Table 1. Number values of the ten Sephiroth in the four Worlds.
The dots will be called “yods,” after the name (yod) of the tenth letter (י) of the Hebrew alphabet, which is somewhat shaped like a dot or point. The inner form of the Tree of Life (Fig. 1) comprises seven enfolded, regular polygons: triangle, square, pentagon, hexagon, octagon, decagon & dodecagon. The last of these is the tenth regular polygon, counting from the simplest one — the triangle. Given the titles “All Perfect,” “God,” & “Kosmos” given by the Pythagoreans to the number 10, which they regarded as the perfect completion of number, it should come as no surprise that the
dodecagon, too, embodies numbers of universal (and therefore scientific) significance. This article discusses how the dodecagon encodes the numbers 168, 248, 336 and 1680 as parameters of the structure and dynamics of superstrings. Of these numbers, only the second — the dimension of the rank-8, exceptional Lie group E8 used in superstring theory — has as yet been recognised by particle physics. According to Table 1, it is the number value of Raziel, the Archangel of Chokmah.
2. Properties of
Rather than give tedious calculations, the properties of both types of dodecagon are listed below for later discussion. For the sake of reference, the number values of the Sephirothic titles, their Godnames, Archangelic Names, Angelic Names and Mundane Chakras are shown in the following table. Numbers in coloured cells either have been already referred to or will appear in later discussion.
Table 2. Gematria number values.
Set out below are the ways in which the Godname numbers prescribe these properties of the dodecagon and two separate or joined dodecagons:
3. Encoding of superstring structural parameter
Further remarkable confirmation that the dodecagon constitutes sacred geometry because its properties are prescribed by Godnames is the fact that, outside their root edge, the two joined, Type B dodecagons
contain 260 (=26×10) yods on the sides of their 72 tetractyses that are not their 48 corners or centres, where 26 is the number value of YAHWEH, the Godname of Chokmah. Compare this with the fact that the seven enfolded, regular polygons contain 260 yods outside their shared root edge (Fig. 8). In the first case, this number is that needed to delineate the edges of their tetractyses outside the root edge, given their corners and centres; in the second case, it is the number of yods required to construct the seven enfolded polygons, starting with the root edge. The ways in which the generative Godname YAHWEH prescribes both geometrical objects are analogous.
It is not coincidental that the two objects possess properties that are quantified by the same sets of numbers listed in Table 2. The dodecagon is the polygonal form of the Tree of Life and will — like any other holistic structure — embody the numbers listed in this table.
there result 248 yods other than corners of tetractyses, that is, yods symbolising the seven Sephiroth of Construction. These symbolise the 248 quantum states of the particle transmitting the unified superstring force described by the gauge symmetry group E8. A dodecagon with its 12 sectors turned into such higher-order tetractyses contains 120 tetractyses, where
showing how the Tetrad determines this number. The number of yods in each sector is
again illustrating the role of the Tetrad. Taking into account that 12 yods on each internal edge of a sector apart from the centre of the dodecagon are shared with adjoining sectors, there are (84–12=72) yods per sector, where 72 is the number of Chesed, the fourth Sephirah from the top of the Tree of Life. Of these, 10 (=1+2+3+4) are corners of tetractyses symbolising Kether, Chokmah and Binah and 62 are hexagonal yods symbolising Sephiroth of Construction, where 62 is the number value of Tzadkiel, the Archangel corresponding to Chesed. Therefore, the number of yods in the 120 tetractyses of a dodecagon = 12×72 + 1 = 865. 13 yods lie along the shared edge of the pair of dodecagons, leaving (865–13=852) yods outside it. The number of yods in the dodecagon outside the root edge that surround the centres of each sector = 852 – 12 = 840,
