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Isomorphism between the geometrical composition of the outer & inner Trees of Life and the root composition of the heterotic superstring symmetry group E8×E8′
As was pointed out in #40, when they are Type B triangles, the 16 separate triangles making up the outer Tree of Life have 496 corners, sides & triangles, where 496 is the dimension of E8×E8 and SO(32), the only two symmetry groups describing the forces between 10-dimensional superstring forces that are free of quantum anomalies. This is because a Type B triangle is composed of 31 corners, sides & triangles, where 31 is the number value of EL ("God"), which is the Godname assigned to Chesed, the first Sephirah of Construction, and 496 = 16×31. 480 geometrical elements (240 corners & triangles, 240 sides) surround the centres of the 16 Type B triangles. These centres correspond to the 16 simple roots of the symmetry group E8×E8, whilst the 480 geometrical elements surrounding them correspond to its 480 roots (240 roots from each E8). Surrounding the centres of the 8 separate Type B triangles in each suitably defined "half" of the outer Tree of Life are 240 geometrical elements. The 8 centres correspond to the 8 simple roots of E8. We shall now show that this 8:240 division reappears in the outer and inner Trees of Life when all polygons are Type A. Moreover, their geometrical composition generates naturally the root composition of the four exceptional subgroups of E8.
Tree of Life basis of the heterotic superstring symmetry group E8×E8′
|The outer Tree of Life (עץ חיים)||
The 7 enfolded polygons in half of the inner Tree of Life. The black line is the root edge and the black dashed line is the only part of their geometry that is shared with the outer Tree of Life.
1. The outer & inner Trees of Life
The outer Tree of Life
The inner Tree of Life
Divided into their 47 sectors, they have the composition: (41,88,47), i.e., 176 geometrical elements. The top and lowest corners of the hexagon and its centre (black dots) and the two vertical, dashed black lines are shared with the outer Tree of Life as its Pillar of Mercy. The composition of the unshared part is (38,86,47). The composition of its unshared part outside its root edge (black line) is (36,85,47), i.e., 168 geometrical elements. This is how the superstring structural parameters 168/1680 determined with micro-psi by C.W. Leadbeater over a century ago is embodied in the geometry of the inner Tree of Life.
2. 496 geometrical elements make up the outer & inner Trees of Life
The number of corners, sides & triangles in a Type A n-gon = 4n + 1. (4n−2) geometrical elements are outside the side it shares with other enfolded polygons. The right-hand corner of the enfolded triangle coincides with Chesed of the outer Tree of Life and the top and bottom corners of the enfolded hexagon coincide with, respectively, Chokmah and Binah, whilst the two interior, vertical sides of its sectors coincide with the Chokmah-Chesed Path and the Chesed-Netzach Path. The table lists the geometrical elements in each of the 7 polygons outside the shared root edge that are unshared with triangles in the outer Tree of Life. The first 5 polygons have 84 geometrical elements, as do the last 2 polygons. This 84:84 division in the 168 unshared geometrical elements is characteristic of holistic systems (for example, see Article 64). The two internal sides of the hexagon shared with the outer Tree comprise 3 corners & 2 sides, whilst the root edge comprises 2 corners & one side. The composition of shared/root edge elements is (5,3,0). We found above that the composition of the 72 elements in each half of the outer Tree of Life is (13,35,24). The composition of the outer Tree of Life and 2 separate halves of the inner Tree of Life is
248 is the dimension of the rank-8 exceptional Lie group E8 and 496 is the dimension of E8×E8. This is one of the two symmetry group found in 1984 to be free of quantum anomalies, a discovery that ushered the so-called "first superstring revolution" in theoretical physics. Every one of the 496 geometrical elements in the outer and inner Trees of Life corresponds to a root of E8×E8. The 248 geometrical elements in either half of the outer Tree and its associated set of 7 enfolded polygons correspond to the 248 roots of E8. The 8 shared elements correspond to its 8 simple roots and the (72+168=240) elements correspond to its 240 roots. They have the composition (49,120,71), i.e., they comprise 120 corners & triangles and 120 sides.
The first 4 enfolded polygons have 54 unshared geometrical elements outside their root edge with the composition (9,28,17). Hence, there are (72+54=126) elements belonging either to each half of the outer Tree of Life and the first 4 enfolded polygons belonging to its associated set of 7 polygons. They correspond to the 126 roots of E7, the largest exceptional subgroup of E8 (for more details, see the discussion in (5) about E7).
