The 7 separate Type B polygons & the Type B dodecagon embody the superstring structural parameter 336

Seven separate Type B polygons

The seven separate types of polygons of the inner Tree of Life have 48 sectors. When they are Type B polygons, they are composed of (3×48=144) triangles with 96 corners & 240 sides, i.e., 336 corners & sides, surrounding their centres. The triangle, square & pentagon have 36 triangles with 84 corners & sides surrounding their centres, as does the dodecagon, and the hexagon, octagon & decagon have 72 triangles with 168 corners & sides surrounding their centres. There are (84+84=168) corners & sides surrounding the centres of the triangle, square, pentagon & dodecagon. As 48 is not a factor of 168, the 336 corners & sides cannot divide equally into two sets of 168 corners & sides that each belong to all the polygons. Instead, this division follows from the unique split of the 48 corners/sectors of the seven polygons into the 24 corners/sectors of the triangle, square, pentagon & dodecagon and the 24 corners/sectors of the hexagon, octagon & decagon. The 84:84 division of the number 168 is characteristic of holistic systems for which it is a defining parameter.

Two joined Type B dodecagons
The number of yods in a Type B n-gon = 15n + 1. A dodecagon (n=12) has 181 yods, of which 12 yods are its corners. As two hexagonal yods lie on the root edge, it shares one of the 169 other yods with the second dodecagon joined to it as part of the (7+7) enfolded polygons of the inner Tree of Life. Hence, the two joined, Type B dodecagons have 336 yods other than their corners, 168 yods being associated with each one.

The (168+168=336) yods other than corners making up the two joined Type B dodecagons correspond to the (168+168) corners & sides of the 144 triangles in the seven separate Type B polygons. The counterpart of the 84:84 division of corners & sides in the dodecagon and the first three polygons is the set of 84 yods in each dodecagon that consist of its centre, the pairs of hexagonal yods on sides of its 12 sectors and the red hexagonal yods at the centres of the three tetractyses in each sector. What better evidence is there of the two joined dodecagons being the single polygonal counterpart of the inner Tree of Life?

Amazingly, just as (168+168) corners & sides surround the centres of the seven separate Type B polygons, so, too, the (7+7) enfolded Type A polygons contain (168+168) geometrical elements outside their root edge that are unshared with the outer Tree of Life.

This is proved as follows: the table shows that the 47 triangular sectors of the seven enfolded polygons have 41 corners & 88 sides. Of the 176 geometrical elements, three green corners (top & bottom of hexagon, outer corner of triangle) coincide with corners of triangles belonging to the outer Tree of Life, whilst the two green vertical sides of sectors in the hexagon are sides of triangles in the outer Tree. Therefore, five corners are either shared between the inner & outer forms of the Tree of Life or are endpoints of the root edge, whilst two sides of the 47 sectors are shared. The number of unshared corners outside the root edge = 41 − 5 = 36. The number of unshared sides outside the root edge = 88 − 2 − 1 = 85. The number of corners, sides & triangles in the seven enfolded polygons = 36 + 85 + 47 = 168. Both sets of seven enfolded polygons have (168+168=336) geometrical elements outside the root edge that are unshared with the outer Tree of Life.

#25 discusses the significance of the number 176 in the context of the subquark state of the E₈×E₈ heterotic superstring.

A Type B n-gon has 10n corners, sides & triangles surrounding its centre. The seven separate Type B polygons with 48 corners have (∑10n = 10∑n = 10×48 = 480) corners, sides & triangles surrounding their centres. The triangle, square, pentagon & dodecagon (let us call this set "S") have 24 sectors that comprise 240 geometrical elements surrounding their centres, as do the hexagon, octagon & decagon (let us call it "S′"). 72 triangles are added to the 168 corners & sides in each of these two sets of polygons. The division:

240 = 72 + 168

is characteristic of holistic systems (see The holistic pattern). Its group-theoretical interpretation is the fact that the superstring symmetry group E₈×E₈′ has (240+240=480) roots, the 240 roots in E₈ or E₈′ consisting of the 72 roots of E₆, the rank-6 exceptional subgroup, and 168 other roots. The following possible correspondences exist:

72 triangles in S (or S′) ↔ 72 roots of E₆ (or E₆′);

168 corners & sides surrounding centres of polygons in S (or S′) ↔ 168 roots of E₈ (or E₈′) not belonging to E₆ (or E₆′).

