Excerpt

## Inhalt

1- Vicinal Water and Quantum Coherence in Microtubules

2- Scaling up the Quantum Property of the Microtubule to the Cerebral Cortex/Brain

3- Symmetry Breaking and Self-Synthesis

4- Phyllotaxis, the Phi Connectome and Bandyopadhyay’s Sphere-Spiral Fractal Structure

5- The Phi Connectome—Formation

7- Information Transmission Delays in the Brain and Consciousness

8- Paradoxical Results in Measurements of the ‘Time’ of Conscious States

9- The Cellular Automaton and Quantum Coherent States

10. Formation of Microtubules in Prokaryotes and Self-Synthesis of the Phi Connectome

11. Microtubule polarity in axons, soma, dendrites, and the phi connectome

12. Phi Connectome Connectivity—Quantum or Electromagnetic Field?

13. Discussion

14. References

This paper proposes:

- A mechanism producing quantum coherence in microtubules.

- How this quantum coherence is scaled up to include the cerebral cortex.

Microtubules are part of the cytoskeleton in all cells. In neurons, they are thought to play a key role in consciousness. ^{1} The Orch-OR theory of consciousness maintains that consciousness—qualia, the feeling of life itself—requires quantum processes, and that Turing machine algorithms, though capable of astonishing calculations, lack insight and understanding. ^{1},^{2} ‘Insight’ as Gödel’s incompleteness argument suggests may not be demonstrable by an algorithmic process. Consciousness (Chalmer’s explanatory gap) may depend on more than calculation. In the current consciousness paradigm, quantum coherent microtubules are considered the primary unit of ‘computation’ rather than neurons and the synapses between neurons.

Roger Penrose demonstrated that 2-dimensional arrangements of certain tile shapes can produce 5-fold symmetry with *non-local* properties such that no algorithm/computation is able to produce this structure. This means that in laying down each *local* tile, one has to consider the positions of *distant* tiles, as if distant parts of the quasicrystal were in quantum linear superposition. Significantly, in one tiling made up of kite-shaped and dart-shaped tiles, the ratio of the number of kites to the number of darts in the tiling is, kites/darts = 1.618 or phi the golden ratio. ^{3}

In 2011 Dan Shechtman received the Nobel prize in chemistry for his work in quasicrystals. These structures show what were thought to be impossible structures, e.g. 5-fold symmetry.

Monte Carlo simulations of confined tetrahedra can form quasicrystals. Given only Brownian motion (‘jiggling’) of tetrahedra in a confined space (without any other forces between the tetrahedra), quasicrystals can still be formed by means of *entropic bonds* rather than chemical, ionic, or hydrogen bonds.^{4}

In a system where potential energy = zero, free energy is *minimum* when entropy is *maximum*.

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Normally, we think of a system with maximum entropy as being *disordered*, but in this case, the system has both order *and* maximum entropy. It is as if randomization produces order—not the usual way we think about entropy. These Monte Carlo simulations have been applied to various whole and truncated tetrahedra as well as soft (‘squishy’) structures ^{4}.

The key point here is that Monte Carlo simulations cause randomization of these tetrahedra as if they were being jiggled about by Brownian motion or heat. Brownian motion, heat, noise, and randomization are taken as one and the same thing. ^{5} Penrose 2-D tilings and Monte Carlo simulations of tetrahedra produce quasicrystals which cannot be formed algorithmically.

Quasicrystals have *a non-local or quantum-like property*, in which different parts of the quasicrystal are correlated such that a change in one part requires a change in a distant part. ^{2} By contrast, in classical crystal growth (which *can* be modeled algorithmically), molecules or ions attach to a continually-growing face of the crystal by means of a *local* process.

Therefore, in the growth of a quasicrystal, we must consider what seems to be an evolving quantum linear superposition of different possible ways of attaching atoms or ions to the growing crystal. That is, there seem to be several alternative atomic arrangements coexisting in complex linear superposition. Moreover, this process starts at a molecular, or smaller, scale and continues at larger and larger scales until a macroscopic quasicrystal is formed. Significantly, the golden ratio, phi = 1.618 = (1 + (5)1/2)/2 is found in the ratio of Penrose’s 2-D tiling. This ubiquitous, *irrational number* found throughout the universe, probably plays an important role in consciousness. Figure 1 below, illustrates the golden ratio inherent in the pentagon.

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**Figure 1** (author’s original figure)

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Figure 2 below illustrates a tetrahedral structure formed by hydrogen bonding among five dipolar water molecules.

