A conjecture concerning the E₈′ counterpart of the E₈-singlet state of the E₈×E₈′ heterotic superstring

Annie Besant and C.W. Leadbeater were locked into a way of thinking about matter that was based upon their mistaken belief that the invisible kind of physical matter they called "etheric matter" was composed ultimately of the particles ("ultimate physical atoms," or UPAs) that their micro-psi vision indicated make up all atoms. They were guided (it would be more accurate to say "misguided") by fashionable, contemporary, scientific notions and conjectures about the existence of a cosmic aether whose vibrations were the electromagnetic waves theoretically established in 1861-1862 by James Clerk Maxwell and demonstrated to exist by Heinrich Hertz a decade or so before Besant & Leadbeater started their micro-psi investigations. As a result, they were eager to establish the existence of etheric matter, which their writings helped to conflate with both the ancient concept of the aether as the fifth Element and Victorian science's necessity for a medium to carry light (and then Hertz's radio) waves, before Albert Einstein disposed of the need in 1905 in his Special Theory of Relativity. This eagerness led them to believe in 1895 that the atoms studied by physicists and chemists were wholly composed of aggregates of indivisible UPAs that — removed from atoms themselves (the notion of "atomic nuclei" had still not been conceived then) — created four grades of "etheric matter,", the least dense of which was a plasma state of free UPAs — or so Besant & Leadbeater conceived. All this seemed to have been revealed by the two Theosophists' purported ability to dissolve the bonds between the particles that they were examining in their field of micro-psi vision. This would immediately allow the particles either to fly apart or to regroup into less complex bound states of UPAs. Starting with what they thought were atoms, Besant & Leadbeater thought that there could be only four stages in their disintegration because Theosophy taught that the plane of physical consciousness has seven subplanes, namely, the three lowest subplanes of life occupied by mineral, vegetable and animal life forms (including humans) and the four highest subplanes, where life forms (so-called "elementals") exist that are invisible to everyone except those with clairvoyant sight. Hence, Besant & Leadbeater believed that they were transferring the pieces of their broken-up atoms to each etheric subplane in turn until the ever smaller fragments finally released free UPAs on the seventh subplane. When Leadbeater examined the structure of the UPA, he thought that he was looking at the basic unit of what Theosophy called "etheric matter" because both this and "dense physical" matter contained them; the distinction between the two kinds of matter was merely one of complexity of the bound states of UPAs. He had no reason (or so he would have argued) to check whether the UPA really was the unit of etheric matter by focussing with his "magnifying clairvoyance" (as he called it) on a sample of what he regarded as etheric matter — such as his own vital body — and seeing whether its basic particles differed in appearance from the UPAs he has seen in the atoms of the chemical elements. It was finally established by Ronald Cowen, the Canadian Buddhist clairvoyant with whom the author collaborated in the 1990s, that, indeed, they were different, although not radically so. However, the visual difference (essentially, amounting to the basic particles of etheric matter having five, not ten whorls) was one that could easily have been missed unless one wanted to check whether the basic particles of ordinary matter and etheric matter at least looked like they were one and the same; as we have stated, Leadbeater believed that they were because he was wedded to the view that physical atoms were composed of etheric matter, and he therefore had no motive to check what amounted to an untested assumption on his part, one that had been created by his Theosophical beliefs — or, rather, by his own understanding of ancient, metaphysical teachings about God, matter and the planes of consciousness.

The diagram below compares the picture of the UPA published in 1951 in the third edition of Occult Chemistry with a sketch, taken from page 141 of Cowen's book "The Path of Love" (FriesenPress, 2015), of the basic building blocks of the invisible (but still physical) matter of the vital body, which Hindus call the "pranamayakosher" and Theosophists call the "etheric double." Cowen spend years clairvoyantly examining the highly complex, microscopic organisation of the vital body and sharing his observations with the author. This culminated in two weeks of intensive investigations in which the author carried out a series of blind and double-blind trials to test

The basic constituent of atomic nuclei is the UPA described by Besant & Leadbeater. As the E8′-singlet, subquark state of the E8×E8′ heterotic superstring, it comprises 10 helices, or "whorls," each with 1680 circular turns, that wind 2½ times around its axis in an outer spiral and then return to the top by making 2½ twists in a much narrower spiral.

