ARTICLE 50 (Part 2)

by

Stephen M. Phillips

Flat 4, Oakwood House, 117-119 West Hill Road. Bournemouth. Dorset BH2 5PH. England.

Website: http://smphillips.mysite.com 

Abstract

Page Index

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The 120 red yods on the boundaries of the seven enfolded polygons symbolize the 120 faces of the disdyakis triacontahedron with 62 vertices. Their 144 internal, blue yods symbolize the 144 faces of a polyhedron with 74 vertices.

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The geometry of the Tree of Life generates the rhombic faces of the rhombic dodecahedron & rhombic triacontahedron.

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Figure 12 Another counterpart of the disdyakis triacontahedron to the inner Tree of Life is as follows: associated with each set of seven enfolded polygons are their 35 corners (34 outside their shared root edge and one corner that is an endpoint of the latter). The 27 sectors of each set of seven polygons in the disdyakis triacontahedron have 34 corners. Including the single C vertex on the axis that is associated with each set, there are 70 vertices and centres surrounding the central polygon, 35 per set of seven polygons & C vertex). The trunk of the outer Tree of Life is: Point (Kether) Line (Path joining Chokmah & Binah) Triangle (triangle with corners at Chesed, Geburah & Tiphareth) Tetrahedron (tetrahedron with vertices at Netzach, Hod, Yesod & Malkuth). Constructed from tetractyses, they have 35 yods. The branches of the Tree of Life are its geometrical components outside its trunk. They, too contain 35 yods. This 35:35 division is characteristic of holistic systems. It is displayed by the 14 polygons and the two C vertices. The latter correspond to the two endpoints of the root edge — the start and end of the unfolding of the polygonal geometry of the inner Tree of Life.

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Figure 13   Each set of seven polygons has 27 corners that are vertices of the disdyakis triacontahedron. Its centre and the 54 vertices of both sets creates a set of 55 points. This number is F10. Each set has 27 sides. Including either the six corners of the central hexagon and a C vertex on the axis or the six sides of the hexagon and the other C vertex creates two sets of 34 points & sides, where 34 is the ninth Fibonacci number F9. The number of points & lines forming the shapes of the 15 polygons = 55 + 2×34 = 123. This is the tenth Lucas number L10. The polygons constitute a geometrical realisation for n = 10 of the general relation between Lucas numbers and Fibonacci numbers: Ln = Fn + 2Fn-1 as L10 = F10 + 2F9. The factor of 2 refers to the upper and lower halves of the polyhedron, each containing seven polygons. This property is the polyhedral counterpart of the 123 geometrical elements in the 3-tree (Fig. 5, Part 1), which has 55 geometrical elements in the lowest Tree of Life and 34 elements in each of the second and third Trees. 76 polyhedral vertices & centres of the 14 polygons surround the centre of the disdyakis triacontahedron. 76 is the ninth Lucas number L9.

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Figure 14 There are 137 corners & sides in the set of 15 polygons and the two diametrically opposite C vertices that form the axis of the disdyakis triacontahedron. This is the single polyhedral counterpart to the 1370 yods in the inner Tree of Life (Fig. 21, Part 1) when its 94 sectors are each divided into three triangles which are then turned into tetractyses. The number 137 determining the fine-structure constant α = e2/ħc is embodied in the arrangement of the vertices of the disdyakis triacontahedron. It is determined by the tenth Lucas number L10 = 123, which is the number of corners & sides other than the centres of the 14 polygons above and below the centre of the polyhedron:   137 = 123 + 14.   It is also determined by the ninth Lucas number L9 = 76 because 76 corners/vertices surround the centre of the polyhedron:   137 = 76 + 60,   where 60 is the number of sides of the 15 polygons. 76 is the number value of YAHWEH ELOHIM, the Godname of Tiphareth.   The number 137 is embodied in the polyhedral Tree of Life as the 137 corners of the 396 internal triangles of the concentric 144 Polyhedron with 74 vertices and the disdyakis triacontahedron with 62 vertices:   137 = 1 + 74 + 62 .

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Figure 15   The 42 triangles surrounding the central triangle in the 2-dimensional Sri Yantra have 68 corners (34 corners in each half). This leaves one other corner of the central, downward-pointing triangle and the bindu point in its middle. The 70 points creating the form of the 2-dimensional Sri Yantra are the counterpart of the 70 yods in the outer Tree of Life when its 16 triangles are tetractyses and the 70 corners of its polygonal, inner form. It demonstrates the equivalence of these sacred geometries. The two sets of 34 corners correspond to the 34 corners of the 27 sectors of each set of seven polygons (Fig. 11), whilst the bindu and lower corner of the central triangle correspond to the two C vertices on the central axis of the disdyakis triacontahedron and to the two endpoints of the root edge shared by the 14 polygons of the inner Tree of Life. Their counterpart in the outer Tree are its top (Kether) and bottom (Malkuth). More details about the analogous features of the Sri Yantra and other holistic systems are discussed in Article 35.

