TOWARDS A GEOMETRY OF PPR ENTANGLEMENT

The symmetry of a tetrahedron

For what follows, a brief detour is required into tetrahedral symmetry, as demonstrated for example, by the chemistry of carbon. A regular tetrahedron (see figure 3a) is ‘a solid figure bounded by four equilateral triangular faces, with four vertices and six edges all of the same length.’⁶⁰ A tetrahedron’s epi-centre is equidistant from its four vertices, and is a centre of symmetry, as long as each of the four vertices are indistinguishable from each other (figure 3a).

Rotational axes and reflective planes of symmetry pass through this centre. These are symmetry elements such that operations involving them return the tetrahedron to a position identical to and indistinguishable from its starting position. For a regular tetrahedron, only rotation and reflection symmetry elements are required to fully define its symmetry.⁶¹

Systematically gathering together all the symmetry elements a shape has, constitutes its point group. This concept from mathematical Group Theory revolutionised the science of chemistry, by helping to elucidate much of what is now known about molecular shape and electronic structures of molecules.⁶¹ The point group of a tetrahedron is written as T_d, and perhaps the best known example is methane (figure 3a). Here, the central carbon atom is the molecule’s centre of symmetry and sits at the epi-centre of the tetrahedron, with four identical hydrogen atoms at each of the tetrahedron’s vertices.

a                                                                                b

Figure 3: a; regular methane tetrahedron.         b; chiral tetrahedra of bromochlorofluorocarbon enantiomers.

Thus, a tetrahedral molecule like methane (figure 3a) may be imagined as having four rotational axes coincident with each of the C-H bonds joining the tetrahedron’s vertices to the centre, and going through the middle of the opposite ‘face’. Rotation of the molecule around one of these axes by 120^o makes it indistinguishable from its starting configuration.

The methane molecule’s regular tetrahedral shape also gives rise to six reflective planes of symmetry going through each of its six edges, including the central carbon atom. Reflection of the tetrahedron in any one of these planes provides a mirror image identical with and indistinguishable from the original.

The symmetry of a tetrahedron, however, undergoes drastic change if its four corners become distinguishable from each other. This is best understood using the chemical example shown in figure 3b, i.e., the molecule bromochlorofluoromethane. Here we see the tetrahedron’s four vertices are each occupied by a different atom. Rotation by 120^o about a symmetry axis now provides something distinguishable from the molecule’s starting position. Full rotation through 360^o i.e., returning to the starting position, is required to provide an identical result. Most importantly however, reflection in a symmetry plane provides a non-superimposable mirror image of the starting configuration. Thus molecules like bromochlorofluoromethane can exist in two chemically identical yet optically distinguishable mirror-image forms like a left and right hand.

In chemistry this property of molecular handedness in known as chirality, and the mirror images are called enantiomers. Chirality is important for all living things as the amino-acids from which proteins are made are all ‘left-handed’, while the sugars used to provide energy and build cell walls are all ‘right-handed’. Molecules of the opposite handedness are at best useless, or at worst damaging or lethal to living organisms. Many antibiotics depend on this principle of opposite handedness for their lethality, and it explains the teratogenic effects of thalidomide.

Tetrahedral symmetry is also important in particle physics where it appears as the SU(4) Lie Group that defines the ‘quark’ quantum number ‘charm’.⁶² The main point, however, is that if for whatever reason the corners of a tetrahedron can all be distinguished from one another other (as in figure 3b), that tetrahedron can then exist in two quite distinguishable chiral forms that are mirror images of each other. We shall see how tetrahedral symmetry impacts on understanding of the PPR entangled state.

PPR entangled state geometry

We begin as in figure 2, by forming a ‘triad of triads’ (figure 4b) out of the individual semiotic triangles for the patient (Px), practitioner (Pr) and remedy (Rx), which are also represented by the wave functions; ?_Px, ?_Pr, and ?_Rx, respectively. This time, however, the larger triangle formed by bringing together the three Px, Pr, and Rx semiotic triangles representing PPR entanglement (figure 4c), leaves not a hexagon (figure 2b), but a triangular space in the middle of equal size to the other three.