yet again illustrating the basic role of the Tetrad in defining properties of sacred geometry with universal significance (as will become evident shortly). A pair of joined dodecagons therefore has (840+840=1680) yods outside their shared edge that surround the centres of their 24 sectors, where 24 = 1×2×3×4 (Fig. 10). This is the number of turns in each of the ten helical whorls (Fig. 11) of the ‘ultimate physical atom,’ or UPA (Fig. 12), observed over 100 years ago by the two Theosophists Annie Besant and C.W. Leadbeater, using a siddhi, or psychic ability, known to Indian yoga. Each whorl makes 2½ outer revolutions about the vertical axis of spin of the UPA and 2½ inner revolutions, spiralling 840 times in circles in each half. We see that each dodecagon containing 840 yods distributed outside the root edge about the centres of its sectors encodes the number of coils in half a whorl; the two identical dodecagons correspond to its inner and outer halves. The ‘Malkuth’ level of the microscopic Tree of Life, that is, each whorl of the superstring, is encoded in the tenth regular polygon and in the last of those constituting the inner form of the Tree of Life. Each one of the 1680 yods both shaping the pair of dodecagons and surrounding the centres of their 24 sectors denotes a circularly polarised oscillation or wave in a whorl. These yods represent the ‘material’ manifestation of the 240 tetractyses of the 24 higher-order tetractyses making up the pair of dodecagons. The question arises: what do these higher-order tetractyses denote? Twenty-four of them are associated with each whorl, that is, 240 higher-order tetractyses are associated with the UPA itself. The gauge symmetry group E8 describing the unified superstring force has 240 so-called ‘generators’ corresponding to the 240 so-called ‘non-zero roots of its Lie algebra.’ To each generator corresponds a kind of charge analogous to the electric charge of a particle. Each charge is the source of a gauge field, i.e., a particular kind of force. Each higher -order tetractys represents one of the 240 gauge charges, and 24 such charges are spread along each whorl, making a total of (10×24=240) for the superstring itself. As 1680 = 24×70 and the Tree of Life comprises 70 yods when its 16 triangles are turned into tetractyses (see Figure 5), this number is the number of yods in 24 separate Trees of Life. This reflects the fact that the 24 gauge charges manifesting in each whorl are all independent and ‘smeared’ along its length in a way analogous to that proposed in the E8×E8 heterotic model of the superstring. 70 is also the number of yods corresponding to Sephiroth of Construction in the higher -order tetractys making up each sector of the dodecagons, showing again that the gauge charges are wholes — complete Tree of Life entities in themselves.
The last statement should answer the following question that may have arisen in the reader’s mind during the discussion above of how the number 1680 was embodied in the pair of dodecagons: what, if any, is the significance of the seemingly arbitrary way in which the 840 yods in each dodecagon were selected — namely, picking out the 840 yods that surround centres of sectors? The yod at the centre of a tetractys denotes Malkuth, the material manifestation of the whole symbolised by the tetractys. The six yods surrounding it at the corners of a hexagon denote the six Sephiroth of Construction above Malkuth. There are 84 yods surrounding the centre of the next higher -order tetractys (see Figure 9). On the cosmic level, these correspond to the 42 subplanes of the six superphysical planes of consciousness and the 42 subplanes of their cosmic counterpart (see Article 5 for more details). On the microcosmic level, they denote the number of circularly polarised waves in a quarter of a revolution about the axis of the UPA, i.e. , a 90° turn in space. In conformity with its ten-fold nature — both in ordinary space and in 10-dimensional space-time — each whorl makes ten half-revolutions, five in an outer twisting and five in a more tightly knit, double helical twist. This 5:5 split corresponds to the division in the Tree of Life between the five uppermost Sephiroth, which span its Upper Face, and the five lowest Sephirah forming its Lower Face. The yod at the centre of a higher-order tetractys denotes the Malkuth level of manifestation of a Tree of Life system and so does not enter the count of the yods symbolising differentiations of Sephiroth beyond Malkuth. Each of the 24 gauge charges spread out along each whorl is that manifestation. What appears at first sight to be merely an ad hoc choice of yods contrived to generate the number 840 in each dodecagon is in fact a selection dictated by the proper, physical interpretation of their higher-order tetractys sectors.
