#2 The E₈-singlet counterpart of the E₈′-singlet state of the E₈×E₈′ heterotic superstring

Theosophical beginnings
Annie Besant and C.W. Leadbeater, two early leaders of the Theosophical Society, claimed to have developed clairvoyant abilities to "see" the human aura, the etheric body, etc. Such was the output and originality of their writings about their discoveries that even the hardened sceptic of the paranormal would have difficulty in dismissing lightly their work as either fraudulent (the cheap option, for which no evidence exists) or self-induced hallucinations, which is a suggestion that stretches the context of this word far beyond the limits of its normal, medical usage, making it a catch-all term that scientists often use when they need an excuse to ignore anything that their field of specialization, or even science in general, cannot explain. However, the issue of whether the micro-psi observations of Besant & Leadbeater were genuine and objective in some sense must be separated from the question of whether they accurately understood the "physics" of what they were seeing. Sceptics of their work have often failed to do this, deluding themselves that, if they can manage to discredit their interpretations or prove that their conclusions contradict scientific facts, they automatically discredit their claim to have remote-viewed subatomic particles. This, of course, is a non sequitur, for all their arguments disprove is that the understanding of Besant & Leadbeater concerning what they were seeing was at fault, not that their accounts were either hallucinatory or fabricated. The proper scientific evaluation of their work must distinguish between those details that are free of their interpretations and descriptions that are subtly embroidered in a language expressing their metaphysical beliefs. It needs to be recognised that the two Theosophists were locked into a way of thinking about matter that was based upon their mistaken belief that the invisible kind of physical matter revealed by their clairvoyance that they called "etheric matter" was composed ultimately of the particles ("ultimate physical atoms," or UPAs) that their micro-psi vision indicated make up as well all physical atoms of the chemical elements (see here). Besant & Leadbeater were guided (it would be more accurate to say "misled") by fashionable, contemporary, scientific notions and conjectures about the existence of a cosmic aether or ether whose vibrations were the electromagnetic waves theoretically established in 1861-1862 by James Clerk Maxwell, the great Scottish, theoretical physicist, and demonstrated to exist by Heinrich Hertz, a German physicist, a decade or so before Besant & Leadbeater started their micro-psi investigations of the chemical elements. They wanted to establish in the same scientific spirit the existence of etheric matter, which their Theosophical writings helped to conflate with both the ancient concept of the aether as the fifth Element and Victorian science's demand 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 to make "occult" notions more scientific in order to be more acceptable to science led them, soon after they commenced their investigation in 1895, to believe that the atoms conceived by physicists and chemists were composed of aggregates of indivisible UPAs. Removed from atoms themselves (the notion of "atomic nuclei" had still not been conceived then), these groups of UPAs created four grades of more rarified "etheric matter," the least dense of which was a plasma state of free UPAs — or so Besant & Leadbeater conceived, for this must now be seen as merely a conjecture on their part, based upon extrapolation from what they had seen on the subatomic scale of their micro-psi vision, when they broke up what they thought were atoms into progressisvely less complex bound states of UPAs. As we shall discover, they made two mistakes:

• They misidentified as atoms the atom-like units they claimed to see with micro-psi vision;
• They wrongly believed that atoms consist of particles of etheric matter, the two kinds of matter differing solely in the complexity and density of the packing of UPAs into numerous types of bound states.