The PPR entangled state triangle is formed in such a way that the remedy (Rx) ‘corner’ of each individual semiotic triangle comprises its corners. Also, each of the smaller triangles now shares two of its corners with its two neighbours (figure 4c). This means that the centre of each side of the larger PPR entangled-state triangle now represents (in clockwise order) correlation between symptoms (Sx) of the patient (Sx(Px)); the symptoms of the disease (Sx(Dx)); and the symptoms of the remedy substance (Sx(Rm)). In other words, the formation of the PPR entangled state identifies for the practitioner two categories of patient-centred symptoms (Sx(Px) and Sx(Dx)) which, by matching with those of a remedial substance (Sx(Rm)), are used to arrive at the curative remedy and potency (Rx).

This may be usefully envisaged by ‘folding’ into the third dimension the larger entangled-state triangle along the edges of the internal inverted triangle, so that each semiotic triangle meets, producing a tetrahedron whose top vertex is the potentised remedy, Rx. The other tetrahedral corners are then Sx(Px), Sx(Dx), and Sx(Rm) (figure 4d). Thus, we have arrived at a 3-D semiotic representation of the PPR entangled state which is a tetrahedron with four identifiably ‘different’ corners, and is equivalent to the entangled-state wave function, I?_PPR> in equation 1.

Now, according to the symmetry rules mentioned earlier, such a tetrahedron must be chiral, i.e., there is another tetrahedral representation that is its totally distinguishable non-superimposable mirror-image reflection. In other words, there are two distinct ways of arranging the corners of such a tetrahedron, and therefore two distinct ways of representing the entangled state. This is shown in inverted form in figure 4f and would be the equivalent of the complex conjugate entangled-state wave function, <?_PPRI in equation 1.

Let us examine these entangled-state semiotic tetrahedra in a little more detail from the point of view of the practitioner. We may imagine the patient notionally at the tetrahedral epi-centres (the black squares in figures 4d and f), exhibiting symptoms Sx which may be interpreted by the practitioner as those of the patient (Sx(Px)) and the disease (Sx(Dx)), which are then matched with the repertorised remedial substance (Sx(Rm)); all of which lead the practitioner to the homeopathic remedy (Rx – hence the direction of the arrows in figures 4d and f)). Note how in this tetrahedral set-up, the practitioner is involved explicitly in two out of the four semiotic faces but is not at the epi-centre: this place is occupied by the patient.

That there are two distinct ways of arranging these semiotic tetrahedra, reveals the essential mirroring activity of the practitioner (?r in figure 4f), and that the PPR entangled state by its very nature is chiral. By acting as a mirror, i.e., providing a therapeutic state space for the patient and assisting in the formation of the PPR entangled state, the practitioner implies the state’s chirality. However, the practitioner is not passively reflecting back to the patient: the coherence produced by the practitioner implies an ‘active’ role in reflecting, represented by the therapeutic ‘mirror plane’ ?r ‘twisting’ the reflected tetrahedron through 60^o or ?/3 radians. This points the patient ultimately in the direction of cure, and leads to the next ‘step’ in the process.

Through the ‘looking glass’…

With the patient at the epi-centres of both semiotic chiral tetrahedra, the process of cure may be envisaged as their being brought together in such a way, one patient-centred ‘state’ is produced. This requires both tetrahedra to move towards each other and merge through the ‘looking glass’ of the therapeutic state space, ‘twisting’ relative to the other by a factor of 60^o or ?/3 radians (figure 4f).