Figure 16   The 3-dimensional Sri Yantra is also equivalent to the inner Tree of Life. As the bindu point is no longer in the plane of the central triangle, as in the 2-dimensional case, the latter constitutes a distinct geometrical element. The eight geometrical elements making up the central triangle and bindu correspond to the eight points that are either corners of the polygons that coincide with Sephiroth of the outer Tree when it is projected onto the plane containing the polygons or centres of polygons. The four sets of triangles have 252 geometrical elements. They correspond to the 252 other yods outside the root edge that make up the 47 tetractyses. YAHWEH, the Godname of Chokmah with number value 26, prescribes both the geometrical composition of the Sri Yantra and the yod population of the inner Tree of Life.

Figure 17   The 252 geometrical elements making up the 42 triangles of the Sri Yantra that surround its central one correspond to the 252 geometrical elements composing seven overlapping Trees of Life. The seven elements in the central triangle that surround the bindu correspond to the seven elements (one point, three lines & three triangles) that are added when the seven Trees become the 7-tree.   The seven Trees of Life represent the seven planes of consciousness corresponding to the seven Sephiroth of Construction.

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Figure 18 Remarkable confirmation of the equivalence of seven Trees of Life and the Sri Yantra is provided by their yod populations when their triangles are tetractyses. There are 384 yods up to the top of the seventh Tree of Life in a set of n overlapping Trees (n>7) and 384 yods that belong either to the central triangle as hexagonal yods or to the 42 triangles surrounding it.

Figure 19   The nine parent triangles that create the Sri Yantra have 27 corners. 26 of them (denoted by small circles) are corners of the 42 triangles that surround its centre. A Type B triangle is a triangle whose sectors are each divided into three sectors. It has 46 yods when the nine sectors are tetractyses. The 42 triangles has (4×42=168 ) internal corners of their (9×42=378) tetractyses. 168 is the number value of the Mundane Chakra of Malkuth. Many previous articles demonstrated that it is a structural parameter of the superstring. The 42 Type B triangles surrounding the centre of the Sri Yantra have 1680 yods other than the 26 corners that belong to the nine primary triangles and the 168 internal corners of tetractyses. This is how the Sri Yantra embodies the number 1680, recorded over a century ago by the Theosophist C.W. Leadbeater as the number of turns in each helical whorl of the particle that he claimed through his remote-viewing ability was the basic unit of matter. The sceptic may argue that the choice of yods is ad hoc and made to deliver the number 1680. This ignores the fact that the corners of tetractyses that are left out of the calculation number 168 — the very superstring structural parameter that numerous articles have proved is embodied in holistic systems. The sceptic’s assumption that this, too, occurs by chance is implausible, especially given the fact that there are 168 yods lining the sides of the 21 triangles in each half of the Sri Yantra, another occurrence of this number which he must also regard as coincidental.

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Figure 20   Imagine the 50 faces of the five Platonic solids divided into their 180 sectors. Then suppose that each sector is a tetractys. There are 48 hexagonal yods in the 12 tetractyses forming the four faces of the tetrahedron, 96 hexagonal yods in the eight faces of the octahedron, 96 hexagonal yods in the six faces of the cube, 240 hexagonal yods in the 12 faces of the icosahedron & 240 hexagonal yods in the 12 faces of the dodecahedron. 48 is the number value of Kokab , the Mundane Chakra of Malkuth. The four Platonic solids associated by the ancient Greeks with the four Elements of Earth, Water, Air & Fire have 480 hexagonal yods (240 hexagonal yods in the tetrahedron, octahedron & cube and 240 hexagonal yods in the icosahedron).

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Figure 21 The counterpart of this property in the outer Tree of Life are the 240 yods added to the lowest Tree when the sectors of its 19 triangles are turned into tetractyses. Its counterpart in the inner form of the Tree of Life are the 240 hexagonal yods that are added to each set of seven separate polygons when their sectors are turned into tetractyses. Each hexagonal yod symbolizes one of the 240 roots of the rank-8, exceptional Lie group E8, the (240+240=480) hexagonal yods in the two sets denoting the 480 roots of E8×E8, which is the symmetry group of the unified interaction of one of the two types of heterotic superstring.