All this seemed at the time to have been revealed by the two Theosophists' purported, psychokinetic ability to dissolve the bonds between the particles inside the atoms that they thought they were observing with their micro-psi vision. This would immediately allow the particles either to fly apart permanently or to regroup into less complex, bound states of UPAs. Whether this actually happened or not, it was certainly what their observations on their face value seemed to indicate. Starting with what they thought were atoms, Besant & Leadbeater believed that there could be only four stages in their disintegration because Theosophy taught that the plane of physical consciousness has seven levels, or subplanes, namely, the three lowest, so-called "dense physical" 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. These etheric subplanes were said to be four in number mainly because of the teachings of Paracelsus who, in his book A Book on Nymphs, Sylphs, Pygmies, and Salamanders, and on the Other Spirits, associated a category of elemental beings with each of the four classical Elements: nymph (Water), sylph (Air), gnome (Earth; Paracelsus used the term "pygmy") and salamander (Fire). As each subplane was thought to be the domain of one type of elemental, there had to be four grades of etheric matter that make up their bodies. As students of many occult traditions, Besant & Leadbeater developed and integrated whenever they could the ideas that underpinned these teachings. In their clairvoyant investigations into matter, they believed that they were transferring the pieces of their broken-up atoms from the dense physical subplane to each etheric subplane in succession until the ever-simpler fragments finally released free UPAs on the seventh subplane, which is the last of the four etheric subplanes belonging to the physical plane. When Leadbeater examined the structure of the UPA, he honestly thought that he was looking at the basic unit of what — mainly through his influence — Theosophy came to call "etheric matter" because both this and what it called "dense physical matter" ultimately consisted of this particle — or so Besant & Leadbeater thought. For the two Theosophists, the distinction between the two kinds of matter was merely one of complexity of the bound states of UPAs that comprised them; it was not one of fundamental differences between their basic constituents. Leadbeater in particular had no reason (or so he would have argued) to check whether the UPA making up atoms of the elements really was (as he thought) the fundamental unit of etheric matter. This would have been a simple matter of focussing with his "magnifying clairvoyance" (as he called it) on a sample of what he regarded as etheric matter — such as his own subtle body (what Theosophists call the "etheric double") — and checking whether its particles looked different from UPAs. For him, they were all the same type of particle because his metaphysical preconceptions told him that they had to be. As far as Leadbeater was concerned, there would have been no need to check the truth of what was for him correct a priori but which in reality was no more than a tacit, working assumption of his that turned out to be wrong.

Ronald D. Cowen
Ronald Cowen was a Canadian clairvoyant and an advanced practitioner of Buddhism with whom the author collaborated in the 1990s. By directly examining his own subtle body, he established that the forms of the two basic particles were different, although not radically so. Indeed, the main visual difference, which amounts to the fundamental particles of etheric matter having five, not ten whorls, could easily go unnoticed by an investigator endowed with micro-psi vision (as Cowen claimed) unless he (like Cowen) felt the need to check whether the basic particles of ordinary matter and etheric matter really were of the same type. For that purpose, in his altered state of consciousness he would have to zoom in close to the particle, steady its spinning (or, rather, "freeze" its image in his field of view) and then count its whorls. As stated earlier, Leadbeater believed that the particles had to be the same because he was wedded to his preconceived, metaphysical view that physical atoms were composed of etheric matter, although this was also likely influenced by contemporary ideas of some physicists. Consequently, he had no motive to check what actually amounted to an untested, working assumption on his part — one that had been created by his Theosophical beliefs, or more accurately, by his own understanding of ancient, metaphysical teachings about God, matter and the higher realms of consciousness. The false notion, fashionable in some areas of Victorian science, that physical matter was different states of the Aether distorted his understanding (as it did for Annie Besant) of what his micro-psi vision was showing him; it led him at times to interpret his genuine experiences in the wrong way, albeit the error was entirely natural for anyone to make prior to the eras of atomic and nuclear physics. Sceptics of the paranormal would, no doubt, argue that it was all nothing but the product of his (and Besant's) imagination, by which they really mean that the two Theosophists either unconsciously or deliberately fabricated all the micro-psi images recorded in their book Occult Chemistry. Accepting that they genuinely perceived them is not permissable to the sceptic because it would be an admission that would only raise (for him) more awkward questions, e.g., how two people could agree on so many details of their own observations of the atoms of each element if their images were nothing but hallucinations. However, statistical tests applied to their investigations of the elements easily discredit that naive view (see Occult Chemistry). They demonstrate a statistically significant correlation with the (at that time, unknown) mass numbers of their nuclei that is much stronger than that with their chemical atomic weights, as measured then and listed in their book. Sceptics of Occult Chemistry like to argue that the latter data could have been used to concoct the micro-psi observations, thereby missing the crucial point that, if Besant & Leadbeater had done this, the correlation with contemporary values of atomic weights would have been stronger (not weaker) than with mass numbers ascertained several decades later by scientists! This very telling feature of their results invalidates the sceptics' alternative explanation of data being fabricated. It is strong evidence for the objective nature of Besant's & Leadbeater's micro-psi observations. Their more accurate correlation with the number of protons and neutrons in atomic nuclei could never have been fabricated, because the notion of atomic nuclei had not been conceived between 1895 and 1908, when most of their work was done (indeed, these particles had not been discovered then, so that the mass numbers of nuclides had not yet been ascertained). Sceptics are also unable to account for the two Theosophists' many anticipations of later scientific discoveries, such as the existence of isotopes (most notably, the neon-22 isotope), certain rare elements, quarks inside atoms and their string-like bonds, etc. Indeed, so unable are they to account for these puzzling anomalies without admitting the reality of micro-psi that they ignore them altogether — as though there were nothing left of the supposed hoax to debunk, when in truth they are hiding their inability to do so! Nevertheless, such features remained baffling to those honest minds who did not have some ideological axe to grind against either Theosophy or the paranormal that would distort their attempts to analyse the observations in an unbiassed way. These features were worthy enough to demand better, more scientific explanation, and this is what the author started to provide in 1979 and has endeavoured to continue doing ever since.

