**Introduction**

Homeopathy’s claims of clinical efficacy and cost effectiveness^{1} are regarded with suspicion and contempt. Evidence other than that framed within the (sometimes biased)^{2-4} reductionism of Evidence-Based Medicine (EBM)^{5, 6} and the double-blind randomised-controlled trial (DBRCT),^{7, 8} is rejected.

EBM and the DBRCT, like much of biomedical science, are rooted in the reductionist philosophy of Logical Positivism combined with Local Realism. The latter states that,* a),* *the universe is real and it exists whether we observe it or not; b)*,* legitimate conclusions and predictions can be drawn from consistent experimental outcomes and* *observations; c), no signal can travel faster than light*.^{9-11}

In questioning a) and c) above, quantum theory transcends Local Realism^{10} and philosophically confounds the reductionism of biomedicine.^{11} Attempts at explaining homeopathy’s efficacy have made use of concepts generalised from the discourses of semiotics^{12, 13} and quantum theory.^{14-16} Thus, non-local entanglement^{17-19} between patient, practitioner, and remedy (PPR entanglement), could form a descriptive basis for the healing interaction.^{20-24} It combines from physics Greenberger- Horne-Zeilinger (GHZ) three-particle entanglement,^{25} and an algebraic generalisation of orthodox quantum theory called Weak Quantum Theory,^{18, 19} with semiotics^{12, 13} to generate a three-way PPR entangled state. ‘Cure’ results from the ‘reflection’ and ‘twisting’ of this state in a notional two-dimensional mirror-like ‘therapeutic state space’^{26} (an analogue of complex Hilbert space more familiar from orthodox quantum theory),^{10} depicted in ‘semiotic geometry’ as a hexagonal bipyramid.^{11, 23, 24}

Though hypothetical, a *post hoc* explanation^{27, 28} of the observed ‘leakage’ between verum and placebo groups during recent double-blind provings of homeopathic remedies,^{29-31} is afforded by the PPR entanglement metaphor, pointing to its possible experimental verification. In addition, and when viewed semiotically, the PPR entangled state’s geometrical projection into a notional ‘therapeutic state space’ has been used to understand the concept of miasms in homeopathy,^{32} and the action of remedies and diseases on the Vital Force.^{33} The purpose of this paper is to further investigate the polyhedral geometry of the PPR entangled state,^{34} and to perhaps deepen insight into the complex therapeutic action and interaction of the patient, practitioner and the remedy on the way to cure.

**ENTANGLEMENT IN THE THERAPEUTIC PROCESS**

*Quantum macro-entanglement*

Entanglement is said to occur in a quantum system when its seemingly separate parts are so holistically matched or correlated, measurement of one part of the system *instantaneously* (i.e., not limited by the speed of light and without classical signal transmission) provides information about all its other parts, *regardless of their separation in space and time, or their size*.^{35} Quantum entanglement between photons was famously demonstrated by the French physicist Alain Aspect and his team in 1982.^{36}

That there is no size limitation on quantum entangled systems is important as quantum theory is thought by many to apply only to sub-atomic particles, atoms and molecules. This is because the equations of quantum theory are dominated by an incredibly small number called Planck’s constant (6.626 x 10^{-34} J s), commensurate with events occurring on such a small scale.

In fact, under certain exceptional circumstances (especially around fluctuating instabilities at critical points when a material is on the verge of changing its physical state),^{37} correlated behaviour between the parts of a quantum system – a keynote of entanglement – can be ‘lifted’ into the macroscopic realm of bulk materials, which then exhibit properties similar to nanoscopic quantum systems, e.g., low-temperature superconductors and super-fluids.^{38}

Such ‘quantum macro-entanglement’ is being used to develop a novel quantum model of energy medicine.^{39} Interestingly, as an explanation of the efficacy of homeopathic remedies, the Memory of Water also relies on macro-entangled coherence, albeit between large numbers of water molecules.^{41, 42 }

Non-local correlation is not the only pre-requisite for entanglement. A quantum system’s processes must also be describable in terms of a ‘non-commuting algebra of complementary observables’.^{10} Thus, when two separate operations of observation are performed sequentially* the result depends on the sequence and what is being measured.* This leads to another key quantum idea: *complementarity*.^{43, 44}

So a single explanation or model might not adequately fit or explain all the different observations that can be made on a quantum system. In order to fully explain quantum phenomena therefore, it is necessary to have two different but complementary concepts. The answer one obtains performing two different sets of observations depends entirely on the order in which they are performed: yet *both* are necessary in order to acquire a complete picture of the system.