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Figure 22   The 62 vertices of the disdyakis triacontahedron are the vertices of 28 Platonic & Catalan solids. 21 polyhedra are copies of the first four Platonic solids. The seven other polyhedra that can be fitted into the disdyakis triacontahedron comprise the dodecahedron, five rhombic dodecahedra and one rhombic triacontahedron.

Figure 23   Suppose that the faces of the 28 Platonic & Catalan solids formed by vertices of the disdyakis triacontahedron are constructed from tetractyses. There are 1680 hexagonal yods in the faces of the 21 copies of the first four Platonic solids and 1680 hexagonal yods in the faces of the other seven polyhedra. The 28 polyhedra therefore have 3360 hexagonal yods

Figure 24 The number 3360 is embodied in the inner Tree of Life as the 3360 yods in its seven enfolded polygons when their 47 sectors are each turned into the so-called ‘2nd-order tetractys.’ 47 is the eight Lucas number L8. This is generated from the Pythagorean tetractys (the 1st-order tetractys) by regarding each of its ten points as a tetractys. This higher-order tetractys has 85 yods, where 85 = 40 + 41 + 42 + 43. 84 yods surround its centre, where  84 = 12 + 32 + 52 + 72. As a parameter of all holistic systems, the number 3360 has the following significance for the structure of the superstring: each of the ten whorls of the unit of matter described by the Theosophists Annie Besant & C.W. Leadbeater is a helix with 1680 turns. It twists five times around the spin axis of the particle, so that it winds (1680/5=336) times in one revolution. The ten whorls wind 3360 times in one revolution and 1680 times in a half-revolution. The 21 copies of the first four Platonic solids and the seven other polyhedra embody the number of circularly polarised oscillations in the ten whorls as they make these two half-revolutions.

Figure 25   The numbers of Platonic solids of the first four types that can be fitted into the disdyakis triacontahedron and which have 1680 hexagonal yods in their faces are the letter values of EYHEH (“I am”), the Godname of Kether, whose number value is 21. It is also the eighth Fibonacci number F8. This one of the ways in which this Godname prescribes the superstring structural parameter 1680.

Figure 26     The number 168 is the number value of Cholem Yesodoth, the Mundane Chakra of Malkuth. Confirmation that it refers to the Malkuth aspect of the Tree of Life is the remarkable fact that 49 overlapping Trees of Life have 1680 corners, sides & triangles. The lowest 49 Trees in CTOL map the 49 subplanes of the cosmic physical plane. Each cosmic plane corresponds to a Sephirah of Construction, and the cosmic physical plane corresponds to Malkuth, the physical level of CTOL in a cosmic sense. 49 is the number value of EL CHAI, the Godname of Yesod.

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Figure 27   Joining the 62 vertices of the disdyakis triacontahedron to its centre creates 180 triangles in its interior with (180×3=540) sectors. There are 60 vertices surrounding an axis drawn through any two opposite vertices. The sectors of the internal triangles have 180 internal corners and (60 + 3×180 = 600) internal sides surrounding the centre of the polyhedron. Its 120 faces are 120 triangles with 180 sides and 60 corners surrounding the axis. 1680 corners/vertices, sides/edges & triangles surround this axis. The disdyakis triacontahedron embodies the superstring structural parameter 1680. Its 900 triangles have 780 corners & sides.

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Figure 28   The 1680 geometrical elements that surround any axis of the disdyakis triacontahedron is the polyhedral realisation of the number value 168 of Cholem Yesodoth, the Mundane Chakra of Malkuth. That this is no coincidence is indicated by the fact that the 780:900 division of geometrical elements reflects, apart from the Tree of Life/tetractys factor of 10, the gematria number values of both Cholem (78) and Yesodoth (90). In accordance with the rules of gematria, the letter value 400 of tav, the final Hebrew letter in Yesodoth , can be contracted to 4.

Figure 29       The disdyakis triacontahedron is unique in that no other Archimedean or Catalan solid has 1680 geometrical elements surrounding an axis that passes through two opposite vertices. This is true for polyhedra with triangular faces whether their faces are regarded as single triangles (case A) or whether they are divided into their sectors (case B). It is also true for polyhedra having faces that are not triangular.