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 some 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 whether

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 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.

the mental images that Cowen supposedly 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 by applying electric and magnetic fields to the specimen that he was examining, thereby indicating that they had to refer to objective, physical reality. 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 analysed, they were found to be statistically significant (p<10⁻⁴). It was also checked whether Cowen's observations 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 by the author 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 to be examining — clues that in principle could have helped him (consciously or unconsciously) to fabricate details that matched scientific facts. 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 during two weeks' of testing was that his micro-psi ability was genuine. The double-blind nature of these tests precluded the possibility of telepathic reception of relevant information from the mind of the author supervising them, whilst the samples that were examined by Cowen with his micro-psi powers provided no visual clues concerning the identity of the element — the paint-covered glass capsules made them totally opaque so that neither person could even know by inspection whether they contained a gas or a solid. Cowen was not even allowed to touch the capsules so as to prevent him from deducing anything by feeling the weight of the capsule in his hand.

Here is an extract from one of Cowen's reports concerning a particle that he had noticed in his own vital/etheric body, which he claimed that he could see clairvoyantly:

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 similar. But the differed from UPAs in various respects. One of these was that they consisted not of ten helical whorls but of five such whorls (he verified this property in close-up observations that followed those quoted above). Each whorl 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 queried, 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 some point but returns to where it started. Therefore, as he was sure that a whole number of revolutions is 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, say, 2.9 times or some other number slightly less than 3. It must be exactly either two times or three times. As — by his own admission —it is nearer to three than to two, the number of revolutions in the core has to be exactly three if his observation that the outer spiral made three complete revolutions is correct (Cowen gave no reason for us to doubt that because he stated unambiguously that it did). His remarks imply that the basic constituent of the matter of the vital/etheric body comprises five whorls, each making six complete revolutions about its axis, i.e., 30 revolutions in total (15 in their outer spirals, 15 in their core). This contrasts with the UPA of ordinary matter, which Besant & Leadbeater said is composed of 10 whorls, each spiralling five times around its axis of rotation, to make a total of 50 revolutions.

Why did Cowen say "almost three times" and not just "three times"? The question is not pointless, for, as any topologist will confirm, the mathematical 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. More specifically, some of the theoretically unexplored physics of E₈′-singlet states of E₈×E₈′ heterotic superstrings might hinge upon such numbers, although we are not claiming that superstrings have to be torus knots. It is, therefore, vital that any reported information about string-like whorls be correctly established from a parapsychological, theory-neutral perspective. Cowen has now died, so he cannot be asked to repeat these micro-psi observations in order 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 some theory or preconceived belief but, merely, to make logical sense of what he said or wrote by eliminating all its possible interpretations that do not make sense. There is a simple answer to the question posed above. Cowen had read Occult Chemistry by the time he had made most of the observations recorded in his various publications, and he knew that its text states 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. Perhaps, because of the greater difficulty in counting revolutions of a more tightly twisting curve, he was unsure whether the correct number was three. Perhaps in the back of his mind was the thought that it ought to be 2½ times in order to match what he knew Besant & Leadbeater had reported for the UPA, 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, however, it could not be between two and three times, because this would not make mathematical sense for a continuous, closed curve, although he did not realise it at the time. There is only one possible conclusion: Cowen should have said that each whorl made three complete revolutions in the core of the UPA, not almost three revolutions. This is not putting words in someone's mouth that he did not intend. Cowen used the word "almost" because he was uncertain whether the number of revolutions was exactly three — not because he was sure that it was less than three! He was always guided by what he saw during the given circumstances, not by what he expected to see or what some theory told him to expect to see. This correction is merely one of extracting the only possible sense from what Cowen reported concerning a feature of the particle that is of prime concern to its interpretation. Its implication is that each whorl revolves six times around the axis of the particle — three times in an outer spiral and three times within its core, whereas each whorl of the UPA spirals five times around its axis — 2½ times in its outer spiral and 2½ times in its inner one. This increase in the number of revolutions by a factor of 6/5 turns out to be highly significant for reasons to be discussed shortly.

