it is not obvious which triangle should be included with the corners and centres. Whichever is the correct one is irrelevant to what is demonstrated here, namely, that the 504 yods in the 15 polygons (including the C vertex) naturally and simply divide into six sets of 84 yods, as should be expected if the disdyakis triacontahedron embodies a map of all superphysical .levels of reality that is analogous to CTOL. The same pattern exists in the distribution of 504 yods that surround the centre of a Type C dodecagon (Fig. 33). Pairs of opposite sectors contribute 84 yods. The 84 light and 84 dark yods symbolize, respectively, the 84 SLs on the Pillar of Mercy above the 49-tree and between the 7-tree and the 49-tree. The light and dark green yods denote the corresponding SLs on the Pillar of Equilibrium and the light and dark blue yods denote the corresponding SLs on the Pillar of Judgement. As the last regular polygon of the Inner TOL, the dodecagon is the single polygonal counterpart of the Tree of Life. This is demonstrated by the fact that the 60 hexagonal yods in a Type A dodecagon are analogous to the 60 hexagonal yods in Figure 35. The sum of the 40 odd integers after 1 that can be assigned to the 40 yods in the tree of Life above the Lower Face (shown shaded) is 1680. The sum of the next 30 odd integers assigned to the 30 yods in the Lower Face is 3360 = 2×1680. Their sum is 5040.

the Tree of Life itself when its 16 triangles are tetractyses (Fig. 34). Only this polygon can have such a number of hexagonal yods. Its Tree of Life character is the reason why 168 yods are needed to turn it into a Type B dodecagon and why 504 yods are needed to convert it to a Type C dodecagon. The former correspond to the 168 SLs on the central pillar of CTOL down to the top of the 7-tree mapping the physical universe. The extra 2 ×168 yods correspond to the 168 SLs on each side pillar (this is an alternative to the representation in Fig. 33 of the 504 SLs). As we have seen, the 168:168:168 division of SLs distributed on the three pillars of CTOL above the 13 7-tree is characteristic of holistic systems. It was encountered in Article 43 in the discussion of the {3,7} tessellation of the 3-torus mapping the 168 automorphisms of the Klein quartic as the 168 hexagonal yods on the edges of its 56 triangles when they are tetractyses and as the 336 hexagonal yods added, generating 504 hexagonal yods, when the sectors of these triangles are tetractyses. The geometry of the tessellated 3-torus was shown to be analogous to the Tree of Life, embodying its parameters. The number 504 and its 168:336 division is embodied arithmetically in the Tree of Life for the following reason: assigning the 70 odd integers after 1 to its 70 yods generates the number 5040 because 2 71 – 1 = 3 + 5 + 7 +… + 141 = 5040. This is the number of yods in 504 tetractyses. There are 40 yods above the Lower Face. Assigning the first 40 odd integers after 1 to them generates the number 1680 because 2 41 – 1 = 3 + 5 + 7 +… + 81 = 1680. This is the number of yods in 168 tetractyses. This means that the 30 integers 83-141 assigned to the 30 yods in the Lower Face add up to 3360, which is the number of yods in 2×168 tetractyses. The distinction between the Lower Face and the rest of the Tree of Life creates in an arithmetic way the 168:2×168 division characteristic of holistic systems. It is remarkable that this purely geometrical distinction divides the 70 yods into precisely the sets of 40 and 30 yods that generate this division when the odd integers after 1 are assigned to them. It is an example of the beautiful harmony between number and geometry displayed by sacred geometries. It manifests in the three uppermost whorls of the UPA shown in Fig. 15, which C.W. Leadbeater described as thicker than the lower seven ones, each whorl being a helix with 1680 circular turns. As the microscopic manifestation in space-time of the Tree of Life, the three whorls correspond to the Supernal Triad of Kether, Chokmah & Binah, whilst the seven other whorls correspond to the seven Sephiroth of Construction. References 1 Phillips, Stephen M. Article 4: “The Godnames http://www.smphillips.8m.com/Article04.pdf, p. 6. 2 Phillips, Stephen M. Article 1: “The Pythagorean nature of superstring and bosonic string theories,” http://www.smphillips.8m.com/Article01.pdf, p. 4. 3 Besant, A. & Leadbeater, C.W.: “Occult Chemistry,” Theosophical Publishing House, Adyar, Chennai, India, 1951, p. 23. 4 Phillips, Stephen M. Article 35: “The Tree of Life nature of the Sri Yantra and some of its scientific meanings,” http://www.smphillips.8m.com/Article35.pdf. 5 See ref. 4 for how other number values of the ten Sephiroth in the four Worlds are present in the Sri Yantra. 6 Proof: Let p = the number of sides of each face (or the number of vertices of each face) and q = the number of faces meeting at each vertex (or the number of edges meeting at each vertex). Then, for a regular polyhedron with C vertices, E edges and F faces, pF = 2E = qC. Substituting these in Euler’s formula for a polyhedron: prescribe the inner Tree of Life,” C – E + F = 2, 2E/q – E + 2E/p = 2, so that 1/q + 1/p = ½ + 1/E. Since E is positive, 1/q + 1/p > ½. Allowed values of p and q are p = 3, q = 3 (tetrahedron), p = 4, q = 3 (cube), p = 3, q = 4 (octahedron), p

= 5, q = 3 (dodecahedron) & p = 3, q = 5 (icosahedron). Phillips, Stephen M. Article 46: “How sacred geometries encode the 64 codons of mRNA and the 64 anticodons of tRNA,” http://www.smphillips.com/Article46.pdf, Fig. 38b. 8 See Table 1 in Article 3 at http://smphillips.8m.com/html/articles.html. 9 Phillips, Stephen M. Article http://www.smphillips.8m.com/Article23.pdf. 10 Proof: n overlapping Trees of Life have (n+1) Faces. Each of the lowest n Faces has 10 yods on the Pillar of Equilibrium when all triangles are tetractyses and the (n+1)th Face has 6 yods. Number of yods on the central pillar of n overlapping Trees of Life ≡Ň(n) = 10n + 6. 11 Proof: the n-tree has (n+1) Faces, each with 10 yods on the central Pillar of Equilibrium. Number of yods on the central pillar of the n-tree = 10(n+1) = 10n + 10. 12 Proof: The seven enfolded polygons have 42 sides. Their 47 sectors have 46 internal sides. When each polygon is Type B, their 47 sectors add (47×3=141) sides. Number of sides of seven enfolded Type B polygons = 42 + 46 + 141 = 229. 13 Phillips, Stephen M. Article 43: “The Tree of Life nature of the {3,7} tessellation of the 168 automorphisms of the Klein quartic on the 3-torus,” http://www.smphillips.8m.com/Article43.pdf, Table 3. 23: 32 “The polyhedral Tree Life,”