According to the Canadian Buddhist clairvoyant, Ron Cowen, the basic unit of the subtle matter of living forms consists of five helices. He never tried to count how many turns each helix has. Unlike those in the UPA of ordinary matter, the five whorls appear to be of the same thickness/brightness and do not separate into two sets of three and seven strands, as the UPA of ordinary matter does, but remain together, following parallel tracks as they twist three times in an outer spiral and then three times in a tighter, inner spiral.

whether the mental images that Cowen experienced whilst he was in a controlled, altered state of consciousness induced by advanced, Buddhist meditative techniques could be independently manipulated by the author without Cowen's knowledge, thereby indicating their objective nature. Tests were made to determine success or failure of observations electrically pre-determined at random by the author behind a screen. When the results were statistically analysed, they were found to be significant (p<10⁻⁴). It was also checked whether they matched scientific facts about various pure samples of chemical elements previously prepared in vacuum-sealed, glass capsules by a professor of chemistry at Manchester University and presented to Cowen in a double-blind way so as to prevent any possibility of either he or the author knowing what element his micro-psi vision was supposed tobe examining. The author's conclusion was that Cowen passed these tests successfully and that the only plausible explanation for the degree of success that he had demonstrated was that his micro-psi ability was genuine.

Here is an extract from one of Cowen's reports concerning a UPA that he had noticed in his own vital body, which he could see, being clairvoyant:

The reason for the author italicizing certain crucial statements in this extract will become apparent shortly. The assertion that needs to be emphasized is that, despite the complexity of organisation of the radically different kind of matter within the vital body revealed by his micro-psi vision, Cowen claimed that its basic particles all looked like the same type. They consisted of five helical whorls (he verified this property in close-up observations that followed those quoted above). Each one spirals three times around its core, then (to quote Cowen) "almost three times" again as it moves up through the latter towards its starting point. However, his last statement must be questioned, for the total number of revolutions made by a whorl in its outer and inner spiralling has to be an integer because it is a continuous curve that does not abruptly stop at any point, but, as he attributes with certainty a whole number of revolutions being made in its outer spiralling, the number it completes in its twisting through the core of the UPA must also be an integer. It cannot be "almost three times," as though it were a fractional number of times. Instead, it must be either exactly two times or three times. As — by his own observation —it was nearer to three than to two, the number of revolutions in the core had to have been exactly three if his observation that the outer spiral made three complete revolutions is correct (there is no logical reason to doubt that). It means, therefore, that Cowen's remarks imply that the basic particle of the matter of the vital body is composed of five whorls, each of which makes six revolutions about its axis, i.e., 30 revolutions in total. This contrasts with the UPA of ordinary matter, which Besant & Leadbeater said is composed of 10 whorls, each making five revolutions about its axis, i.e., a total of 50 revolutions.

Why did Cowen say "almost three times" and not just "three times"? The question is not trivial, because, as any topologist will confirm, the properties of a torus knot depend upon how many times it twists around a torus as well as how many revolutions it makes around the axis of the torus during these twists. Some of the theoretically unexplored physics of E₈′-singlet states of E₈×E₈′ heterotic superstrings might hinge upon this number, so it is vital that it is accurately ascertained from the parapsychological perspective. Cowen has now died, so these micro-psi observations cannot be repeated to check their accuracy. If he misspoke or erred in an observation, we have to infer what he should have said — not so that it might confirm a preconceived belief by others but merely to make logical sense of what was said. There is a simple answer to the question posed above. Cowen had read Occult Chemistry by the time he made most of the observations recorded in his various publications, and he knew that the whorls of the UPA of ordinary matter spiral 2½ times around its axis in their outer motion and 2½ times in their twisting around its narrow core, for that it what its text states. Perhaps because of the greater difficulty in counting revolutions in a tightly twisting curve, he was unsure whether the correct number was two or three. Perhaps the thought that it ought to be 2½ times was in the back of his mind and this made him hide his uncertainty and play safe by saying that each whorl revolved almost three times. Obviously, it did not revolve two times because his stated impression is closer to three times than to twice. At the same time, it could not be between two and three times, because this would not make mathematical sense, although he did not realise this at the time. There is therefore only one possible conclusion, namely, that Cowen should have said that each whorl made three revolutions in the core of the UPA. This is not putting words in someone's mouth that he did not mean. It is merely extracting the only possible sense out of what Cowen reported in his book.