According to the table of number weights discussed on page 17 of Superstrings as sacred geometry/Disdyakis triacontahedron, 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 also (12×60=720) corners & sides in its faces that surround the axis. This makes a total of 2400 geometrical elements. This number is the number of components of the 10-d gauge fields associated with the 240 roots/gauge charges of E₈, the rank-8, exceptional Lie group, whilst 720 is the number of gauge field components associated with the 72 roots/gauge charges of its exceptional subgroup E₆:

240 = 72 + 168.

Listed below are the different types of the 240 roots as 8-tuples defined in the 8-d linear vector root space of E₈:

We found on page 17 linked to above 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 (its so-called "positive" and "negative" versions have whorls with the same number of turns). The UPA is the lightest 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 gauge charges of E₈ are spread 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 = 7×240×10 = 7(72+168)×10 = 7×3×24×10 + 7×168×10 = 3×7×240 + 7×1680 = 3×1680 + 7×1680.

The first term expresses the three major whorls, each with 1680 turns, whose differentiation from the seven minor whorls (see here for details) is the manifestation of symmetry-breakdown from E₈ to E₆, one of its exceptional subgroups, whose 72 gauge charges are spread out along (70×72=5040) circular turns of these three heical whorls, 24 gauge charges per whorl (hence the factorization: 72 = 3×24 that appears in the equation above).

For the 144 Polyhedron, the second polyhedron in the Polyhedral Tree of Life, V = 72 = (6/5)×60. In fact, every corresponding combination of geometrical elements in the two polyhedra is in the ratio 6/5. We have seen on page 17 at the link given above that this augmentation factor 6/5 reflects the fact that the 144 Polyhedron represents the "branches" of the Polyhedral Tree of Life, whilst the disdyakis triacontahedron represents its trunk. This is because 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 suppose that, just as the number weight 28 generates a number (28×60=1680) 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 the same weight generates a number (28×72=2016) for the 144 Polyhedron which is the number of turns in a whorl of the E₈-singlet state of the E₈×E₈′ heterotic superstring. In other words, if the UPA of ordinary matter is the specific microscopic realisation of the trunk of the Tree of Life blueprint, perhaps the basic particle of (let us call it "shadow matter" instead of Leadbeater's inaccurate term "etheric matter") is the realisation of its branches. Given that the trunk and branches of the Tree of Life and their polyhedral counterparts display the same proportion of 6/5 as the ratio 2016/1680, it seems a plausible conjecture whose implications need exploring in a search for evidence that confirms it. If it really is the case that the shadow matter particle constitutes the branches of the microscopic Tree of Life, the number of turns in a whorl of this particle = 28×72 = 28×(6/5)×60 = (6/5)×1680 = 6×336 = 12×168 = 2016, which compares with 10×168 = 5×336 for a whorl of the UPA/subquark. This reveals the 6:5 ratio of the number of turns in the whorl of each type of superstring. The question then arises: how many whorls does the particle 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 minimally 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 = 6×1680. In Kabbalah, the vital body is associated with Yesod, the sixth Sephirah of Construction. The five whorls of its basic building block described by Cowen constitute the branches of the microscopic Tree of Life, as embodied in the 144 Polyhedron, whereas the 10 whorls of the UPA described by Besant & Leadbeater form its trunk, as expressed by the disdyakis triacontahedron.

