1
Table 1. Number values of the ten Sephiroth in the four Worlds.
2
248 of
the superstring constituent of up and down quarks (the latter embodiment was also discussed
in Article 1 (2)). Although the relevance of these numbers to the physics of the universe
was, of course, unknown to the early Pythagoreans, it illustrates in a remarkable way their
profound intuition about the fundamental importance of the number 4 to the study of the
natural world. This principle, which the author has called the “Tetrad Principle,” was
formally postulated in Article 1. But the number 10 was also central to Pythagorean
mathematics because it was symbolised as the fourth triangular number by the
tetractys: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 dodecagon, too, embodies numbers of universal (and therefore scientific)
significance. This article discusses how the dodecagon encodes the numbers
______________________ 3
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:
4
5
168 yods other
than corners surrounding its centre (Fig. 6). In other words, 168 new yods are needed to
transform its sectors into tetractyses. Compare this with what was found for the square in
Article 9 (5): the Type B square with three tetractyses as each sector has 60 yods
surrounding its centre, whilst the Type C square with nine tetractyses as each sector has 168 yods surrounding its centre. Polygons of
Type A, B, C, etc represent successive levels of complexity in their construction from
tetractyses. What is so remarkable and significant in the context of the special emphasis
given by the Pythagoreans to the Decad and to the Tetrad symbolised by the square is that
both the square and the tenth regular polygon embody the same pair of
numbers, although differently. 168 is just the number of extra yods
required to turn the twelve sectors of a dodecagon into tetractyses. In the case of the
outer form of the Tree of Life, there are 60 extra yods needed to construct it from
tetractyses. The dodecagon bears to the first six polygons the same relation as Malkuth
bears to the six higher Sephiroth of Construction. This is suggested by the fact that it
contains as many hexagonal yods as the Tree of Life — only its skeletal (Malkuth) boundary
is different. It is confirmed by the fact that there are
155 hexagonal yods associated with each of the two joined, Type B
dodecagons (Fig. 7), whilst it has 168 yods other than corners of
its sectors, where 155 is the number value of ADONAI MELEKH, the
Godname of Malkuth, and 168 is the number value of Cholem
Yesodeth, the Mundane Chakra of this Sephirah. EL, the Godname of Chesed, prescribes
the pair of joined dodecagons because they contain
(155+155=310=31×10) hexagonal yods, where
31 is its number value.
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 6
contain 260 (= 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 showing how the Tetrad determines this number. The number of yods in each sector is 7
8
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= where and 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 E
9 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 Another similarity between the powers of the square and dodecagon to embody
various superstring parameters like
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 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 10
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 E 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 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
11
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 (
12 13
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= 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 14
the root edge on the sides of the first (6+6) enfolded polygons is their
2. Phillips, Stephen M. Article 1: “The Pythagorean nature of superstring and bosonic string theories,” (WEB, PDF), p. 4. 3. The number of yods in a polygon with n corners is: N = 6n + 1 (Type A); N = 15n + 1 (Type B). A Type A dodecagon (n=12) has
73 yods. A Type B dodecagon has 181 yods.4. Formulae for a polygon with n corners:
5. Ref. 1. 15 |