What is remarkable is that the geometrical compositions of S and S′ are analogous to not only the root composition of E₈×E₈′ but also the composition of at least one of the exceptional subgroups of E₈ that have been widely discussed by string theorists as a possible stage in its breakdown into the symmetries U(1)×SU(2)×SU(3) of the Standard Model of particle physics. The conclusion is inescapable that this correspondence exists simply because E₈×E₈′ heterotic superstrings conform to the cosmic blueprint of the inner Tree of Life. Notice that the 72 triangles in S or S′ consist of 24 sets of 3 (the three triangles in each of the 24 sectors) or, alternatively, three sets of 24. Notice also that the 168 corners & sides consist of 24 sets of seven (the two corners & five sides per sector) or, alternatively, seven sets of 24 corners & sides. The 240 geometrical elements in S or S′ can, therefore, be regarded as divided into ten groups of 24. Each group corresponds to one of the ten classes of geometrical elements that make up a sector when it is a Type B triangle:

1. corner of polygon;
2. side of polygon;
3. internal side of sector;
4. internal corner of 3 triangles in sector;
5. internal side of triangle;
6. internal side of triangle;
7. internal side of triangle;
8. triangle;
9. triangle;
10. triangle.

and which are repeated 24 times over the 24 sectors of S or S′. This ten-fold division of the basic elements of a sacred geometry is discussed at length in Article 53. It is the sacred-geometrical basis of the ten whorls of the UPA/subquark superstring, along each of which are spread 24 gauge charges of E₈. The (3×24=72) gauge charges carried by its three major whorls are those associated with the 72 roots of E₆, and the (7×24=168) gauge charges carried by its seven minor whorls are those associated with the remaining 168 roots of E₈. The following two questions arise:

1. Does E₈ correspond to S and E₈′ correspond to S′ (as stated above without justification)? Or does E₈ correspond to S′ and E₈′ correspond to S?

2. What three classes of geometrical elements correspond to the three major whorls and what seven classes of geometrical elements correspond to the seven minor whorls?

We shall consider the second question first. We were forced to associate the 72 triangles in either set of polygons with the 72 roots of E₆ because we had already counted 168 corners & sides of the 24 sectors in each set. However, the 10 geometrical elements per sector can be divided into any group of three elements and any group of seven elements. In the absence of an a priori reason for a specific choice, we are at liberty to choose not three triangles but a different group to associate with the 72 roots. This ambiguity, of course, is unsatisfactory. We really do need such a reason in order to remove it. The following intuitive/metaphysical consideration may not satisfy the mathematician but it does at least provide one: we know that the group of three geometrical elements are associated with the three major whorls of the UPA which, because the ten whorls are associated with the ten Sephiroth, are in turn associated with the Supernal Triad of Kether, Chokmah & Binah, whilst the group of seven elements are associated with the seven minor whorls, which are in turn associated with the seven Sephiroth of Construction. What three of the 10 geometrical elements per sector of a Type B polygon play the same defining role as the Supernal Triad? Clearly, it is the corner, side & one internal side of that sector, for these determine the shape of a polygon divided into its sectors. They exist whether the polygon is Type A or Type B, so they have a more basic function in shaping the polygons of the inner Tree of Life than the remaining seven geometrical elements added by regarding a sector as a Type A triangle. A choice is therefore possible that makes intuitive sense and which removes ambiguity of association of the 10 geometrical elements per sector with the ten whorls. Classes 1, 2 & 3 must be associated with the 72 roots of E₆ and classes 4–10 must be associated with the 168 remaining roots of E₈. The correspondences between classes of geometrical elements, roots, whorls & Sephiroth are set out below for either S or S′:

