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#14 The fourfold nature of the superstring structural parameters 336, 672, 1680 & 2016
The 2ndorder tetractys is the next stage of differentiation of the 1storder tetractys, the Pythagorean representation of holistic systems. It has 85 yods, where
85 = 4^{0} + 4^{1} + 4^{2} + 4^{3}.
Below its apex are 84 yods, where
84 = 4^{1} + 4^{2} + 4^{3} = 1^{2} + 3^{2} + 5^{2} + 7^{2}.
A crossshaped array of four triangles turned into 2ndorder tetractyses has (4×84=336) yods surrounding their shared apices, where
336 = 4^{2} + 4^{3} + 4^{4} = 2^{2}(1^{2} + 3^{2} + 5^{2} + 7^{2}) = 2^{2} + 6^{2} + 10^{2} + 14^{2},
i.e., 336 is the sum of the squares of the first four even integers that start with 2 and are four units apart. This array is one of the variants of the cross pattée that was used by the Knights Templars on their tunics and shields:

The Knights Templar cross. Triangular arms with straight sides are a variant of this cross pattée. 
It has a connection to Freemasonry that goes beyond its historical connections with the Knights Templars. To discover this, first note that, if each triangular arm is turned into a 1storder tetractys, there are (4×9 =36) yods surrounding their point of connection, where 36 is the gematria number value of ELOHA, the Godname of Geburah, the fifth Sephirah. This is also the number of yods that line the three sides of a 2ndorder tetractys. The attributes of this Sephirah, whose meaning is "severity," are mostly martial in character, although this connotation must be understood in its proper, metaphysical sense, abstracted from the realm of war and soldiers, because it refers to its positive virtues of justice, correction and mental discrimination (what can be thought of as "sorting out"). In the Golden Dawn branch of modern Kabbalah, a 'Magical Image' is associated with each Sephirah. The Magical Image for Geburah is "a mighty warrior in his chariot." It seems appropriate that the Godname of this Sephirah should prescribe the symbol used by the Knights Templars, who protected Christian pilgrims travelling to faraway sites of worship. The number of yods lining the four 1storder tetractyses is 33, a number that is known in Freemasonry as its highest degree of initiation:

As they all lie on its boundary, the 33 yods are the least number of yods that are required to create the shape of the cross pattée. The sacred significance of this number is not that it is generally believed to be the age of Jesus when he died — there can be nothing remarkable or sacred about a number that was someone's age, whoever that person might have been! Rather, it has a sacredgeometrical significance in being the smallest number that quantifies through the template of the tetractys the shape of this sacredgeometrical object, which embodies a number of universal significance because 336 is the number of circularly polarised oscillations in a whorl of both the subquark and shadow matter particle, as deduced earlier in this section. The sacredness of this figure has nothing to do with its being one of the many types of cross associated with one of the world's religions. It has all to do with its archetypal character, embodying the same number as the yod or geometrical compositions of the first four Platonic solids, the inner Tree of Life, the 3d Sri Yantra, the 24cell, the disdyakis triacontahedron and the 4_{21} polytope. As a geometrical figure, the cross pattée is composed of 21 points & lines, where 21 is the number value of EHYEH ("I am"), the Godname of Kether. This illustrates the prescriptive power of this Godname in determining the shape of a cross whose arms are triangles, the cross being the simplest geometrical symbol of the Tetrad (the number 4), other than the square. The 2ndorder tetractys consists of 15 points, 30 lines & 16 triangles, i.e., 61 geometrical elements. The number of geometrical elements in the cross pattée composed of four 2ndorder tetractyses = 1 + 4×60 = 1 + 240. In other words, 240 points, lines & triangles surround its centre. This number is also a structural parameter of holistic systems, e.g.,
As the 2ndorder tetractys contains 60 hexagonal yods other than centres of 1storder tetracyses, the cross pattée contains 240 hexagonal yods that symbolise the six Sephiroth of Construction above Malkuth, which is symbolised by these centres. The cross pattée embodies not only the structural parameter 336 of E_{8}×E_{8}′ heterotic superstrings but also their groupdynamical parameter 240, this being the number of roots/gauge charges of E_{8}/E_{8}.′ Replacing each of the 336 yods that surround the centre of the cross pattée by the number 10 generates the number 3360. This is the number of circularly polarised waves in one revolution of the 10 whorls of the UPA. As 3360 = 336×2×5, it is also the number of plane waves that compose one revolution of the five helical whorls of the shadow matter particle. Like the number 336, it is a structural parameter of both basic states of the E_{8}×E_{8}′ heterotic superstring because, according to the micropsi observations of Ronald Cowen (see earlier pages), the number of whorls in the UPA is twice that of its shadow matter counterpart.
Shown below are various ways in which the number 4 (Tetrad) determines structural parameters of the whorls in the subquark and shadow matter particle:
a. The number of yods surrounding the centre of the Type C ngon = 42n. Surrounding the centre
of the Type C square are (42×4=168) yods. 168 is the number value of
Cholem Yesodoth, the Mundane Chakra of Malkuth. As the symbol of the four Elements, the
square embodies the number of circularly polarised oscillations in half a revolution of each whorl of the subquark
superstring. Weighted with the Decad, the 168 yods generate the number 1680, which is the
number of circularly polarised oscillations in each whorl of the UPA;
b. Weighted with the Tetrad, the 168 yods surrounding the centre of the Type C square
generate the number 168×4 = 672. This is the number of plane wave oscillations in one revolution
of each whorl in either the UPA or the shadow matter particle;
c. The Type C triangle has (3×42=126) yods surrounding its centre. This is also the number of yods outside one
corner of the triangle. Surrounding the centre of the cross pattée whose arms are Type C triangles are (4×126=504)
yods. Weighted with the Tetrad, they generate the number 4×504 = 2016. This is the number of circularly polarised
oscillations in each whorl of the shadow matter particle.
The sum (S_{n}) of the squares of the first n integers is
S_{n} = n(n+1)(2n+1)/6.
Therefore,
S_{15} ≡ 1^{2} + 2^{2} + 3^{2} + 4^{2} + 5^{2} + 6^{2} + 7^{2} + 8^{2} + 9^{2} + 10^{2} + 11^{2} + 12^{2} + 13^{2} + 14^{2} + 15^{2} = 1240.
This sum can be represented as a triangular array of the integers 115. A cross pattée whose arms are these arrays consisting of 120 integers generates the number 4960 as the sum of these 480 integers:
4960 =  
The number value 15 of YAH (יה), the shorter Godname of Chokmah, determines the dimension 496 of E_{8}×E_{8}′. 
The superstring significance of this is that 496 is the crucial dimension of E_{8}×E_{8}′ and SO(32), the two symmetry groups that allow 10dimensional, heterotic superstrings to have interactions that are free of quantum anomalies. As each gauge charge is coupled to its own 10d vector gauge field, the number 4960 is the number of spacetime components of the 496 gauge fields of these two symmetry groups. Remarkably, the number 496 is the gematria number value of Malkuth (מלכות), "Kingdom," signifying the physical universe composed of particles and their forces. It is, of course, not due to chance that it takes this value. As part of the arcane wisdom of the Perennial Philosophy, Kabbalah describes what is, not what a scientific theory like superstring theory proposes may exist. The mathematical Kabbalah is not the pet idea of some wellknown theoretical physicist, nor is it the doctrine of a learned rabbi. It is not even a theory. Instead, it is the interplay of mathematics within a metaphysical context.
There are (1+2+3+...+15=120) integers making up each arm of this cross pattée representation of 4960, where
3 