The 72 geometrical elements in each half of the outer Tree of Life correspond to the 72 roots of E6, another exceptional subgroup of E8, because the 126 elements corresponding to the 126 roots of E7 contain the 72 elements, which is consistent with E6 being a subgroup of E7.
When the original (8+8=16) triangles are divided into their (24+24=48) sectors, (24+24=48) sides are added. The 48 sides & triangles in each half of the outer Tree of Life are the counterpart of the 48 roots of F4, the rank-4 exceptional subgroup of E6 (see the discussion below about F4).
3. Correspondence between the geometrical
Whilst studying and classifying the regular and semi-regular polytopes (higher-dimensional versions of the 3-d polyhedra), the English amateur mathematician Thorold Gosset (1869-1962) discovered in 1900 the 421 polytope, a semi-regular 8-polytope. The position vectors of its 240 vertices in 8-d space are the root vectors of the 240 roots of E8. Its Petrie polygon is the triacontagon, a regular polygon with 30 corners. The E8 Coxeter plane projection of the 421 polytope consists of 8 concentric triacontagons.
Each half of the outer Tree of Life comprises 8 triangles with 5 corners and 11 sides. When they are Type A, they consist of (8×3=24) sectors with (5+8=13) corners and (11 + 8×3 = 35) sides — a total of 72 geometrical elements.
The 7 enfolded Type A polygons comprise 41 corners and 88 sides of 47 sectors. Five of these 176 geometrical elements (3 corners & two sides of sectors of the hexagon) are shared with two triangles whose bases are either the left-hand or the right-hand Pillars of the outer Tree of Life. Therefore, (176−8=168) geometrical elements outside their root edge belong exclusively to each set of 7 enfolded polygons. They are intrinsic to them.
Two separate sets of 7 enfolded polygons
Two combined sets of 7 enfolded polygons
4. Correspondence between the 168 unshared
The outer Tree of Life with 16 Type A triangles consists of 48 triangles with 26 corners & 70 sides. Seventy-two of these 144 geometrical elements can be associated with each "half" of the outer Tree of Life.
Each set of 7 enfolded Type A polygons contains 176 geometrical elements. Three corners of triangles and two sides (coloured green in the diagram opposite) in each side pillar of the outer Tree of Life coincide, respectively, with the topmost, centre & bottom of a hexagon and the two internal vertical sides of its sectors. This means that 67 of the 72 geometrical elements in each half of the outer Tree are intrinsic to it, where 67 is the number value of Binah. (176−5−3=168) geometrical elements outside the root edge are intrinsic to each set of 7 enfolded polygons.
The number of geometrical elements in the separate outer and inner Trees of Life = 176 + 144 + 176 = 496. Each half has (72+168+8=248) geometrical elements. This is the dimension of E8. The 496 geometrical elements correspond to the 496 roots of E8×E8′. The 8 geometrical elements that are either shared or belong to the root edge correspond to its 8 simple roots. The 72 elements in each half of the outer Tree of Life correspond to the 72 roots of E6, the rank-6 exceptional subgroup of E8. The 168 elements that are intrinsic to each set of 7 enfolded polygons correspond to the 168 roots of E8 or E8′ that are not roots of E6 (in the latter case, it is written E6′ in the diagram). Here is unambiguous evidence that the geometrical composition of the outer & inner Trees of Life is analogous to the root composition of E8×E8 rather than to SO(32) — the only other gauge symmetry group which, because it has the crucial dimension 496, is free of quantum anomalies.
The composition of the 240 geometrical elements outside the root edge in each half of the combined outer & inner Trees of Life is (49,120,71). EL CHAI, the Godname of Yesod with number value 49, prescribes the number of corners of the 71 triangles in each half of the combination. ELOHA, the Godname of Geburah with number value 36, prescribes the 36 corners of the 47 triangles in each set of 7 enfolded polygons that are outside their root edge and unshared with the outer Tree of Life.
5. Tree of Life counterpart of the 126 roots of E7
In 7-dimensional geometry, 231 is a uniform polytope, constructed from the E7 group. Its Petrie polygon is the octodecagon. Its 126 vertices represent the 126 root vectors of the simple, exceptional Lie group E7. These 126 roots have as follows their counterparts in the outer & inner Trees of Life:
the first 4 polygons are the triangle, square, pentagon & hexagon. Article 48 analyses the holistic nature of this subset of the 7 regular polygons making up the inner form of the Tree of Life. The diagram in (2) shows that they have 54 unshared geometrical elements outside their shared root edge. Hence, each half of the outer Tree of Life and its associated set of the first 4 enfolded polygons have (72+54=126) geometrical elements. They correspond to the 126 roots of E7, the largest exceptional subgroup of E8.