As a collection of geometrical elements, the branches of the outer Tree of Life contain none of its 10 Sephirothic points that are corners of its 16 triangles but which belong to its trunk made up of a point, line, triangle & tetrahedron, the geometrical representation of the integers 1, 2, 3 & 4. The three major whorls correspond to the Supernal Triad and the seven minor whorls correspond to the seven Sephiroth of Construction. None of the five whorls of the shadow matter UPA is a major whorl in the sense used by Leadbeater & Besant of appearing "thicker" than the others. This is because these five whorls are the source of the gauge fields of E₈′, the symmetry break-down of which does not mirror that of E₈, which breaks down into its exceptional subgroup E₆ (which is what is responsible for the augmentation of the three major whorls carrying the 72 gauge charges of E₆ relative to the seven minor whorls, which carry the remaining 168 gauge charges). 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), a 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 each 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). The boundary and internal yods are analogous to the membrane of the cell and the DNA in its nucleus. The disdyakis triacontahedron encodes the formative aspect of the Polyhedral Tree of Life, the core, or trunk, of the Tree of Life — what makes up its outer form, realised in the subatomic world as the subquarks making up the protons and neutrons in atomic nuclei. On the other hand, the 144 Polyhedron encodes the life-supporting, or vital, aspect, just as the leaves in the branches of a tree provide through photosynthesis fuel for its growth, etc. The two polyhedra encapsulate the two complementary principles of matter and the lifeforce. The invisible unit of shadow matter is insensitive to every force operating in the material universe except gravity. According to Cowen (see his book "The Path of Love"), all energy transactions of this particle within the vital body generate physical consciousness; their termination ends physical awareness, but not consciousness in general. The hardware of the human computer is the body of flesh and bone. Its operating system is the vital body.

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, or 168 turns per half-revolution. Each whorl of the basic particle 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:

This factor arises because the basic particles of ordinary matter and shadow matter represent the microscopic manifestation of, respectively, the trunk and the branches of the Tree of Life; the former contains 10 straight lines ("Paths"), whilst the latter have 12 lines, and 12/10 = 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 basic particle 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 Pythagorean Tetrad (4) expresses the structural parameter that is common to both kinds of particle 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 in this infinite array, it is surrounded by nine number weights that line a tetractys and add up to 14N:

This is because any number weight N is the centre of a tetractys array of 10 number weights that are N/6 times their counterpart in the Lambda Tetractys (see here). As any musical 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 in the tetractys add up to (14×24 =336):

 (4+8+16+32+108) + 12 + 36 + 48 + 72 = 168 + 168 = 336.

Purely as a number, 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 of the Pythagorean scale, where 33 = 1! + 2! + 3! + 4!, and 10 = 1 + 2 + 3 + 4. Notice that the nine musical weights consist of two combinations that add up to two 168s, just as 336 is the sum of the 168 turns in two half-revolutions of a whorl. The significance of this vis-à-vis the Tree of Life is that in the Pythagorean scale there are 22 inharmonic partials up to the tenth overtone, just as there are 22 Paths connecting the 10 Sephiroth:

The 32 overtones and partials up to the perfect fifth of the fifth octave constitute a Tree of Life pattern because the ten overtones are the musical counterparts of the ten Sephiroth of the outer Tree of Life and the 22 partials are the counterparts of the 22 Paths that connect the Sephiroth. The sequential assignment of the partials to the Paths follows the traditional numbering (11-32) of the 22 Paths.

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 on page 17 of Superstrings as sacred geometry/Disdyakis triacontahedron. 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 in the general Lambda Tetractys with N at its centre. 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, involving additive numbers whose meaning lies entirely outside the context of music. The generalised Lambda Tetractys transcends the context for which Plato intended it in Timaeus, his treatise on Pythagorean cosomology! 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 this is a musical number weight, being also the second note (D) of the seventh octave in the scale of C major. The sum of the nine weights in the Lambda Tetractys 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, that n is a number weight). Every overtone of the Pythagorean scale 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 two sides forming the shape of 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.

As there are 672 yods in the first four Platonic solids (tetrahedron, octahedron, cube & icosahedron) when they are constructed from tetractyses (see here), 336 yods are required to build a set of their halves. The counterpart of this in the basic particles of ordinary and shadow matter is the 168 turns in half a revolution of a whorl. As we shall discuss in the next page, a turn of a helical whorl is a circularly polarised wave composed of two plane waves that oscillate in perpendicular directions 90° out of phase with each other. This means that a half-revolution of a whorl comprises 168 pairs of plane waves, i.e., 336 plane waves. It is how the first four Platonic solids, which the ancient Greeks thought were the shapes of the particles of the four physical Elements Fire, Air, Earth & Water, embody the superstring structural parameter 336. It does not need to apply for all excited states of a superstring; we claim only that it applies to the basic superstring constituents of ordinary matter and shadow matter in their ground states. That is why this number always appears in the structure of various sacred geometries.