In geometrical terms, this represents equation 1, i.e., the combination of <?_PPRI and I?_PPR> via the homeopathic operator ?r, leading to a change in symptoms, ?Sx (the product <?_PPRI?_PPR> = I?_PPRI² presumably representing the probability of cure; a concept that will be dealt with at a later date). The ‘twisting’ effect of the therapeutic state space has been noted in previous papers of this series,^{11, 24, 33, 55} so it is compelling that, albeit in a slightly different context, it arises from the treatment provided here.

This effect could also be inherent early in the entanglement process as folding the three semiotic triangles shown in figure 4c completely flat leads to the inverted entangled-state triangular representation, shown in figure 4e. Here, each corner of the entangled-state ‘flat pack’ now represents the individual ‘wave functions’ for the remedy, practitioner, or patient; the whole entangled state being turned, again, through 60^o or ?/3 radians, compared to the larger triangle in figure 4c.

The merging of the two semiotic tetrahedra in figure 4g provides a 3-D object called a stellated octahedron, or stellar octangula. First described by Pacioli in the early 16^th century, the stella octangula was named by the mathematician and astronomer Johannes Kepler one hundred years later.³⁴ Compared to a single tetrahedron, the combined tetrahedra of a stellated octahedron essentially has cubic symmetry, the eight corners coinciding with those of a cube in which it is inscribed, while the edges of the tetrahedra form diagonals of the cube’s square faces. However, of particular interest for the semiotic application of this geometry to entanglement in the therapeutic process, is that the stellar octangula can be viewed as a 3-D version of the 2-D Star of David configuration obtained in the previous investigation into the geometry of the PPR entangled state (figure 2e).²⁴

The five convex regular polyhedra called Platonic solids include the tetrahedron, cube, octahedron, icosohedron, and dodecahedron. Interestingly, in more than three dimensions, these polyhedra generalize to what are known as multi-dimensional polytopes, with higher-dimensional convex regular polytopes being the equivalents of the three-dimensional Platonic solids. In all dimensions higher than four, there are only three convex regular polytopes; the simplex, the hypercube, and the cross-polytope whose projections in three dimensions, correspond with the tetrahedron, the cube, and the octahedron, respectively.^{34, 63}

Since the time of their discovery by Pythagoras, there are those who have regarded the Platonic solids as imbued with mystical significance, four of them being used to represent the classical four elements. Thus the cube was associated with the earth; the octahedron with air; the icosahedron with water; and the tetrahedron with fire. The dodecahedron represented spirit or the universe. It is in this context that the stella octangula, has a long history in esoteric thought, especially in Jewish Hasidic philosophy where it is known as the Merkabah or ‘chariot’.⁶⁴

Here, the Merkabah was considered a multi-layered analogy that offered insight into the nature of man and his relationship to God, the ecosystem, and the world. In current ‘New Age’ circles, the stella octangula is thought of as a ‘vehicle’ for transporting consciousness between different dimensions. The two superimposed tetrahedra are imagined as counter-rotating, which when combined with specific breathing techniques, certain eye movements and prayer, are supposed to generate a spine-anchored energy field around the human body. This, when activated, is supposedly the carrier of consciousness directly to higher dimensions.⁶⁴

More in keeping with modern ideas of quantum entanglement and its use in quantum information theory,⁶⁵ the stella octangula has recently been invoked by Aravind to explain in the context of group theory a so-called ‘twirl’ or twist operation.⁶⁶ This was first introduced by Bennett,⁶⁷ in order to allow more convenient discussion of entanglement purification protocols,⁶⁸ used in distilling pure entangled states from a larger number of impure states (perhaps shared through a noisy channel), and so more reliably transmit quantum information via quantum teleportation.⁶⁹

Given the already-mentioned multi-dimensional nature of the Vf discussed in this and other papers,³³ especially in describing the Vf’s behavior in terms of notional quantised gyroscopic dynamics,^{11, 55} then perhaps the semiotic formulation of the curative state as a stella octangular (and all it implies in terms of rotation, consciousness expansion, and the use of entanglement in the quantum teleportation of information) could be an appropriate and compelling metaphor with which to represent healing during the homeopathic process. What is noteworthy here in this geometric representation, is that while it is the practitioner who provides the mirror-like therapeutic state-space ?r, it is the patient who has to make the movement ‘through the looking glass’ towards cure, which the practitioner may facilitate but does not control.