According to the table of number weights discussed on page 17, the number weight for the number of corners, sides & triangles in a polyhedron with (2+V) vertices, 3V edges & 2V triangular faces is 40. The number weight for corners & sides in faces is 12, so that the number weight for geometrical elements other than corners & sides in faces = 40 − 12  = 28. For the disdyakis triacontahedron, one of the two polyhedra in the Polyhedral Tree of Life, V = 60 and it has (28×60=1680) such geometrical elements surrounding its axis, there being (12×60=720) corners & sides in its faces that surround the axis. This makes a total of 2400 geometrical elements. These numbers are the number of components of the 10-d gauge fields of E₈ and its exceptional subgroup E₆ of order 72:

240 = 72 + 168

We found earlier that the orders of the exceptional subgroups of E₆ are (apart from the common factor of 10) numbers of various combinations of geometrical elements surrounding the axis of the disdyakis triacontahedron. The number 1680 is the number of circular turns in a helical whorl of the UPA of ordinary matter. The UPA is the least massive, subquark state of the E₈′-singlet state of the E₈×E₈′ heterotic superstring. Its 10 whorls have 16800 circular turns representing circularly polarised oscillations. The 240 E₈ gauge charges are smeared along the lengths of the 10 whorls, 24 to a whorl, so that one E₈ gauge charge corresponds to 16800/240 = 70 turns/oscillations. Therefore,

16800 = 7×2400 = 7×720 + 7×1680 = 7×3×240 + 7×1680 = 3×1680 + 7×1680.

The first term expresses the three major whorls, whose differentiation from the minor whorls is the manifestation of symmetry-breakdown from E₈ to E₆, whose 72 gauge charges are spread out along (70×72=5040) turns of three whorls, 24 gauge charges per whorl.

For the 144 Polyhedron, V = 72 = (6/5)×60. In fact, every corresponding combination of geometrical elements in the two polyhedra are in this ratio 6/5. We have seen in previous pages that the augmentation factor 6/5 reflects the fact that this polyhedron represents the "branches" of the Polyhedral Tree of Life, whilst the disdyakis triacontahedron represents its trunk, for the branches of the outer Tree of Life consist of 11 triangles with 12 sides, whilst its trunk has five triangles with 10 sides, where 12/10 = 6/5.  Let us now conjecture that, just as the number weight 28 generates a number for the disdyakis triacontahedron which is the number of turns in a whorl of the E₈′-singlet state of the E₈×E₈′ heterotic superstring, so it generates a number for the 144 Polyhedron which is the number of turns in a whorl of the E₈-singlet state. In other words, if the UPA of ordinary matter is the microscopic realisation of the trunk of the Tree of Life blueprint, perhaps the UPA of (let us call it "shadow matter" instead of Leadbeater's inaccurate term "etheric matter") is the realisation of its branches. If this is the case, the number of turns in a whorl of the UPA of shadow matter = 28×72 = (6/5)×60×28 = (6/5)×1680 = 6×336 = 12×168 = 2016. The question then arises: how many whorls does it have?  Suppose it has N whorls. The 240 gauge charges of E₈′ must be spread along its 2016N turns. One E₈′ gauge charge corresponds to 2016N/240 = 42N/5 turns/oscillations. As this is an integer,  N must be an integer multiple of 5, i.e., N = 5p (p = 1, 2, 3, etc). Its smallest value is 5. The UPA of shadow matter comprises five whorls, each with 2016 turns, making a total number of 10080 turns, which is the number of turns in six whorls of the UPA of ordinary matter: 10080 = 1680×6. In Kabbalah, the vital body is associated with Yesod, the sixth Sephirah of Construction. The five whorls of its basic building block constitute the branches of the microscpic Tree of Life, whereas the 10 whorls of the UPA described by Besant & Leadbeater form its trunk.

