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#12 How the sacred number 7 expresses various superstring structural parameters
We have just seen that there are 252 yods associated with each of the five star-shaped arrays of six sectors that form a triacontagon. If we pair an array chosen from a triacontagon in one set of four triacontagons belonging to a 600-cell with its counterpart in the set of four triacontagons that belong to the other 600-cell, the 12 sectors have the same yod population as a Type C dodecagon, which has (12×42=504) yods surrounding its centre. As 504 = 7×72, this is the number of yods surrounding a heptagon with 2nd-order tetractyses as its sectors, there being 72 yods per sector, where 72 is the number of Chesed, the first Sephirah of Construction:
504 = |
The heptagon represents seven overlapping Trees of Life, its Type A version having 43 yods that symbolise the 43 yods up to Chesed of the 7th Tree. When its sectors are 2nd-order tetractyses, the 72 yods per sector correspond to the 72 corners of the 94 sectors of the (7+7) polygons making up the inner form of successive Trees of Life that are intrinsic to it because they are not corners of triangles in the outer Tree:
The (7+7) enfolded Type A polygons in the inner Tree of Life have 94 sectors with 80 corners. Of these, the topmost corners of the two hexagons coincide with the lowest corners of the two hexagons belonging to the inner form of the next higher Tree of Life. Therefore, the number of corners of the 94n sectors of the 14n polygons enfolded in the n-tree = 78n + 2, where "2" denotes the two topmost corners of the hexagons belonging to the inner form of the nth Tree. As the inner Tree of Life lies in the plane formed by the Pillar of Mercy and the Pillar of Judgement, Chokmah, Binah, Chesed, Geburah, Netzach & Hod of each overlapping Tree are located either at the centre of each hexagon or at its top and bottom corners. They are indicated by the black yods in the adjacent diagram depicting the lowest seven Trees of Life that map the seven subplanes of the physical plane. Successive Trees add four black yods, so that the number of SLs in the n-tree that are shared with its inner form = 4n + 2. The number of intrinsic corners of the 14n polygons = 78n + 2 − 4n − 2 = 74n. Outside their root edges with white yods at their ends are 72n corners (36n in each set of 7n polygons). Each successive Tree of Life has an inner form with 72 intrinsic corners of the 94 sectors of its (7+7) polygons. The 7-tree maps the seven subplanes of the physical plane. The (7×14=98) polygons of its inner form have (7×94=658) sectors with (7×72=504) intrinsic corners (252 for each set of 49 polygons). This is the inner Tree of Life counterpart of the (7×72=504) yods that surround the centre of the heptagon with 2nd-order tetractyses as its sectors. The (7+7) polygons in the inner form of a single Tree of Life have 504 intrinsic yods (none belonging also to its outer form). The (4×504=2016) yods intrinsic to the inner form of the four uppermost Trees of the 7-tree symbolise the 2016 turns predicted to be in a whorl of the shadow matter superstring. These Trees map the four "etheric subplanes" of the physical plane described by Theosophy. Here is the vital, mathematical connection between the Theosophical teachings concerning the seven planes of consciousness and the basic particle of shadow matter that is predicted by E_{8}×E_{8}′ heterotic superstring theory and which was first observed with micro-psi vision by Ronald Cowen in the early 1990s. The factor of 4 in the factorisation: 2016 = 4×504 corresponds to the four Type C triacontagons in each 600-cell formed by the E_{8} Coxeter plane projection of the 4_{21} polytope each having five sets of six sectors, each set containing (6×42=252) yods that are repeated four times for each 600-cell; these correspond to the two halves of the inner form of the uppermost four Trees that map the four etheric subplanes of the physical plane mapped by the 7-tree: 4×504 (4 Trees) = 4×252 + 4×252 ((4+4) triacontagons). The remaining factor of 5 due to the 5-fold symmetry of each triacontagon corresponds to the five helical whorls of the shadow matter particle, each with 2016 circular turns. The 7-tree has 47 SLs. Its apex is the 504th SL from the top of CTOL, which has 550 SLs. The Tree of Life structural parameter 504 is also a parameter of CTOL because it demarcates the domain of space-time mapped by the 7-tree from all superphysical levels of reality, mapped by 84 higher Trees. |
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The 7-tree has 30 SLs on its side pillars that are shared with its inner form. |
The inner form of the 7-tree consists of 658 sectors of 98 polygons with 504 intrinsic corners outside root edges. Its total number of corners = 504 + 30 = 584. |
This type of polygon is absent from the seven types of regular polygons making up the inner form of the Tree of Life. However, we saw on page 2 that the number 504 is a structural parameter of the inner Tree of Life, being the number of yods outside the root edge that belong solely to it because they are not part of the outer Tree of Life when its triangles are tetractyses. In fact, we found that the (4×504=2016) intrinsic yods in the inner form of the four uppermost Trees of the 7-tree that map Theosophy's four etheric subplanes is the number of turns in each whorl of the particle described by Ron Cowen to be the basic constituent of the matter pervading these subplanes — thereby confirming this interpretation and the validity of his description. This embodiment of a structural parameter of the inner Tree of Life in the polygon that symbolises the number 7 is just one example of the sacred quality of this number. The fact that the Type A heptagon has 42 yods surrounding its centre, where 42 is the number of turns per E_{8}′ gauge charge, is another example of this power to embody such parameters, whilst the fact that two joined Type B heptagons embody the number (206) of bones in the adult human skeleton, its different types of yods symbolising the 80 bones in the axial skeleton and the 126 bones in the appendicular skeleton, is a third example (see diagram above). Listed below are ways in which the number 7 expresses properties of the two basic states of the E_{8}×E_{8}′ heterotic superstring:
Basic shadow matter particle (E_{8}-singlet) |
UPA/subquark (E_{8}′-singlet) |
1. Number of turns = 10080 = 30×336,
where number of revolutions = 30 =
1^{2} +
2^{2} +
3^{2} +
4^{2} |
1. Number of turns = 16800
= 50×336, where number of revolutions
= 50 = 49 + 1 =
7^{2} + 7^{0} and
number of turns per revolution = 336 =
7^{3} −
7^{1}. |
* The partition function p(n) is the number of possible partitions of a non-negative integer n. For example, p(3) = 3 because the integer 3 has the partitions 3 = 1 + 1 + 1, 1 + 2 and 3. The partition function for the integer 10 (the Decad, symbolised by the tetractys) takes the value 42, which is the number of turns in each revolution of a whorl of the shadow matter particle. The partition function has the value 7 for the integer 5 and the value 15 for the number 7. This is the number value of YAH, the shorter version of the Godname of Chokmah. The number 7 is the only integer whose partition function is the gematria number value of the Godname of a Sephirah.
Cowen never attempted to check whether the turns ("1st-order spirillae) of the helical whorls of the shadow matter particle comprise seven smaller turns ("2nd-order spirillae") in a ever-smaller series of six higher orders of spirillae, as Besant & Leadbeater noticed for the UPA. Although as yet unconfirmed by micro-psi observation (Cowen is unavailable to check this because he passed away last year), it ought to be the case because the higher dimensions of superstring space-time are compactified to form the same 6-dimensional space (a 6-d torus) for both types of superstrings. The seven orders of spirillae have their basis in the seven Sephiroth of Construction. The seven Trees of Life in the 7-tree mapping the physical plane and the tetractys-constructed heptagon that is isomorphic to them are mathematical representations of these seven Sephiroth, their Godnames prescribing through their gematria number values the properties of these Trees. The number 7 does not explicitly describe the 3-dimensional form of the shadow matter particle, unlike with the UPA, which comprises three "major whorls" and seven "minor whorls" — a 3:7 division of its 10 whorls that reflects the difference between the Supernal Triad and the seven Sephiroth of Construction of the Tree of Life. However, the partition function (defined in the footnote of the table shown above) for the number 7 is 15, which is the number of purely bosonic string dimensions of the 26-dimensional space-time prescribed by YAHWEH, the complete Godname of Chokmah with number value 26. As the shadow matter particle and the UPA exist in the same 11-dimensional space-time predicted by M-theory, albeit in 10-dimensional sheets separated by a small gap extending along the 10th dimension of space that the theory requires, there are seven compactified dimensions. The six compactified, superstring dimensions generate the six higher orders of spirillae (each being the 7-fold winding of a closed curve around the surface of a torus). The total number of compactified dimensions is
22 = 7 + 15,
where (as we have just seen) 15 is the value of the partition function for the number 7.
We discussed above how the n-gon with the 2nd-order tetractys as its sectors is isomorphic to the inner form of the n-tree in terms of the corners of the (7+7) polygons enfolded in each Tree that are intrinsic to them. Its 10 tetractyses have 85 yods, where
85 = 4^{0} + 4^{1} + 4^{2} + 4^{3}.
"4^{0}" (= 1) denotes the yod at the centre of its central 1st-order tetractys. It corresponds to the lower end of the root edge that joins the two sets of seven polygons. The six 1st-order tetractyses that have corners shared with the central tetractys contain (6×6=36) hexagonal yods that surround their centres. Including the central tetractys creates a set of 42 hexagonal yods. As the 2nd-order tetractys has 84 yods surrounding its centre, it can be seen to consist of two distinct sets of 42 yods. The discussion of CTOL in Maps of reality revealed that these two sets refer to the two sets of six superphysical planes of consciousness and the division of planes into seven subplanes: 84 = 6×7 + 6×7. The 84 yods are depicted in the diagram below as 42 red yods and 42 blue yods:
This 42:42 pattern of yods appears in the seven-pointed star when it is constructed from 1st-order tetractyses. Comparison with the shadow matter particle, whose whorls have one E_{8}′ gauge charge spread over every successive sequence of 42 turns reveals how it conforms to the universal blueprint whose Pythagorean expression is the 1st- and 2nd-order tetractyses. It exists because the number of turns per E_{8}′ gauge charge is exactly half of 84 — the number of yods that surround the centre of the 2nd-order tetractys. This means that its five whorls can each make the same number of revolutions in their outer spirals as in their inner spirals. This is needed because there have to be two halves, each carrying 120 gauge charges, in order to conform to the 4-d projection of the 240 vertices of the 4_{21} polytope as the compound of two 600-cells, each with 120 vertices.