5  7  
120 = 11^{2} − 1 = 
9 
11 
13 

15  17  19  21. 
There are (4×120=480) integers in the cross, where
12 

20  28  
480 = 4×120 = 
36 
44 
52 

60  68  76  84. 
This number is the number of roots of E_{8}×E_{8}′, each group having 240 such roots. If we associate one pair of arms with E_{8} and the other pair with E_{8}′ because of the analogy between the 240 integers of each pair and the 240 roots of each group, the sum:
240 = 120 + 120,
which is the number of integers in a pair of arms, has its counterpart in the fact that the 240 vertices of the 4_{21} polytope representing the 240 roots of E_{8} or E_{8}′ have as their H_{4} Coxeter plane projection the (120+120=240) vertices of two 600cells, one inside the other (see diagram on page 10). It is as though the cross pattée of (240+240=480) integers is expressing the 480 nonzero roots of E_{8}×E_{8}′ and their representations by the vertices of four 600cells. The 240 integers in each pair of arms add up to 2480. This is the number of spacetime components of the 248 10d gauge fields associated with each group. The number 248 is the gematria number value of Raziel, the Archangel associated with Chokmah. The number values of both the Godname and the Archangel of this Sephirah are present in the cross pattée. In fact, the number value 140 of Masloth, the Mundane Chakra of Chokmah, is also present when its arms are 2ndorder tetractyses. The number value 73 of Chokmah is present as well because there are 140 yods lining its arms and surrounding its centre when each arm is a Type A triangle with 19 yods, so that the number of yods in four Type A triangles linked at one corner = 1 + 4×18 = 73:
The gematria number value 73 of Chokmah is the number of yods needed to create a cross pattée whose arms are Type A triangles.  There are 73 yods in a hexagram
built from tetractyses. 
There are 73 yods up to Chokmah in the 1tree. 
The Type A dodecagon has 73 yods. 
Notice that the letter values of the word "Chokmah" (חכמה), meaning "wisdom," are the numbers of various classes of yods in the cross pattée. A hexagram constructed from tetractyses also has 73 yods, as has the Type A dodecagon. The number value of Chokmah is determined by the position of this Sephirah in the 1tree (as is also the number value 67 of Binah), for there are, counting from the bottom of the 1tree, 73 yods up to Chokmah and 67 yods below Binah. It is implausible that such matching properties could exist by chance. Instead, they demonstrate that the gematria number values of the names of the Sephiroth, their Archangels, Angelic Orders and Mundane Chakras have a basis in the geometry of the Tree of Life.
The cross pattée embodies various structural parameters of the inner Tree of Life. For example, if its arms are triangular arrays of 10 dots, i.e., the tenth triangular number, 55, becomes a representation of the number 217:
1 + 4×54 = 217.
This is the number of yods that line the 47 tetractyses making up the seven enfolded Type A polygons:
Moreover, the 41 black corners of these tetractyses are symbolised by the 41 black yods either at the centre of the cross or lining the vertical and horizontal sides of its arms. One of these corners is unique in that, as the topmost corner of the hexagon, it coincides with the lowest corner of the hexagon enfolded in the inner form of the next higher Tree of Life. This single point of contact between polygons of adjacent, overlapping Trees is denoted by the centre of the cross pattée. It is surrounded by 216 dots, where 216 = 6^{3} is the number value of Geburah, the sixth Sephirah of Construction from Malkuth. The 176 red dots in the cross pattée correspond to the 176 hexagonal yods that line the 47 tetractyses making up the inner Tree of Life. The number of yods in a Type C mgon = 42m + 1 (see Table 5, Article 65, for the case n = 3). A Type C triangle (m = 3) contains 127 yods, of which 16 are corners of tetractyses, leaving 111 hexagonal yods. A cross pattée whose arms are Type C triangles has (4×111=444) hexagonal yods. This is the number of hexagonal yods in the (7+7) enfolded polygons of the inner Tree of Life. The cross pattée embodies both its structural parameters 217 and 444.
In conclusion, the cross pattée embodies 1. the structural parameter 336 of both the subquarks bound together in up and down quarks and the shadow matter particle, 2. the structural parameter 2016 of the latter, 3. the holistic parameter 240 (e.g., the number of roots of E_{8}/E_{8}′) and 4. the dimension 496 of E_{8}×E_{8}′ (as well as, of course, SO(32)). The real reason for regarding as sacred this religious symbol, still found on the flags of some countries, is not its connection to Christianity but its archetypal character in embodying various structural parameters of superstrings and dynamical parameters of the symmetry group describing their unified forces — numbers of cosmic (not just human) significance that express part of the mathematical content of God's blueprint. Transcending the historical origins of the symbol, this reason should be acceptable to adherents of all religions, not to those of just one. That the cross pattée is crossshaped is irrelevant from this universal perspective because its four arms symbolise not the cross of Calvary but the Pythagorean Tetrad, which determines and expresses basic cosmic parameters, including those that relate to the structure of superstrings. As Article 1 discusses in detail, this sacredgeometrical figure is but one of the geometrical expressions of the Tetrad Principle that was at the heart of Pythagorean number mysticism. The neoPythagorean Nichomachus said that the Pythagoreans called the number 4 "the greatest miracle," "a God after another manner," "a manifold divinity," the "fountain of Nature," and its "key bearer." True sacredness necessarily transcends all historical, religious contexts, which merely convey the illusion of this quality without revealing its essence. Truely sacred symbols do not represent events in human history or aspects of religious dogma, for they can be sacred for this reason only to those who believe them; instead, they express the mathematical nature of God, whose understanding requires not faith but reason and intuition.