6. Outer Tree of Life basis of the 72 roots of E6
The I22 polytope is a 6-d uniform polytope with 72 vertices. Its Petrie polygon is the dodecagon. Its vertices represent the 72 roots of the rank-6 exceptional group E6, which is an exceptional subgroup of E7 & E8.
Each half of the outer Tree of Life has 8 triangles with 5 corners & 11 sides. When they are Type A, their (8×3=24) sectors have (5+8=13) corners & (11 + 8×3 = 35) sides. Each half has 72 geometrical elements. They comprise (13+11=24) corners and original sides, 24 new sides & 24 triangles.
The outer Tree of Life basis of the two F4 groups in E8×E8′
7. Outer Tree of Life basis of the 48 roots of F4
E6 has a rank-4 exceptional subgroup called F4. It has 48 roots. The Coxeter plane projection of the 24 vertices of a 24-cell and the 24 vertices of its dual configuration are its 48 root vectors.
As found in (6),
the 72 geometrical elements in the 8 Type A triangles in each half of the outer
Tree of Life comprise 3 sets of 24 elements. As the 48 roots of
with 2 sets of 24 vertices of 24-cells (see diagram), we expect their geometrical counterpart in
each half of the outer Tree of Life to comprise 2 sets of 24 geometrical elements. But which pairs
of 24 corners & original sides, 24 new sides and 24 triangles? An equivalent form of this
question is: which set of 24 geometrical elements (let us call it 24 and underline all numbers
representing sets of geometrical elements) corresponds to the 24 roots in
E6 that are not roots of F4? There are 3
Just as the 480 8-dimensional root vectors of E8×E8′ have (8×480=3840=384×10) coordinates, so the 48 root vectors of F4 have (8×48=384) coordinates. E8×E8′ has (192+192=384) roots other than those in the F4 groups belonging to E8 and E8’. The appearance of the holistic parameters 192 & 384 (see here) in F4 and the fact that 48 is another holistic parameter indicate the central importance of this symmetry group for heterotic E8×E8′ superstrings as a step in the breakdown of their E8×E8′ & E8 symmetries.
8. Outer Tree of Life basis of the 12 roots of G2
1. Three of the 8 Type A triangles associated with each half of the outer Tree of Life span the central Pillar of Equilibrium. Of those associated with its right-hand side, only the two blue Type A triangles (Kether-Chokmah-Binah & Binah-Chokmah-Tiphareth are shared with the next higher, overlapping Tree of Life because they form the Upper Face, which is the Lower Face of the next higher Tree. Similarly, of those associated with its left-hand side, only the two red Type A triangles (Tiphareth-Netzach-Hod & Hod-Netzach-Malkuth) are shared with the next lower Tree of Life because they form the Lower Face, which is the Upper Face of the next lower Tree. Inside the two shared blue Type A triangles are (2×3=6) sides and (2×3=6) triangles, i.e., 12 sides & triangles. Similarly for the two red Type A triangles. We found in (7) that, for either half of the outer Tree of Life, 48 = (0,24,24). Therefore, 48 = (0,6+18,6+18) = (0,6,6) + (0,18,18). The 12 sides & triangles in the former set correspond to the 12 roots of the smallest exceptional group G2, which is a subgroup of F4. The division:
48 = 12 sides & triangles + 36 sides & triangles
is another manifestation of the 12:36 division of the
48 corners of the 7 separate polygons forming each half of the inner Tree of
Life into the 12 corners of the dodecagon and the 36 corners of the first 6
The 7 enfolded Type A polygons have 264 yods. Seven of them are their centres (coloured white). Above or below the centre of the hexagon are 3 black yods that lie on the right-hand pillar and are part of the outer Tree of Life when its 16 triangles are tetractyses. Therefore, (7+6=13) yods are either centres or shared with the outer Tree of Life. Similarly for the mirror-image set of 7 enfolded polygons on the other side of the root edge. (13+13=26) yods are either centres of the (7+7) enfolded polygons or shared with the outer Tree. The number 26 of YAHWEH determines the 498 yods surrounding the centres of the 14 polygons that are intrinsic to them. Two of them are the endpoints of the root edge, leaving 496 yods that are intrinsic to the polygons and surround their centres. 248 yods are associated with each half of the inner Tree of Life. This number is the dimension of E8. In confirmation of what has been shown in previous discussions, the symmetry group E8×E8 conforms to the inner form of the Tree of Life, its direct product character reflecting the fact that this inner form is composed of two halves, one of which is the mirror image of the other. The 13:13 division defines this equal distribution of the 496 intrinsic yods that surround centres of polygons.