Another similarity between the powers of the square and dodecagon to embody various superstring parameters like 248 and 168 is the fact that the latter number is the sum of the first 12 odd integers after 1 (Fig. 13) and that the shapes of both polygons are defined by the number 12 because a square divided into tetractyses has 12 yods along its boundary, whilst a dodecagon is delineated by its 12 corners. As the template for constructing objects possessing sacred geometry, the tetractys unveils a beautiful harmony between geometry and arithmetic that exists only in such objects.
5. Encoding of 336 in
the pair of dodecagons
Its 24 (=1×2×3×4) sectors have 24 corners. This illustrates once more how the integers 1, 2, 3, & 4 express properties of the dodecagon. As there are 22 compactified dimensions in 26-dimensional space-time, each corner of a dodecagon can be regarded as symbolising the higher dimensions of space. The ten corners outside the root edge of one dodecagon symbolise the ten curled-up dimensions generating the ten string- like components of the superstring (see Article 2). The twelve corners of the other dodecagon denote the twelve remaining compactified dimensions. These consist of the five dimensions that define a compactified space whose symmetry generates the superstring gauge group E8 and the seven curled-up dimensions predicted by supergravity theory. The centres of the two dodecagons symbolise the two transverse dimensions of 26-dimensional strings.
Notice that the division:
22 = 3 + 7 + 12
of the 22 letters of the Hebrew alphabet into the three mothers: aleph, mem & shin, the seven double
consonants: beth, gimel, daleth, caph, pe, resh, & tau, and the twelve simple consonants has a remarkable geometrical counterpart in the 22 corners of the pair of joined dodecagons. This is because the three mother letters correspond to three corners symbolising the curled-up dimensions beyond supergravity space-time that generate the three major whorls of the UPA, the seven double consonants correspond to seven corners that denote the curled-up dimensions generating its seven minor whorls and the twelve simple consonants correspond to the corners of the other dodecagon that
symbolises the five E8-generating dimensions and the seven curled-up, supergravity dimensions.
Property number 22 in the list given in Section 2 states that the number of yods in two joined dodecagons other than their 22 corners is 336 (Fig. 14), where
Starting with the Tetrad, 22= 4, the sum of the squares of the four integers 2, 6, 10 & 14 spaced four units apart is 336. It was stated in the last section that the inner and outer halves of a whorl makes 2½ revolutions. This means that its 1680 turns are spread over five revolutions, 336 turns per revolution. Each turn in a revolution of a whorl is symbolised by a yod in the two joined, Type B dodecagons other than their 22 corners and two centres, that is, new yods generated by their construction from tetractyses. The 168 such yods in each dodecagon denote the number of circularly
polarised oscillations made during the traverse of either half of one revolution of a whorl.
With their sectors turned into the next higher-order tetractys after the Pythagorean tetractys, each dodecagon was found earlier to contain 840 yods outside their root edge surrounding their centres. Each dodecagon represents half of a whorl made up of 840 coils (Fig. 15). Enfolded in each Tree of Life belonging to CTOL are the two sets of seven regular polygons. The lowest ten Trees of Life have 140 enfolded polygons, where 140 is the number value of Masloth, the Mundane Chakra of Chokmah. Their 20 dodecagons contain (10×1680=16,800) yods outside their root edges surrounding the centres of their 240 sectors representing the 240 gauge charges of E8. This Tree of Life representation of the superstring shows that the Godname ADONAI prescribes the number 16,800 because its number value 65 is the number of SLs in the lowest ten trees of CTOL. The number of corners of the 70 polygons enfolded on either side of these trees is 351, which is the number value of Ashim, the Order of Angels assigned to Malkuth. 351 is also the sum of the first 26 integers, showing how the Godname YAHWEH with number value 26 prescribes the ten overlapping Trees of Life representing the ten whorls of the superstring. Each dodecagon has ten
corners outside its root edge. The ten dodecagons enfolded in the lowest ten trees have (10×10=100) external corners. This means that the 60 polygons enfolded on either side of the ten trees that are not dodecagons have (351–100=251) corners. Article 5 (6) discussed the significance of the number 251 in relation to the superstring. The concurrence in the same context (the lowest ten trees) of this structural parameter encoded in the first six types of polygons with the number 16,800 encoded in the seventh type is remarkable evidence for the Kabbalistic basis of superstring theory and the author’s identification of the UPA as a superstring. Notice that the proportion of the dodecagons to the first six types enfolded in ten trees, namely, 10:60, corresponds in the tetractys-transformed Tree of Life to the 10:60 pattern of yods created by the ten Sephiroth and the 60 hexagonal yods. Indeed, the inner form of the Tree of Life has the same pattern, namely, the ten corners of a dodecagon outside the root edge that it shares with the other 13 polygons with 60 corners. Just as the points in space where the ten Sephiroth are located define the basic geometrical aspect of the Tree of Life — its Malkuth level — so their counterpart in its inner form — the pair of dodecagons — quantitatively embodies the physical nature of the Tree of Life in the subatomic world as measured by the 1680 coils in each helical whorl that makes up the UPA/superstring. One can only affirm the hermetic axiom: “as above, so below.”