The branches of the Tree of Life contain none of the 10 Sephirothic points that are corners of the 16 triangles making up the outer Tree of Life and which belong to its trunk. The five whorls should be thought of as expressing the two sets of five Sephiroth, now paired, rather than the ten whorls corresponding to the Supernal Triad as the three major whorls and the seven Sephiroth of Construction as the seven minor whorls. This is why none of the five whorls of the shadow matter UPA are major whorls in the sense of appearing thicker than the others. An E₈′ gauge charge corresponds to 42 turns/oscillations, so that (2016/42=48) E₈′ gauge charges are spread along each whorl. Notice that, as 1680 = 7×240 = 7×(120+120), and 2016 = 7×288 = 7×(144+144), the whorl of the E₈′-singlet superstring has as many turns as there are yods lining the two sets of seven enfolded polygons that make up the inner forms of seven separate Trees of Life, whilst the whorl of the E₈-singlet superstring has as many turns as there are yods inside their polygons. This connects these two structural parameters of the two basic types of E₈×E₈′ superstring to the Tree of Life mapping of the seven subplanes of the physical plane. However, notice that the Trees are regarded as separate, not overlapping, just as the two sets of seven enfolded polygons are viewed in this context as separate, not joined (see also below).

Each whorl of the UPA of ordinary matter revolves fives times around its axis of spin; there are (1680/5=336) turns per 360° revolution, 168 turns per half-revolution. Each whorl of the UPA of shadow matter revolves six times around its axis, so that there are still (2016/6=336) turns per revolution, 168 turns per half-revolution. This property is the same in both cases because the number of turns and the number of revolutions have been increased by the same factor of 6/5:

The UPA of ordinary matter is formed by 50 revolutions of its 10 helical whorls, each revolution being made up of 336 circular turns of a helix. The UPA of shadow matter is formed by 30 revolutions of its five helical whorls, each revolution being made up of 336 turns:

1680/5 = 2016/6 = 336.

The structural parameter that is common to both kinds of UPA is expresed by the Pythagorean Tetrad (4) because 336 = 4×84, where 84 = 4¹ + 4² + 4³, so that 336 = 4² + 4³ + 4⁴. Also, 84 = 1² + 3² + 5² + 7², so that 336 = 2²(1² + 3² + 5² + 7²) = 2² + 6² + 10² + 14², i.e., 336 is the sum of the squares of the first four even integers that are four units apart. This number has a musical context that is unique in the following way. The tone ratios of the notes of the Pythagorean musical scale are the ratios of number weights that form an infinite hexagonal lattice. If one chooses any number weight N, it is surrounded by nine number weights that add up to 14N:

As any number weight is of the form 2^p3^q, where p = ±1, ±2, etc, and q = ±1, ±2, etc, any tone ratio that is the ratio of two number weights is also a number weight. Conversely, any number weight is also a tone ratio of a note in some octave. The 33rd note in the Pythagorean scale is the perfect fifth of the fifth octave. It has the tone ratio 24, where 24 = 4!, and it is the tenth overtone. Its nine nearest neighbours add up to (14×24 =336). This superstring structural parameter (it applies to both singlet states of the E₈×E₈′ heterotic superstring) is therefore associated with the 33rd note and tenth overtone, where 33 = 1! + 2! + 3! + 4!, and 10 = 1 + 2 + 3 + 4.

In music, it makes no sense to add up tone ratios or number weights; they can be only divided by one another to generate new musical intervals, all measured in terms of some fundamental frequency that has been set by convention. However, when the deeper meanings of Plato's Lambda and the Lambda Tetractys are taken into account (see Plato's Lambda),  i.e., they are seen as the purely arithmetic expression of the holistic patterns defining sacred geometries, it then becomes meaningful to add musical number weights. An example of this is provided by the polyhedron with (2+V) vertices, 3V edges & 2V triangular faces discussed in the previous page. When its external and internal triangles are tetractyses, the table of number weights given there indicates that the sum of the number weights for the yod population is 14, just as the sum of the nine number weights surrounding N is 14N, and it is made up of:

We see that, when calculating the number weights for some combination of yods, we are, effectively, adding together some of the nine number weights surrounding the central number weight N. For example, the number weight for yods on sides of tetractyses = 1 + 6 + 2 = 9, which compares with the sum (9N) of the seven numbers that form the lambda shape. Just as all musical weights are generated from the numbers 2 and 3 and their integer powers, so, too, all geometrical or yod number weights in a polyhedron with (2+V) vertices, 3V edges and 2V faces are calculated from the same two numbers and their higher powers. We can add musical number weights because — in the context of sacred geometry such as the Polyhedral Tree of Life, whose polyhedra conform to this type of polyhedron — we are dealing with a geometrical counterpart of the musical number weights, which can be additive numbers whose meaning lies entirely outside the context of music. The generalised Lambda Tetractys transcends the context for which Plato intended it! Of course, in the context of the Polyhedral Tree of Life, N = 60 for the disdyakis triacontahedron, but this is not a musical number weight. However, N = 72 for the 144 Polyhedron and it is a musical number weight. The sum of the nine weights surrounding it = 14×72 = 1008 = 3×336, which is the number of turns calculated above in the outer or inner half of each helical whorl of the UPA of shadow matter. In the case of polygons, 14N yods are needed to turn an N-gon defined by just its corners into a Type B N-gon. 5N yods are needed for a Type A N-gon, so 9N more yods are required. This 5:9 division corresponds in a general Lambda Tetractys to the two numbers 2N and 3N adding to 5N that appear on its base in the extrapolation of the seven numbers lining the lambda itself and to the sum (9N) of the latter. When we add the yods surrounding the centres of several Type B polygons with, say, a total of n corners, this is equivalent to adding the nine number weights in a tetractys surrounding the number weight n (assuming, of course, n is a number weight). Every Pythagorean overtone can be said to define a polygon with n corners whose 5n added yods, when it is Type A, is the sum of the two extrapolated number weights in the Lambda Tetractys having this note at its centre and whose 9n additional yods when the polygon is Type B equal the sum (9n) of the seven number weights that line the lambda.

The fundamental structural parameter 336 that is common to the basic units of ordinary matter and shadow matter has been discussed in this website in the context of many sacred geometries. For example, it is: 1. the number of yods that line the 42 triangles of the Sri Yantra when they are tetractyses; 2. the number of dots in a hexagram when formed from triangular arays of eight dots, and 3. the number of yods in two joined, Type B dodecagons other than their corners (this type of polygon is the last of the seven regular polygons that make up the inner form of the Tree of Life):

336 yods line the 126 sides of the 42 triangles of the 3-d Sri Yantra when they are tetractyses.	336 dots make up a hexagram constructed from triangular arrays of 8 dots.
Two joined, Type B dodecagons have 336 yods other than their corners.

Evidence connecting the structural parameter 2016 to the four etheric subplanes/Trees of Life
Neither the UPA described by Besant & Leadbeater nor its hypothesized E₈′ counterpart analysed by Ron Cowen has been the basis of any model or theory that has appeared in the research journals of particle physics. One can at present only theorize about them and search for evidence for these speculations in various sacred geometries. For some, such sources may seem a dubious way of testing the truth of ideas that as yet are unsupported by solid, scientific data. If, however, these sacred geometries, truly, are blueprints of what exists in nature at a fundamental level, we should expect their mathematical properties to be quantified by the same numbers that lie at the core of their observations. The pages of this website have amply proved this expectation in the case of Besant & Leadbeater, and now we are beginning to discover the same matchings of numbers occur for Cowen's observations. Surely, that counts as 'evidence' in the non-scientific sense of this word? Here is another example of the appearance of these numbers that sceptics cannot dismiss convincingly as due to chance.

If the whorl of the E₈′ counterpart of the subquark state of the E₈×E₈′ heterotic superstring really does have 2016 circularly polarised oscillations  as the string-like manifestation of 48 gauge charges of E₈,′  this number (or perhaps the number for all five whorls: 5×2016=10080, should appear in the context of the Trees of Life that map the very subplanes of consciousness whose forms of matter these invisible particles are supposed to constitute. It must be emphasized here that shadow matter, made up of all possible states of the E₈-singlet state of this type of heterotic superstring, occupies exactly the same 9-d space that ordinary matter does. Yet it never intermingles with it exactly because it is confined to a 10-d space-time sheet that is ever separated by a small gap from the 10-d space-time sheet occupied by all E₈′-singlet states. The finite segment extends along the 10th dimension of space required by M-theory (see picture below), which is the as yet undiscovered scheme that unifies supergravity theories and superstring theory. The two basic forms of superstring matter occupy the same 9-d space (three large-scale dimensions, six compactified) but different regions of 10-d space, which cannot be traversed by them, so that they can interact only gravitationally. Actually, there are 15 higher dimensions of space, but only certain modes of vibration of both types occur in them. What Besant & Leadbeater (and Theosophy in general) called "etheric subplanes" refer to a different part of the physical universe that cannot be traversed as though one were taking a journey. It intermingles with the material world, yet is invisible and still separate from it, requiring a shift of consciousness to make it perceptible. Just as the UPA of ordinary matter is the 'trunk' of the microscopic Tree of Life, its most material manifestation, so the basic particle of shadow matter (the Theosophists' etheric matter) is its 'branches', that is, its interactions with similar particles sustain life by feeding it energy for it to function. This is not biochemical, as with a real tree, but subtle, involving energies that are unfamiliar to physicists and chemists, although not to yogis who practise pranayama or to students of qi gong who can manipulate chi (qi), the subtle energy that permeates the vital, or etheric body, which exists solely in this disjoint region of space-time.