Figure 30       ADONAI, the Godname of Malkuth (the physical universe), has number value 65 . It prescribes the ten lowest Trees of Life in CTOL because they have 65 SLs. These Trees map the ten spatial dimensions predicted by M-theory, the general theory being currently being sought by physicists to explain the basic properties of physical matter and the forces of nature. A decagon within a square is a representation of the ten Trees because the object has 65 yods when the ten sectors of the decagon are tetractyses. The four corners of the square correspond to Kether, Chokmah, Binah & Daath of the tenth Tree and the 61 yods in the decagon correspond to the 61 SLs up to Chesed of this Tree. The letter values of ADONAI are the numbers of different yods.

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Figure 31 When the triangles in the 10-tree prescribed by ADONAI are divided into their sectors and each sector turned into a tetractys, there are 1680 yods below the apex of the tenth Tree of Life (65th SL), including those outside it that belong to the eleventh Tree. Each yod denotes one of the circular turns in a whorl of the heterotic superstring described by the Theosophists Besant & Leadbeater with the aid of a form of remote-viewing known in yoga as one of the siddhis, or paranormal abilities. The same number has been encountered in this article as: the 1680 yods in the 42 so-called ‘Type B’ triangles of the Sri Yantra other than internal corners of tetractyses and the corners of its nine parent triangles; the 1680 points, lines & triangles making up the disdyakis triacontahedron that surround an axis passing through any two diametrically opposite vertices;  the 1680 hexagonal yods in the faces of the five Platonic solids, the rhombic triacontahedron & the disdyakis triacontahedron;  the 1680 hexagonal yods in the faces of the 21  copies of the first four Platonic solids whose vertices belong to the disdyakis triacontahedron.  The n-tree has (12n+7) triangles with (16n+9) sides. The number of sides in their (36n + 21) sectors is 16n + 9 + 3(12n+7) = 52n + 30.  The 10-tree has 361 sectors with 550 sides. This is the counterpart of the 550 geometrical elements in the 50 faces of the five Platonic solids, the 550 Sephiroth in CTOL and the 550 unshared geometrical elements in the 15 polygons of the disdyakis triacontahedron. The tenth Fibonacci number 55 determines the form of all such holistic systems. In particular, it determines the form of each of the ten helices that comprise the E8×E8 heterotic superstring.

Figure 32     Tree levels are defined as the formative stages in the Fibonacci number-governed growth of successive Trees of Life in CTOL. The number of tree levels in the n-tree is:   t(n) = 3n + 4.   t(1) = 7, which is the fourth Lucas number, t(3) = 13, which is the seventh Fibonacci number and t(10) = 34, which is the ninth Fibonacci number. This number has appeared in this article:   1. in the outer Tree of Life as the number of geometrical elements in the Upper Face of the 1-tree and as the number of elements added by successive Trees; 2. in the inner Tree of Life as the 34 corners associated with each set of seven enfolded polygons that are intrinsic to them because they do not coincide with those of polygons enfolded in the next higher Tree; 3. as the 34 single bones of the axial skeleton of the human body; 4. as the 34 vertices & centres of the first four Platonic solids; 5. in the disdyakis triacontahedron as the 34 corners of the 27 sectors of the seven polygons either above or below the central hexagon that are formed by vertices of this polyhedron.   Remarkably, it determines the very number of overlapping Trees of Life that have a yod population of 1680. In this way, it determines the very helical structure of each whorl of the heterotic superstring. The sum 34 = 21 + 13 differentiates between large-scale, 3-dimensional space, which is mapped by the 3-tree with 13 tree levels, and the next 21 tree levels that span seven Trees up to the 34th, which marks the top of the 10-tree mapping the ten spatial dimensions. In other words, the eighth Fibonacci number 21 measures the seven compactified dimensions predicted by M-theory. The tenth Fibonacci number 55 determines the 10-tree (and therefore the superstring structural parameter 1680) because their 127 triangles have 361 sectors with 550 (=55 ×10) sides.

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References Phillips, Stephen M. Article 27: “How the disdyakis triacontahedron embodies the structural parameter 1680 of the E8×E8 heterotic superstring,” (WEB, PDF). Besant, A. & Leadbeater, C.W. “Occult Chemistry,” 3rd ed., Theosophical Publishing House, Adyar, Madras, India, 1951. Phillips, Stephen M. Article 36: “The Sri Yantra-like pattern of the 15 layers of vertices in the disdyakis triacontahedron and its scientific meaning,” (WEB, PDF). Phillips, Stephen M. Article 49: “How some sacred geometries are equivalent maps of all levels of reality,” (WEB, PDF). Phillips, Stephen M. Article 35: “The Tree of Life nature of the Sri Yantra and some of its scientific meanings,” (WEB, PDF).