The dimension 496 of E8×E8 is embodied in the combined outer & inner Trees of Life as their 496 hexagonal yods (see third diagram here).
A. The 9 primary triangles that generate the Sri Yantra have 26 corners surrounding the downward-pointing triangle at its centre (their 27th corner is the lowest corner of this triangle). The 5 downward-pointing triangles expressing the Shakti (feminine) principle have 14 white corners surrounding the central triangle. The 4 upward-pointing triangles representing the Shiva (masculine) principle have 12 black corners. The 14 corners correspond to the 14 centres of the (7+7) enfolded polygons and the 12 corners correspond to the 12 shared yods on the two side pillars of the outer Tree of Life other than those at Chesed and Geburah, which coincide with the centres of the two hexagons.
B. The 3-d Sri Yantra comprises 4 layers of triangles. The lowest layer has 14 triangles whose 14 white tips correspond to the 14 centres of the (7+7) enfolded polygons. The 3 other layers have 28 triangles, of which 12 have the black corners of the 9 parent triangles. The (7+6) white/black corners in each half of the Sri Yantra correspond to the 7 centres & 6 shared yods in each half of the inner Tree of Life.
C. The Sri Yantra can be divided into two vertical halves, each containing 13 corners of the 9 parent triangles whose intersection generates the 43 triangles. Every violet yod has its mirror-image, reflected across the vertical mirror plane symbolised by the vertical, dashed line. The 8 tips of the 4 upward-pointing triangles and the 4 downward-pointing triangles lie on this line. The upper 4 green corners on this line can be associated with the right-hand set of corners and the lower 4 violet corners on the line can be associated with the left-hand set.
Including the octave, the 7 diatonic musical scales have 13 types of rising intervals.* They have 13 falling counterparts. Therefore, there are 26 types of rising & falling intervals between their notes. Formally counting the octave as a Pythagorean note because it belongs to the Pythagorean scale, there are (7+7=14) types of rising & falling, Pythagorean intervals (written in red in the diagram opposite) and (6+6=12) types of rising & falling, non-Pythagorean intervals (written in black).** As we have pointed out in the previous discussion, this 14:12 division in the number 26 appears in the Sri Yantra as the 14 corners of the 5 downward-pointing (Shakti) triangles that surround the centre triangle and the 12 corners of the 4 upward-pointing (Shiva) triangles. The 42 triangles surrounding the central triangle have 26 points provided by the 9 parent triangles, leaving 16 other points. This 26:12 division appears in the 7 diatonic scales as the 26 Pythagorean notes and 16 non-Pythagorean notes that make up the (7×6=42) notes between their tonics and octaves. The Divine Name YAHWEH prescribes both features because the Sri Yantra and the 7 diatonic scales are analogous, holistic systems, the points of the 42 triangles in the former denoting their 42 notes. The gematria number values of the 4 letters of this Divine Name are the numbers of Pythagorean notes in various scales. Two alternative schemes to that shown are possible:
1. D & E scales → 6 (=V) and A, B, & C scales → 10 (=Y);
The depicted scheme seems the most natural because it has more letters with number values that denote numbers of Pythagorean notes in a single scale.
* The rising interval between two notes with tone ratios m & n (m>n) is m/n. Their falling interval is n/m. The tonic has tone ratio 1, so that a falling interval 1/n can be associated with every note with tone ratio n.
** A "falling Pythagorean interval" is not necessarily an actual note of the next lower octave of the Pythagorean scale. In fact, only the perfect fourth with tone ratio 4/3 and the perfect fifth with tone ratio 3/2 have this property. Proof: the tone ratios of a note (n) and its inversion (m) are related by nm = 2. Therefore, 1/n = ½×m. A note with tone ratio 1/n is, respectively, Pythagorean or non-Pythagorean only if m is Pythagorean or non-Pythagorean. The only Pythagorean notes having tone ratios whose reciprocals are tone ratios of notes in the next lower octave of the Pythagorean scale are found by inspection to be the perfect fourth and perfect fifth: 3/4 = ½×3/2 and 2/3 = ½×4/3. Therefore, a "falling Pythagorean interval" does not mean a falling interval that is a note of the next lower octave of the Pythagorean scale, It simply means the reciprocal of the tone ratio of a note belonging to this scale.
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