The (7+7) enfolded Type A polygons that make up the inner form of the Tree of Life contain 524 yods (see here). Seven black yods in each hexagon line a side pillar and

the centre of the triangle coincides with a hexagonal yod on the horizontal line (Path) that joins Chesed and Geburah. Outside the four white yods that line the root edge are (524−4−8−8=504) yods that are unshared with the outer Tree of Life. They are intrinsic to the inner form of each Tree. The 7-tree maps the seven subplanes of the physical plane. Its four uppermost Trees map the four etheric subplanes and the three lowest Trees map the three "dense physical" subplanes. They are the two universes, or space-time sheets, that contain either ordinary matter, composed of E₈′-singlet states of superstrings), and shadow matter, composed of E₈-singlet states. As 2016 = 4×504, the number of yods intrinsic to the inner form of the four highest Trees in the 7-tree is equal to the number of turns in each whorl of the basic particle that Ron Cowen described making up etheric matter. The number 504 is the number of turns in 1½ revolutions of a whorl. As CTOL has 550 SLs, the top of the 7-tree (47th SL) is the 504th SL from its top. In other words, this number determines the very point of emergence of the physical plane from the next higher plane. The Type C n-gon has 42n yods surrounding its centre. The Type C dodecagon (n=12) has 504 yods surrounding its centre:

504 yods surround the centre of the Type C dodecagon.

The total number of turns in the five helical whorls = 5×2016 = 10080. The outer/inner halves of the basic particle of shadow matter contain 5040 turns. We can generate this number by assigning the number 10 (Decad) to each yod in the Type C dodecagon. This demonstrates the designing power of 10, for the dodecagon is the tenth type of regular polygon. The number 5040 is also the number of turns in the three major whorls of the UPA: 3×1680 = 5040. It is 7! = 1×2×3×4×5×6×7, which is the order of S₇, the symmetric group of rank 7. There are 7! turns in the 15 revolutions of the outer half of the particle and 7! turns in the 15 revolutions of its inner half. As

71² − 1 = 5040 = 3 + 5 + 7 + ... + 141,

the number 5040 is the sum of the first 70 odd integers after 1. As the Tree of Life with its 16 triangles turned into tetractyses has 70 yods:

5040 = 7! =

assigning successive odd integers 3, 5, 7, etc to these yods generates the number of turns in each half of the basic particle of shadow matter. As 5040 = 72×70, the number 72 is the arithmetic mean of these 70 odd integers after 1.  It is the number of yods surrounding the centre of the Type A dodecagon and the number of vertices surrounding the axis of the 144 Polyhedron that is the polyhedral counterpart of the basic unit of shadow matter. The musical nature of the structural parameter 2016 is seen when we realise that the sum of the nine musical number weights belonging to a tetractys array of such weights with 72 (major 2nd of the seventh octave) at its centre:

is 14×72 = 1008, which is the number of turns in the outer or inner half of each whorl in the basic E₈-singlet unit of shadow matter. It is an illustration of how the number weights making up Plato's Lambda and the Lambda Tetractys have a far more profound meaning than the historical context of music. They are the arithmetic expression of the archetypal patterns and divisions within sacred geometries. And not only these, for the same numbers and patterns appear in a certain world-famous, megalithic representation of God, as will be reported later this year.

We have come full circle..... 

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