Conclusions

Quantum macro-entanglement between various combinations of Px, Pr, and Rx permits alternative descriptions (compared to the bio-molecular and biomedical sciences) of the therapeutic encounter in homeopathy.^19-22 In particular, three-way PPR entanglement^{23, 24} combined with semiotics^{21, 45, 46} could provide geometrical insights into the nature of the curative interaction. The formation of the PPR entangled state, involving the patient (Px), practitioner (Pr), and the homeopathic similimum (Rx) may be seen in semiotic terms as two chiral tetrahedra produced by reflection in an active mirror-like ‘therapeutic state-space’, ?r, provided by the practitioner. The corners of each of these semiotic tetrahedra are derived from their component Px, Pr, and Rx semiotic triangles such that they represent symptoms of the patient Sx(Px), symptoms of the disease Sx(Dx), symptoms of the remedy material Sx(Rm), and the homeopathic similimum, Rx. The ‘epi-centre’ of each tetrahedron is ‘occupied’ by the patient, whose ‘work’ of cure may be represented as the combining of the two tetrahedra through the therapeutic state-space ‘looking glass’, into one patient-centred curative semiotic polyhedron called a stella octangula. This may be seen as the geometrical equivalent of the equation:

<?_PPRI?rI?_PPR> = <?Sx>

For this to occur successfully, however, the practitioner may notionally facilitate formation of the tetrahedral entangled PPR and curative stella octangula states but not, in semiotic terms, be at their epi-centres. These ‘places’ are reserved exclusively for the patient.

ACKNOWLEDGEMENTS

The author wishes to express his gratitude to fellow homeopath, Ms. Anne Vervacke, and to an old friend, Ms. Susan Norman, for providing the initial and final inspirations for this work.

__________________________

†This is part 11 in a series of papers entitled Patient-Practitioner-Remedy (PPR) entanglement.

‡Address: Department of Chemistry, Imperial College London, South Kensington, SW7 2AZ, UK; e-mail: l.milgrom@ic.ac.uk

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FIGURE LEGENDS

Figure 1. Walach’s double entanglement model: two semiotic contexts (triangles) linked by the Law of Similars. Left, object = remedy substance, R?; sign = remedy, Rx; meaning = remedy picture, i.e., symptoms produced during provings, Sx. Right, object = patient’s ‘disease’, Dx; sign = patient’s symptoms, Sx; meaning = required remedy, Rx.^{12, 13}

Figure 2. PPR entanglement represented geometrically. In a, Walach’s two semiotic triangles for remedy and patient (also wave functions, I?_Rx> and I?_Px>) are joined by a third for the practitioner I?_Pr>, which are entangled into the PPR ‘state’ represented by I?_PPR> in b. The multi-dimensional geometry of this state is represented in c through to e and shows the action of the homeopathic operator ?r in ‘reflecting’ this state (d). But the reflection is not passive: by opening out the polyhedra in d and superimposing them, it is seen that the reflecting plane also twists the reflection through 60^o (e). The ‘space’ in which these wave functions and ‘operations’ take place is a therapeutic state space created by the homeopathic operator ?r, which also functions within it.^{23, 24}

Figure 3. a; regular methane tetrahedron. b; chiral tetrahedra of bromochlorofluorocarbon enantiomers.

Figure 4. A semiotic representation of PPR entanglement. Note the difference in the triad of triads compared to Figure 2: (Rm)Sx = symptoms of remedial substance; Sx(Px) = symptoms of patient; Sx(Dx) = symptoms of disease.