Such notions of complementarity and entanglement can be broadened far beyond their specific meaning in the orthodox quantum theory of particles, atoms and molecules: they might be useful in describing the behaviour of what are also in essence macroscopic systems, i.e., the patient, practitioner, and remedy.** **Examples have been cited of complementarity and entanglement in engineering, information dynamics, philosophy and the cognitive sciences, especially psychology.^{18} On this basis, Atmanspacher *et al* formalised a more general version of quantum theory which relaxes several of its nanoscopically-limiting axioms, including dependence on Planck’s constant. Called Weak Quantum Theory (WQT),^{18} it differs from orthodox quantum theory in that:-

? Complementarity and entanglement are not restricted by a constant like Planck’s constant:

? WQT has no interpretation in terms of probabilities:

? Complementarity and indeterminacy are epistemological not ontological in origin.

As a result, WQT explicitly allows quantum theory’s application into the above-mentioned macroscopic areas and, by implication, into possible explanations of the dynamics of healing.^{19, 21, 23, 24}

*PPR entanglement and semiotics*

Semiotics is the study of signs and symbols. A sign can be anything as long as it is interpreted as a sign, i.e., that it signifies something. Without signification by someone, a sign of itself has no intrinsic meaning. From this perspective, it is the meaningful use of signs that concerns semiotics.^{45} Its two dominant models are those of the French linguist Ferdinand de Saussure, and American pragmaticist philosopher, Charles Sanders Peirce.

De Saussure’s semiotics is dyadic, i.e., a sign is thought to consist of two parts: a *signifier*, which is the form of the sign; and that which is *signified*, or the *concept* the sign represents. Peirce introduced a third element, i.e., the *sense* made of the sign. Thus, Peirce’s semiotics is essentially triadic.

Modern semiotics impacts on our understanding of how and in what media we communicate, and among other things, influenced the development of post-Modern philosophies.^{46} Semiotics, however, has very ancient roots. The word itself derives from the ancient Greek shmeiwtikoz – semeiotikos, meaning an interpreter of signs; and both Plato and Aristotle are known to have considered the relationship between signs and the world of phenomena. Thus without calling it such, semiotics has long been associated with Western philosophical thought.

Henry Stubbes first used the word ‘semiotics’ to denote precisely the branch of medical science relating to the interpretation of signs and symptoms; an idea taken up by the late 17^{th} century philosopher John Locke, ^{47} and still being examined today.^{48-50} Locke’s writings on the value of signs and symptoms seem to pre-date those of Hahnemann. Consider, for example Locke’s, “*Nor is there any thing to be relied upon in Physick but an exact knowledge of medicinal physiology (founded on observation not principles), semeiotics, method of curing, and tried (not ex-cogitated, not commanding) medicines.*“^{47 }Then compare this with Hahnemann’s Organon of Medicine; article 6, written about one hundred years later:^{51} “*The unprejudiced observer is well aware of the futility of transcendental speculations which can receive no confirmation from experience. Be his powers of perspicacity even very great, he can take note of nothing in every individual disease, except the changes in the health of the body and of the mind (morbid phenomena, accidents, symptoms) which can be perceived externally by means of the senses…. All these signs represent the disease in its whole extent, that is, together they form the true and only conceivable portrait of the disease.*”

Walach specifically applies modern semiotics^{52, 53} to homeopathy by making some quite revolutionary assumptions.^{12, 13} Thus, the usual supposed local, causal (i.e., its pharmacological activity, regardless of the absence of molecules of the substance) effects of a potentised homeopathic remedy are dropped. Instead, Walach adopts the semiotic notion that the homeopathic remedy is a ‘sign’ working simultaneously in and for two different but connected meaningful contexts: 1) the symptoms of a sick person signify a certain disease state (first meaningful context), while simultaneously signifying, 2) a homeopathic remedy in the materia medica (second meaningful context), the two contexts of illness and remedies being connected by the Law of Similars (Figure 1).

**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}

Walach then uses WQT to demonstrate how semiosis illustrates homeopathy as two instances of generalised entanglement: one between the potentised remedy (Rx) and the original unpotentised substance (R?); the other between the individual symptoms of a patient and the general symptoms of the substance produced during homeopathic provings. This generalised double entanglement is thought to be analogous to the cryptographic and teleportation applications of ordinary quantum entanglement.^{21} Walach goes on to derive what he considers are testable predictions of his double entanglement model.