## 1- Vicinal Water and Quantum Coherence in Microtubules

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**Figure 2** [Wikipedia Properties of Water—labels added by author]

Five dipolar water molecules form a tetrahedral structure in which the van der Waals forces are balanced such that the tetrahedron has an overall neutral charge.^{6} Figures 3,4, and 5 illustrate the interaction of vicinal water tetrahedra with tubulin molecules.

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**Figure 3** [author’s original figure]

Illustrating how microtubules become quantum coherent. **1**. Vicinal water tetrahedra are formed by relatively weak van der Waals/hydrogen bonding between bipolar water molecules. **2**. Distortion of the tetrahedron, by femtosecond asymmetries of the tetrahedral shape and/or interaction between the tetrahedron and tubulin molecules, results in a **3**. *mutual change in shape* and electrical neutrality of the tetrahedron, and polarization of the tubulin molecule. Hammeroff has pointed out, that the hydrophobic ends of protein molecules are protected from ‘noise’ in external hydrophilic ionic compartments because they are buried within hydrophobic regions of the folded tubulin molecule. Therefore, it may be that the mutual change in shape of the vicinal water tetrahedron and tubulin molecule is more important than a change in electrical neutrality of the vicinal water tetrahedron. **4** Metabolic heat ‘jiggles’ or randomizes the tetrahedra, **5**. producing a quasicrystal in which distant parts of the quasicrystal are correlated, i.e. in quantum linear superposition. **6** Mutual interactions between vicinal water tetrahedra and tubulin molecules result in a quantum coherent microtubule, i.e. a macroscopic coherent quantum object in which the individual tubulin molecules are in linear superposition, analogous to a Bose-Einstein condensate, as found in lasers, superfluidity, and superconductivity. It is probably important that vicinal water tetrahedra present a flattened 2-D face to tubulin molecules. This 2-dimensional relationship may be important in order to permit quantum correlation. Despite possible collapse of the quantum state of the microtubule by, e.g., action potentials, microtubular quantum coherence is *continually maintained* by metabolic heat.^{4} It has been said that the brain is too warm, wet, and noisy to be a quantum mechanical system. Nonetheless, it is this very combination of metabolic heat Brownian motion noise, and vicinal water ‘wetness’ that enable and maintain quantum coherence of the microtubule.^{5} Figure 4 below illustrates the relationship of vicinal water to graphene in a confined tube. One side of the tube is shown.^{7}

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**Figure 4** [author’s original figure]

This figure shows the relationship of vicinal water tetrahedra to a graphene surface in a *confined* space.^{7} Immediately adjacent to the graphene surface, tetrahedra present a flat 2-D face. Slightly farther away from the graphene surface, in the second layer, water tetrahedra align such that the apex of the tetrahedron points towards the graphene. The third and interior layers within the confined space show a disordered orientation. In the confined space within a microtubule there is probably a similar relationship of vicinal water to the tubulin layer. One might say that vicinal water ‘wets’ graphene and tubulin molecules. Since vicinal water tetrahedra are small compared to the tubulin molecule, several vicinal water tetrahedra probably interact with each tubulin molecule. The coherent non-locality and non-algorithmic nature of quasicrystals induces the same coherent, non-locality in the microtubule, so that the microtubule is capable of non-algorithmic/quantum computation. Note that vicinal water is ‘pure,’ i.e. does not contain other charged particles such as ions. Perhaps the confined volume of the microtubule excludes ions.

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**Figure 5** [author’s original figure]

Figure 5 illustrates diagrammatically, (**a**) an undistorted vicinal water tetrahedron in which the van der Waals forces are balanced so that there is no net charge on the structure (shown as ‘0’ charge). Actually, several vicinal water tetrahedra interact with each tubulin molecule. If vicinal water has no ions and no charge, then it forms a hydrophobic compartment and may distort a tubulin molecule by interacting with its hydrophilic outer ‘shell,’ causing a *change in* polarization, i.e. electron distribution in the aromatic rings of proteins buried within the hydrophobic interior of tubulin are in quantum superposition. Another possible mechanism involves distortion of the vicinal water tetrahedron (**b**) such that the van der Waals/hydrogen-bonding forces are not balanced resulting in a net charge on the tetrahedron (shown as + charges). This net charge or change in shape interacts *reciprocally* with a tubulin molecule producing a conformation or polarization change in the tubulin molecule that leads to a coherent quantum state for the microtubule. This means that there is a coupling between vicinal water and tubulin protein. There is a mutual interaction between vicinal water tetrahedra and tubulin molecules, each distorting the other. The system exhibits entropic bonding. Perhaps there are also interactions between tubulins (dashed line with ‘?’). Thus, **heat-induced, non-local vicinal water quasicrystals produce the same non-local coherence of tubulin molecules**. Consequently, the microtubule exhibits quantum correlation at body temperature (approximately 370 C = 3100 K), and is capable of non-algorithmic computation. The electrical properties of the microtubule are independent of its length.^{8} The atomic water core resonantly integrates all proteins around it such that the microtubule functions like a single protein molecule irrespective of its size. **[12]** Significantly, if the vicinal water is removed, then this correlation is lost wherein each tubulin molecule becomes independent, and the quantum property is lost.^{8} Anesthetics are thought to act on the pi resonance of the hydrophobic interior of the tubulin molecules, and might also work by interfering with vicinal water van der Waals bonding and vicinal water tetrahedral-tubulin reciprocal interactions illustrated in figure 5.^{1}