24 corners of polygons ↔ 24 roots of E₆ ↔ 1st major whorl ↔ Kether;

24 sides of polygons ↔ 24 roots of E₆ ↔ 2nd major whorl ↔ Chokmah;

24 internal sides of sectors ↔ 24 roots of E₆ ↔ 3rd major whorl ↔ Binah;

24 internal sides of triangles ↔ 24 roots of E₈ ↔ 1st minor whorl ↔ Chesed;

24 internal sides of triangles ↔ 24 roots of E₈ ↔ 2nd minor whorl ↔ Geburah;

24 internal sides of triangles ↔ 24 roots of E₈ ↔ 3rd minor whorl ↔ Tiphareth;

24 triangles ↔ 24 roots of E₈ ↔ 4th minor whorl ↔ Netzach;

24 triangles ↔ 24 roots of E₈ ↔ 5th minor whorl ↔ Hod;

24 triangles ↔ 24 roots of E₈ ↔ 6th minor whorl ↔ Yesod;

24 internal corners of triangles in sectors ↔ 24 roots of E₈ ↔ 7th minor whorl ↔ Malkuth.

Notice that the triplets of internal sides and the triplets of triangles in each sector correspond to minor whorls that correspond to the two triads of Sephiroth of Construction: Chesed-Geburah-Tiphareth and Netzach-Hod-Yesod Notice also that the corner shared by all three triangles inside each sector (the only one of the three types of geometrical elements: point, line & triangle, that occurs once in a given sector) corresponds to the last minor whorl, which is associated with Malkuth, the last Sephirah of the Tree of Life that stands on its own, geometrically speaking. In this way, the chosen associations reflect the 3:3:3:1 pattern of Sephiroth in the Tree of Life.

Returning to the first question, it seems more correct to associate the 240 geometrical elements surrounding the centres of the four polygons in S with the 240 roots of E₈ rather than E₈′. The reason for this is as follows: in terms of the formal analogy between the seven Sephiroth of Construction and the seven types of polygons in the inner Tree of Life, the dodecagon corresponds to Malkuth and the triangle, square & pentagon correspond to Netzach, Hod & Yesod, so that S corresponds to the lowest four Sephiroth of Construction, making S′ correspond to the triad of Chesed, Geburah & Tiphareth. In traditional Kabbalah, the last four Sephirah are associated with the four Elements of Earth, Water, Air & Fire, whilst Tiphareth is associated with Aether and Chesed and Geburah are associated with the last two of the seven Elements. If we associate ordinary matter described by E₈ with the four physical Elements and invisible shadow matter described by E₈′ with the "superphysical" Elements, then E₈ has to be associated with S and E₈′ has to be associated with S′. Does this mean that the basic spin-½ particles of shadow matter have 10 whorls? Not necessarily. The 10 classes of geometrical elements per sector consist of five corners & triangles and five sides, so that the 240 geometrical elements surrounding the centres of the three polygons in S′ can be grouped not as 10 groups of 24 elements but as five groups of 48 elements:

1. 24 corners & 24 sides of polygons;
2. 24 internal corners of triangles & 24 internal sides of sectors;
3. 24 internal sides of triangles & 24 triangles;
4. 24 internal sides of triangles & 24 triangles;
5. 24 internal sides of triangles & 24 triangles.