The idea of PPR entanglement on the other hand, envisages the patient (Px), practitioner (Pr), and remedy (Rx) achieving a potentially therapeutic macro-entangled state.^{23} This is described by a state function **I****? _{PPR}**

**>**

**(as are each of the patient, I?**

_{Px}

**>**; the practitioner, I?

_{Pr}

**>**; and the remedy, I?

_{Rx}

**>**) in a ‘therapeutic state space’. Though more detailed mathematical proofs are needed to flesh out the meaning of state functions and hypothetical spaces used in therapeutic contexts, PPR macro-entanglement can lead to a generalised quantum system.

**Figure 2**. Semeotic PPR entanglement represented geometrically. In **a**, Walach’s two semiotic triangles for remedy and patient (also wave functions, I**? _{Rx}**> and I

**?**>) are joined by a third for the practitioner I

_{Px}**?**>, which are entangled into the PPR ‘state’ represented by

_{Pr}**I?**in

_{PPR}>**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}

It is useful to ‘map’ the patient, practitioner, remedy, and PPR entangled state wave functions onto Walach’s semiotic triangles. In order to do this a third semiotic triangle needs to be constructed for the practitioner (who in Walach’s model, creates entanglement by applying the Law of Similars, see figure 1).^{24} The large triangle in figure 2b is then the semiotic equivalent of combining the Px, Pr, and Rx state functions, I?_{Px}>, I?_{Pr}>, and I?_{Rx}>, into the PPR entangled state function, **I****? _{PPR}**

**>**. By constructing a ‘triad of triads’, in this way the semiotic analysis has more in common with Peirce’s model of semiotics than de Saussure’s. This ‘mapping’ was extended into a third dimension by folding the corners of the large triangle to create a pyramid with an hexagonal base, the solid semiotic figure again representing the PPR entangled state function

**I**

**?**

_{PPR}**>**in three dimensions (figures 2c and 2d).

Now, in quantum theory, a wave function describing a quantum state I?**> **consists of complex numbers,^{54} i.e., x + *i*y, where x and y are ‘real’ and *i* is the ‘imaginary’ number, ?-1. Any complex number defines its complex conjugate – its ‘mirror image’, x – *i*y, such their product gives only a real number (i.e., x^{2} + y^{2}). Similarly, a complex wave function defining a quantum state I?**> **automatically defines its mirror image complex conjugate, <?I, the product <?I?**> =** I?I^{2} being ‘real’ and interpreted as the probability of finding the quantum state in a defined volume of space.

Such ideas can used to interpret part of a practitioner’s role, which is to be an ‘active’ mirror, reflecting back to the patient the possibility of cure. Thus, the PPR entangled state, **I****? _{PPR}**

**>**suggests its mirror image complex conjugate,

**<**

**?**

_{PPR}**I**arises out of the active reflecting activity of the practitioner acting as homeopathic operator,

**?r**(see figure 2d), while at the same time being part of the entangled PPR state. In other words, the practitioner creates the conditions for cure (i.e., the therapeutic state space), and then operates within that space as the homeopathic operator,

**?r**, and as part of the PPR entangled state. Finally, bringing the PPR entangled state and its reflection together, results in a change in symptoms (leading to cure), as shown in the equation:-

**<****? _{PPR}**

**I**

**?r**

**I**

**?**

_{PPR}**> = <?**

**Sx**

**>**

**1**

This is represented in figure 2d where the PPR entangled state hexagonal pyramid is reflected in the active mirror therapeutic state space (?r) and gives its complex conjugate inverted hexagonal pyramid. Opening out the two hexagonal pyramids by ‘projecting’ them onto the mirror plane defined by ?r shows that the PPR entangled state is twisted through 60^{o} relative to its complex conjugate (giving the Star of David configuration in figure 2e); suggesting the practitioner is not a passive ‘reflector’, but is active in a way that may be represented metaphorically as a twisting motion.^{55}

In this paper, a different approach is adopted to the geometrical interpretation of the PPR entangled state and its complex conjugate. Ultimately this leads to a more detailed appraisal of the roles of patient and practitioner in the therapeutic process.

*On the nature of the therapeutic ‘state space’: the Kochen-Specker theorem*

The mirroring activity of the homeopathic operator **?r** takes place in a ‘therapeutic state space’, which may be understood generally by considering the nature of the space in which quantum systems operate.