Evidence connecting the structural parameter 2016 to the four etheric subplanes/Trees of Life
Apart from a theory of particles and forces proposed in 1979 by the author, neither the UPA described by Besant & Leadbeater nor its hypothesized E₈′ counterpart analysed by Ron Cowen has been explained by any model or theory that has appeared in the research journals of particle physics. This is hardly surprising, given that the author identified the UPA as the as-yet undiscovered constituent of quarks, an idea which few physicists have explored. One can at present only theorize about these particles and search within various sacred geometries for evidence to support such speculations. 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 isomorphic blueprints of what exists in nature at a fundamental level, we should expect their mathematical properties to be quantified by not only the same numbers but also the numbers revealed by their micro-psi observation. The pages of this website have amply proved this expectation in the case of Besant & Leadbeater, and now we are beginning to discover that the same matchings of numbers occur for Cowen's observations. Surely, that counts as 'evidence' even though it would not win Nobel Prizes? If the same number appears in so many different contexts as to make this happening by chance extremely improbable (as is the case here), what right does anyone have to ignore all this prima facie evidence of a universal pattern of parameters for holistic systems on the grounds that it is not buried within the computer readouts of a multi-billion dollar particle accelerator? Here is another example of the appearance of these numbers that sceptics cannot dismiss convincingly as due to chance. It has not been cherry-picked in any way, for the reader will see that the context in which the number 2016 appears matches exactly the meaning of this number.

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 wholly intermingles with it 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); this is the as yet undiscovered "theory of everything" that unifies supergravity theories and the five types of superstrings. The two basic classes of superstring matter occupy the same 9-d space (three large-scale dimensions, six compactified) but different points along a tenth dimension of space, which cannot be traversed by them, so that they can interact only gravitationally because the gauge fields transmitting their forces are also 10-d superstrings and therefore cannot travel along this extra dimension. 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 to what has been called the "third eye." 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 11-d 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 of the Tree 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 that are often referred to in Theosophical literature. It is proposed here that they are the two universes, or space-time sheets required by E₈×E₈′ heterotic superstring theory, 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 that given in accounts of the history of music in the West. They are the arithmetic expression of the archetypal patterns and divisions within sacred geometries. And not only these, for precisely the same numbers and their patterns appear in a certain world-famous, megalithic representation of God, as will be reported here and elsewhere later this year.

 In the section Plato's Lambda, the general, 3-dimensional version of the Lambda Tetractys was described. In the cosmological account presented by Plato, the Demiurge marked out along a strip of some cosmic substance sections of lengths given by the successive terms of the two geometric series:

1, 2, 4, 8, ....
1, 3, 9, 27,...

He then bent the strip at the starting point of His measurement, creating the shape of the ancient Greek letter lambda (Λ). This "Lambda" formation of seven numbers is but two sides of a tetractys array of 10 positive integers that can be selected from an infinite lattice of numbers, or musical weights, whose ratios determine the tone ratios of the notes of the Pythagorean musical scale. The general form of such a tetractys whose central number weight is N was given earlier. This tetractys is but the first face of a tetrahedral array of 20 weights selected from an infinite, 3-dimensional lattice of such weights. Called the "Tetrahedral Lambda," its four faces are tetractyses of number weights with N at the centre of the 1st face and 4N at the centre of the 4th face:

The general Tetrahedral Lambda. The colour of the text identifying each face is that of its edges.

#15 discusses the way in the Godnames assigned to the Sephiroth of the Tree of Life prescribe the properties of this archetypal object. The sum of the 20 number weights making up the Tetrahedral Lambda is 350N/6. This can be written as 50×(7N/6), where 50 is the gematria number value of ELOHIM, the Godname assigned to Binah, the third member of the Supernal Triad. Is there a number weight N such that this sum is 16800 — the number of spirillae in the UPA? Yes, it is 288. In order of increasing size, it is the 28th weight in the list of number weights. It is the central weight in the following 1st face of the Tetrahedral Lambda whose apex is 48:

The sum of the nine weights surrounding it = 14×288 = 4032 = 2×2016. As we shall see in the next page, this is the number of plane waves making up each whorl of the shadow matter particle described by Ron Cowen. So the 1st face of the Tetrahedral Lambda with 288 at its centre generates a major structural parameter of the E₈-singlet state of E₈×E₈′ heterotic superstrings! Even more remarkable, the complete Tetrahedral Lambda generates the number 50×7×288/6 = 50×7×48 = 50×336 = 16800, which is the number of turns in the 50 revolutions of the 10 whorls of the UPA, each revolution comprising 336 turns. This will be be discussed on page 5 and related to the inner form of the 7-tree mapping the physical plane, or space-time continuum.

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