Such a grouping suggests that the basic spin-½ particles of shadow matter that correspond to UPAs (the superstrings making up ordinary matter paranormally described by Besant & Leadbeater), have five (not 10) whorls. The question then arises: why should the geometrical elements in S′ rather than in S be regarded as forming five groups? In other words, why do E₈×E₈ heterotic superstrings of ordinary matter have ten whorls instead of the five whorls making up their shadow matter counterparts? When the 48 sectors of the seven polygons are tetractyses, their 240 hexagonal yods consist of five types of hexagonal yods, so that there are 48 repetitions of each type. Here, the reason for dividing the 240 hexagonal yods into five groups is clear. It reflects the fact that the five hexagonal yods per sector symbolize the five Sephiroth of Construction up to Tiphareth. But this case also allows the 240 hexagonal yods to be divided into five groups of hexagonal yods belonging to the 24 sectors in S and five groups belonging to the 24 sectors in S′, i.e., it permits them to be regarded as 10 groups of 24 hexagonal yods. The difference between the two possible divisions amounts to either ignoring the separation of the seven polygons into S and S′ or taking it into account, namely, the division: 48 = 24 + 24 that is characteristic of holistic systems, for example, the 480 hexagonal yods in the two sets of seven separate polygons that make up the complete inner Tree of Life. Ignoring this division is tantamount to ignoring the essential difference between the two sets of polygons, which is one of being mirror images of each other, so that it amounts to ignoring the degree of freedom created by the chiral nature of each set. It reminds us of how the 48 symmetries of the full octahedral group O_h consist of the 24 proper rotational symmetries (the identity transformation plus 23 rotations) and 24 rotations followed by inversion. The tetrahedral group T_d of order 24 is the point group of symmetries of the tetrahedron, including the inversion operation. It has a pure rotation subgroup T of order 12 (they are the 12 ways that a tetrahedron can be rotated so as to preserve its orientation). The octahedral and tetrahedral symmetry groups are part of the description of holistic systems, and we expect the division: 24 = 12 + 12 found in such systems to manifest in both S and S′. Well, they do for S because its four polygons have 3, 4, 5 & 12 sectors and 3 + 4 + 5 = 12. But they do not for S′ at a polygonal level because its three polygons have 6, 8 & 10 sectors and none of these numbers, or combination thereof, equals 12. The only way S′ can display both holistic divisions is for its 240 geometrical elements to form five groups of 48 elements, each group consisting of two sets of 24 that belong to all three polygons. The polygons themselves cannot be sorted into two sets, each of which has 12 sectors. Instead, each set of 24 geometrical elements divides up into 12 elements and their mirror-image counterparts in the sectors on the opposite side of each polygon. This is the simple reason why, given that S′ must be associated with E₈′ for the reason given above, the unit of shadow matter must have five (not 10) whorls, each whorl being a closed curve that carries 48 gauge charges of E₈′ that correspond to each of the five groups of 48 geometrical elements listed above. However, the 24 geometrical elements of a given type in S do not all divide into 12 elements and their inversions, reflected across the centre of each polygon, because the triangle and the pentagon are not their own mirror images. Consistency of interpretation of the (12+12) division of geometrical elements for both S and S′ is not possible. This means that we do have to separate the dodecagon from the triangle, square & pentagon in order to reproduce the 12:12 division in S — there is no way to split the triangle and the pentagon in half. Whilst the 10 sets of 24 elements in S′ can be regarded as five sets of 48 elements, each set being two groups of 12 pairs of an element and its counterpart in the opposite sector, the 10 sets of 24 elements in S can only be considered as divided further into 10 sets of 12 in the triangle, square & pentagon and 10 sets of 12 in the dodecagon. The 24:24 and 12:12 divisions appear in both S and S′, but they have to be in different ways.

In the case of the 240 geometrical elements of S associated with the 240 roots of E₈, the 72 corners & sides of the 24 sectors of the four polygons that surround their centres correspond to the 72 gauge charges carried by the three major whorls; each type of geometrical element corresponds to a whorl. In the case of S′, however, group 1 consists of the corners & sides of the three polygons (what correspond in S to Kether & Chokmah) but group 2 consists of internal sides of sectors and internal corners of the triangles making up sectors. In other words, the geometrical elements in group 2 correspond to what in S are Binah and Chesed. This mixing of Sephirothic correspondences implies that none of the five whorls can be associated with a unique Sephirah. Rather, they are associated with a pair of Sephiroth. This implies that the major/minor distinction that Besant & Leadbeater made for the whorls in the UPA (see here) does not exist for the whorls in the basic unit of shadow matter. None of its whorls should be augmented relative to the others.