A fully specified quantum system is usually represented by a statistical (as opposed to definitive) description derived from experimental observations, called a *state vector*. Its governing equation contains complex numbers (see above). Thus, the ‘space’ of a state vector cannot be our ordinary space and time. It is an abstract complex mathematical ‘configuration space’ called a Hilbert ‘state space’.^{56}

Hilbert space is a higher level of abstraction than our ordinary geometric space. It is a mathematical device for arranging pieces of information, whose ‘coordinates’ are complex. Each complex ‘coordinate’ (which can be infinite in number) is not geometric: it represents the possibility for a given quantum state to exist that might correspond to something definite in our ‘real’ space.^{56}

Before a measurement/observation is made, all possible states ‘co-exist’ and contribute to the wave function. Thus, a wave function does not have inherent properties, only incompletely defined ‘potentialities’. Put another way, *a wave function contains within it all that can be known about a system by observation, not its ontological reality ‘out there’, separate from the observer*.^{57}

When an experiment is performed on a quantum system, a real number called an eigenvalue is returned. An eigenvalue is not a property of a quantum system: it is what we observe during an experiment; so it is ‘classical’ and it consists of real numbers. On the other hand, a quantum system is described by a state vector which consists of a set of complex numbers. Therefore, it is extremely difficult to assign definite eigenvalues from ‘our’ space to quantum systems in Hilbert space.

Partly, this is because complex numbers cannot be contained within real numbers. In just the same way that a 3-dimensional object cannot be squeezed into its 2-dimenstional projection, something always gets lost ‘in translation’ when trying to map eigenvalues onto quantum states. And this is not due to any carelessness on our part: it is a basic limitation in our form of observation.

This has profound consequences. Prior to quantum theory it was assumed everything physical is measurable or observable. It is why quantum theory is so difficult to grasp. For *without forsaking the requirement of empirical evidence for knowledge of physical characteristics, it is possible for quantum properties (e.g., a particle’s wave function) to be physical yet not directly observable or measurable*.

It is for this reason the *discourse* of quantum theory might be useful in describing the homeopathic process,^{14, 15, 18} where signs and symptoms of disease are considered observable manifestations of an ‘invisible’ disturbed vital force, Vf. Thus, ‘*Eigenvalues are analogous to symptoms of a disease, which are disturbances of the body that show up and indicate something that does not show up. Just as a cold persists though its symptoms are suppressed, so a quantum system’s wave function has a definite amplitude, even though it has no eigenvalue …*‘:^{10} a physicist’s words, not those of a homeopath.

In quantum theory, observables are represented by special mathematical operators called *self-adjoint operators*. Like mathematical functions, they are transformations or mappings, but unlike functions, the range of operators is not just the line representing real numbers: it is the whole complex state space. This applies to self-adjoint operators even though they consist only of real numbers.

The Kochen-Specker theorem indicates *that in most cases*, eigenvalues cannot be considered as properties of quantum systems if there is a one-to-one mapping between self-adjoint operators in a Hilbert space and observables in ‘real’ space. But the theorem shows that this only leads to contradictions if the dimension of the Hilbert space is greater than 2.

This is why the Kochen-Specker theorem might be significant for quantum theoretical models of the therapeutic process. Thus, previous papers^{24, 32, 58} have argued the ‘therapeutic state space’ is best likened to an active ‘mirror’; something that tallies with practitioners’ experience of the therapeutic process. Mirrors are essentially planar, i.e., 2-dimensional. The homeopathic operator not only operates in this mirror plane but is an essential feature in its creation. According to the Kochen-Specker theorem therefore, it should be possible for a meaningful and non-contradictory mapping to be made from observables (i.e., signs and symptoms) by the homeopathic operator **?r** in its 2-D therapeutic state space onto the patient’s Vf. For this reason*, *the contradictions in orthodox quantum theory arising from one-to-one mappings of ‘classical’ observables onto physical properties* appear not to arise in quantum models and metaphors of the homeopathic process.* This, in my view, legitimises the assumption of symptoms (‘eigenvalues’) of disease being considered properties of an unobservable ‘quantum property’, in this case, the Vf. Thus, the Kochen-Specker theorem delivers back in terms of a quantum discourse, what homeopaths already know:* that signs and symptoms of disease are the manifestations of a Vf that is not in itself apprehensible or, indeed, comprehensible in purely materialistic terms.*^{59}

**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.

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. |

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, **<****? _{PPR}**

**I**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 **<****? _{PPR}**

**I**and

**I**

**?**

_{PPR}**>**via the homeopathic operator

**?r**, leading to a change in symptoms,

**?**

**Sx**(the product

**<**

**?**

_{PPR}**I**

**?**

_{PPR}**> =**I

**?**I

_{PPR}^{2}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:

**<****? _{PPR}**

**I**

**?r**

**I**

**?**

_{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.

__________________________

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

â€¡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

**?**>) are joined by a third for the practitioner I

_{Px}**?**>, which are entangled into the PPR ‘state’ represented by

_{Pr}**I?**in

_{PPR}>**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.

Reads like a mathematical thought experiment with numerous erroneous assumptions.

Completely amateur and worthless.