## 2- Scaling up the Quantum Property of the Microtubule to the Cerebral Cortex/Brain

According to the Orch-OR (Orchestrated-Objective Reduction) theory of consciousness ^{1},^{10}, the objective reduction of the quantum state at the microtubule level is scaled up in some way so that the brain (cerebral cortex) acts as a macroscopic (Orchestrated) quantum object. Figure 6 illustrates Fibonacci (Golden) spirals/helices in a microtubule. In the microtubule we see intertwined/interpenetrating (clockwise and counterclockwise), two adjacent Fibonacci spirals/helices with neighboring Fibonacci numbers 5 and 8. Microtubules with 13 (= 5+ 8) columns of tubulin molecules are common in mammals. We see similar interpenetrating Fibonacci spirals in sunflowers and many other natural objects.

The pairing of adjacent Fibonacci numbers and spirals is probably significant. Scaling up microtubule quantum coherence depends on self-synthesis of the ‘ *phi connectome* ’ (discussed below). In phyllotaxis, leaves on a plant stem are disposed at 137.50 from one another forming a Fibonacci (logarithmic) spiral or spira mirabilis which allows a *maximum* amount of sunlight and water to reach every leaf on the stem because this angle results in a *minimum* amount of overlap (interference) among all leaves. Think of sunlight and water as ‘information.’

Figure 6 below illustrates a microtubule, with two ‘opposed’ interlocking/interpenetrating (head-to-toe) spirals, which implies that there is bidirectional maximum information flow with minimum flow interference. If the microtubule is a quantum coherent object, then it is not surprising that information flows in both—and even all—directions producing very complex patterns of microtubule activity.

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**Figure 6** [author’s original figure]

Figure 6 illustrates the ‘ *paired/interpenetrating’ phi connectome* where φ2/φ1 = 1.618… The Fibonacci helices are *interpenetrated*, where each tubulin is part of both a right-handed and left-handed helix. If a helix is ‘flattened’ it becomes a golden spiral. See Figure 7. The interpenetration of left and right-handed spirals is seen in the disposition of seeds in a sunflower. This paired/interpenetrating phi connectome is probably manifest from the microtubule to the cerebral cortex. Could it be that the same relationship is also to be found from the Planck to the cosmic scale? If the phi-connectome is a macroscopic quantum object, then information can be said to travel in either direction through it. Within the microtubule, bi-directional information flow produces very complicated patterns that seem to mimic cellular automata and ‘gliders’ where all these patterns represent superposed quantum states that undergo objective reduction into some classical pattern represented as a moment of proto-consciousness, a ‘bing.’ As the phi connectome self-synthesizes, these moments of proto-consciousness are Orchestrated into a moment of what we recognize as one of our conscious moments.^{1} Glotzer quasicrystals and Penrose tiles have Fibonacci numbers. For example, the ratio of kites to darts in one of the Penrose tilings is 1.618. Figure below illustrates the bi-directional form of the phi connectome seen in the microtubule and scaled up in the brain. Axon, MAP, length MT counterclockwise clockwise (cross section dendrite, soma)

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**Figure 7** (figure from WordPress.com—labels added by author)

Illustrates counterclockwise and clockwise logarithmic spirals combined, giving the pattern of sunflower seed arrangements. This pattern is seen in the *phi connectome*. Does mixed polarity of microtubules in cortical pyramidal cells indicate homochirality, i.e. *unidirectional* clockwise or counterclockwise spirals) for the final output of these cells (Figure 21)? Microtubule associated proteins (MAPs)^{1} form spirals among many microtubules (dotted connector between two ‘rows’ of microtubules). Figure 8 below illustrates the geometry of the phi connectome fractal in three ways (**A**, **B**, and **C**).