A profound analogy can now be seen to emerge between the inner and outer halves of the shadow matter particle carrying 240 gauge charges and the two sets of six superphysical planes of consciousness, 42 subplanes being in each set:
42 turns/one gauge charge in outer revolution ↔
42 superphysical subplanes;
42 turns/one gauge charge in inner revolution ↔ 42 cosmic superphysical
subplanes,
where 240×42 = 10080, the predicted number of turns/oscillations in the five whorls of this E_{8}-singlet state of the E_{8}×E_{8}′ heterotic superstring. Just as the cosmic physical plane (the lowest of the seven cosmic planes) is the counterpart of the physical plane, so the four highest planes of the former are the counterpart of the four highest subplanes of the latter. In Theosophical terminology, they are the Adi (or Divine) plane, the Anupadaka (or Monadic) plane, the Atmic plane and the Buddhic plane (see here). The (4×7=28) subplanes of these four planes of existence are mapped by 28 Trees of Life, where 28 is the seventh triangular number and the second perfect number**. The number of SLs in the n-tree is given by
S(n) = 6n + 5,
so that the number of SLs in every 28 Trees is
S(n+28) − S(n) = 6×28 = 168.
The appearance of the gematria number value 168 of the Mundane Chakra of Malkuth (the material universe) in characterising four successive planes (in particular, the four highest planes in the cosmic physical plane)* is a sign that their Tree of Life representation has some property which quantifies the form of a whorl of the basic particle of the shadow matter universe corresponding to those physical subplanes that are the counterpart of these four planes. The property in question is the 72 corners of the 94 sectors of the inner Tree of Life that are not also corners of its outer form, i.e., its 72 intrinsic corners, which were discussed above. The number of such corners in the (28×14=392) polygons enfolded in the inner form of these 28 Trees of Life = 28×72 = 2016. This is the number of turns in each helical whorl of the shadow matter particle. It is a striking, mathematical illustration of the ancient, Hermetic principle: "As above, so below" (see here). As the (7+7) enfolded polygons have 504 intrinsic yods (see the discussion above), 72 of which are corners of sectors, we see that the four highest Trees/subplanes of the 7-tree/physical plane have an inner form that is made up of 56 polygons containing (4×504=2016) intrinsic yods, whilst the four highest planes of the 49-tree/cosmic physical plane has an inner form that contains (4×7×72=2016) intrinsic corners of sectors of polygons. The same number is embodied in two analogous sections of CTOL, namely, (to use Theosophical terminology) the four "etheric subplanes" and the four "cosmic etheric planes": the Adi plane, the Anupadaka plane, the Atmic plane & the Buddhic plane. Here is compelling, independent evidence that the analysis of Cowen's particle given in previous pages is correct — compelling not simply because properties of two sections of CTOL are quantified by the same number but (more significantly) because these sections are analogous and refer to the shadow matter universe predicted by E_{8}×E_{8}′ heterotic superstring theory. It is ludicrous to dismiss this agreement between numbers as due to chance. It is not merely the fortuitous conjunction of the same number quantifying properties of unrelated sections of CTOL; these two sections are conceptually related because they are metaphysically analogous to each other. This sort of conjunction cannot be reasonably dismissed as a coincidence. Instead, it is the undeniable manifestation of logically coherent design — a mathematical example of the Hermetic principle: "As above, so below". Indeed, historically speaking, the interpretations that have been made of this great principle of cosmic correspondence have been varied, inaccurate and vague. We claim that it was always really a statement of mathematical isomorphism between metaphysically analogous levels of reality. Its true meaning only becomes revealed by a correct mathematical representation of all levels of reality. It is not that these representations have been missing up till now. They have existed for centuries (thousands of years in the case of the Sri Yantra). But the lack of a proof of isomorphism between them, combined with a rigid adherence to canonical formulations that inhibited any correlation that was quantitative as well as qualitative, meant that the mathematical meaning of this ancient principle has never been revealed until now. The demonstrations of isomorphism between various sacred geometries given in this website now make it possible to penetrate beyond the superficial meaning that students of Hermeticism have understood to reveal the principle as a statement that analogous levels of reality are described by the same mathematical patterns between the same numerical parameters. These numbers include the gematria number values of the 10 Sephiroth, their Godnames, Archangels, Orders of Angels & Mundane Chakras.
* The top of the highest Tree of Life in the 7-tree is the 168th SL on the central Pillar of
Equilibrium of CTOL, counting from its apex.
** A perfect number is a positive integer that is the sum of its divisors, e.g., 28 = 1 + 2 + 4 + 7 + 14, where 28
= 1×28 = 2×14 = 4×7.
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