Confirmation
The analysis presented above makes two predictions:

1. the superstrings of shadow matter are composed of five whorls;

2. none of the whorls are augmented relative to the others.

During 18–28 June, 1992, the author carried out a series of preliminary investigations of the micro-psi faculty claimed by a Canadian Buddhist. As well as being able to magnify UPAs, he also claimed to have examined with micro-psi his own bio-energetic fields, including his chakras. This subtle matter is composed of what he calls "consciousness units." Their nature and structure is too complex to be described here. All that needs to be reported in this context is that his vision reveals these units to be composed ultimately of particles that are similar to the UPA except that they are many orders of magnitude smaller and contain only five whorls, none of which are thicker than the others. Each particle is slightly flatter but contains the same characteristic twists and turns and overall geometry of the larger UPA. It, too, is ultimately composed of strings of what Annie Besant and C.W. Leadbeater called "bubbles in koilon" (see here). Although similarly shaped like a ring doughnut (torus), they differ from the bubbles in koilon making up UPAs by being much smaller and more "solid" (this, obviously, has to be understood in its intuitive sense). Below are compared the UPA of ordinary matter, as depicted in the third edition of Occult Chemistry by Besant & Leadbeater, and the basic constituent of bio-energetic fields, as drawn by the Canadian clairvoyant:

The two chiral types of UPA (from Occult Chemistry, 3rd ed., 1951)	Micro-psi depiction of the five whorls of the basic particle making up human bio-energetic fields (private communication).

As with the UPA, the whorls twist approximately twice around the axis of symmetry. But, crucially, they number five, not 10, according to the clairvoyant, and none appear to his micro-psi vision to be thicker than the others. This confirms the two predictions. He established the number of whorls a couple of years after the investigations were carried out (there was no follow-up to them). The author had no reason to believe that shadow matter superstrings had five whorls and so could not have influenced the clairvoyant's observations. However, the author has known about this particular observation for about 18 years and the skeptic may feel inclined to argue that the confirmation lacks significance by suggesting that the above analysis was slanted in order to confirm them. But proposing hypotheses and theories to explain already established, experimental facts is a normal part of scientific methodology. The analysis presented above is simply that. Therefore, such criticism is misconceived. The crucial facts that the analysis uses are

that there are 10 geometrical elements per sector of a Type B polygon and that they naturally divide into either sets of three & seven or two sets of five. There are no other divisions that are as natural, for the set of three refers to the corners & sides of sectors of Type A polygons that surround their centres and the set of seven refers to the extra seven geometrical elements per sector needed to turn a Type A polygon into a Type B polygon. So there is nothing ad hoc about this proposed geometrical correlation between the seven polygons and the whorl structures of the E₈×E₈′ heterotic superstrings that make up ordinary matter and shadow matter, according to two people who have claimed to exercize micro-psi vision. Indeed, the same 5×48 pattern appears in the seven separate Type A polygons because their 48 sectors have five hexagonal yods per sector when the latter is a tetractys:

Here is the explicit inner Tree of Life basis for the 5-fold distribution of the 240 E₈′ gauge charges implied in what has been identified as the shadow matter superstring. See also #30 & #33 for how the 240 hexagonal yods in the seven Type A polygons correspond to the 240 roots of E₈ and to the 10 whorls of the UPA. See Article 53 for the proof that five sacred geometries display a 10-fold distribution of their 240 structural elements. The way in which the dodecahedron, the Platonic solid that the ancient Greeks associated with Aether, the fifth Element, embodies the 5-fold constituent of bio-energy fields is explained here.

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