6. Encoding of 248
& 168 in Type A & Type B dodecagons
Chakra of Hod, the Sephirah that signifies mental activity and communication. Previous articles discussed how this number is a parameter of the Tree of Life, being the number of corners of the seven separate, regular polygons making up its inner form and the number of corners, edges & triangles making up its outer form. Its superstring interpretation is as follows: as discussed earlier, each of the 24 gauge charges carried by a string component of the superstring/UPA manifests as a circularly polarised standing wave. Each such wave has two orthogonal, plane wave components that are 90º out of phase. Each whorl therefore consists of (2×24=48) independent standing plane waves. This 24:24 division manifests geometrically in the Type A dodecagon as the 24 vertices & edges on its boundary and the 24 edges and triangles inside it. The pattern appears in the first (6+6) enfolded polygons as the 24 corners intrinsic to each set. As future articles will demonstrate, it is a characteristic of any holistic system possessing sacred geometry. The ten whorls of the UPA/superstring comprise (10×48=480) plane waves. The encoding of these in the inner form of the Tree of Life is the set of 240 hexagonal yods in either set of seven separate regular polygons, i.e., their 480 hexagonal yods (Fig. 17). In the case of the first (6+6) enfolded polygons enfolded in ten Trees of Life, the two sets of 240 plane waves are the counterpart of the (240+240) corners of the (60+60) polygons of the first six types. Every hexagonal yod or corner in one set is the mirror image of its counterpart in the other set. Every such pair denotes the two orthogonal plane waves making up each of the 24 circularly polarised oscillations in each of the ten whorls of the superstring constituent of up and down
quarks paranormally described over a century ago by the Theosophists Annie Besant and C.W. Leadbeater.
The numbers of corners, edges & triangles surrounding the centres of the separate Type A and Type B dodecagons are:
There are 84 edges and 84 corners & triangles. This 84:84 division of the 168 geometrical elements in the two types of dodecagons is characteristic of holistic systems. Later articles will provide numerous examples. Its remarkable consequence is that, if we consider the ten dodecagons enfolded on one side of the central pillar of ten overlapping Trees of Life as Type A and their counterparts on the other side as Type B, the ten pairs of separate dodecagons have 840 edges and 840 corners and triangles. This is the same 840:840 division as was found for the yods surrounding the centres of the 24 sectors of two enfolded polygons when each sector is a higher-order tetractys. It manifests physically as the 840 circularly polarised waves in each half of a whorl of the E8×E8 heterotic superstring. Here is clear evidence that the superstring’s oscillatory form, as described by Besant & Leadbeater, conforms to the geometry of the pair of dodecagons in the inner Tree of Life. The Type A dodecagon has 48 elements surrounding its centre and the Type B has 120 elements surrounding its centre. The counterpart of this 48:120 division in a Type B dodecagon with 168 yods other than the 13 corners of its 12 sectors is the 48 yods at corners & centres of tetractyses and the 120 hexagonal yods on their 60 edges (Fig. 18). Its counterpart in the 168 yods outside
the root edge on the sides of the first (6+6) enfolded polygons is their 48 corners and the 120 hexagonal yods on their 60 sides. Equivalent holistic structures always display analogous patterns.