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**Figure 8** [author’s original figure]

Figures 8 illustrates the phi connectome. 8 **A** Illustrates the phi—also phy(*llotaxis*) connectome fractal, from the microtubule to the cortex. Figure 8 **B** illustrates the gap junction connection among all cells. Gap junctions may also allow quantum tunneling. Skin and liver cells have microtubules, but somatic cells are not part of the phi connectome. 8 **C** illustrates the same phi connectome in terms of phyllotaxis. The arrow indicates that the phi connectome is bi-directional, i.e. representing two interpenetrating logarithmic spirals with adjacent Fibonacci numbers. The final output of pyramidal cells of the cerebral cortex may be homochiral, counterclockwise *or* clockwise (Figure 7). Golden spiral in polar coordinates is: r = aebθ. If a>0 the spiral is counterclockwise; if a<0, the spiral is clockwise. Mixed microtubule polarity in dendrites and neuron cell bodies, especially in giant pyramidal cells of the cortex may allow positive-feedback amplification essential for consciousness (see section 11, Figure 21). All levels from microtubule to MAP, etc. vibrate at their respective frequencies such that the ratios of frequencies are *not* rational.^{11}

The whole body is a single oscillator. Scale Sx represents the Fibonacci spiral interconnection of several tubulin molecules. Scale Sx+1 represents the Fibonacci spiral interconnections of several microtubules by microtubule associated proteins (MAPs) forming a Fibonacci spiral at a larger scale within each neuron; then, at the next larger scale, several of these neuronal axons form a Fibonacci spiral/helix as they interconnect with dendritic spines of post-synaptic neurons, and so on up the fractal scale. This hierarchical relation continues at larger and larger scales, eventually encompassing the entire brain, particularly the cerebral cortex. From ‘seed’ layer to the cortex, the phi connectome is a self-synthesizing frequency fractal. In other words, the macroscopic quantum coherence of the entire system reflects and depends on the quantum coherence of the microtubule, and the quantum coherence of the microtubule depends on metabolic heat-induced quasicrystal formation in microtubule interactions between tubulin molecules and vibrating vicinal water (Figures 3, 4, 5, 6, and 8). Figure 8 illustrates the Phi Connectome Fractal (PCF). The microtubule contains adjacent interpenetrating Fibonacci numbers, 5 and 8. As mentioned above, at the next level of the PCF, microtubules are interconnected in a Fibonacci spiral *within* a neuron by microtubule associated proteins MAPs. At the next higher scale of the PCF, several axon knobs connect in a golden spiral to dendritic spines of post synaptic neurons, and so forth to larger and larger scales to encompass the cerebral cortex. The microtubule is self-synthesized. It is A. Bandyopadthay’s ‘seed’ which gives rise to the whole ‘clocks within clocks’ structure. It is important to recognize that the phi connectome fractal is a base, and this base changes, i.e. the fractal is modified, as consciousness/learning develops. See Figure 13. An analogy involves a rubber band loop. Keep one end fixed, and rotate the other end while stretching the band to prevent its collapse into a ball. If the twist and tension are balanced, then the frequency of the loops is in the phi ratio. This structure is seen in the DNA helix. Continue twisting until the initial loops become secondary loops at the next larger scale of the fractal. Eventually, twisting while stretching to prevent collapse into a ball, one ends up with a loop within a loop within a loop, resulting in a model of a phi connectome. The loop within a loop within a loop structure of the twisted rubber band is like the Zhang fractal design for a flexible electronic circuit.^{21} Let each scale of these models vibrate at a frequency related to its scale in the fractal, analogous to Bandyopadhyay’s clocks within clocks model.^{12} Narrow gap junctions in all cells of the body may allow quantum tunneling. Even so, inasmuch as all cells in the body share a common metabolic heat bath (Figure 9 below), heat-induced quantum coherence in microtubules suggests the cells in the somatic and central nervous system are entangled. Except for the brain, they are not part of a phi connectome at scales larger than microtubules in individual cells, and so they do not, at their level of organization, participate in orchestrated consciousness (BINGs), but only simple objective reductions, i.e. proto-conscious ORs (‘bings’). Nonetheless, the brain and all cells constitute a *whole*.

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**Figure 9** [author’s original figure]

Different cells of the body and regions of the brain are immersed in a common metabolic heat bath which drives oscillation of vicinal water tetrahedra in microtubules so that all cells in the body are entangled. Somatic cells of the skin or liver, e.g. communicate by gap junctions (an electrical ‘synapse’) that, when open, allows the passage of ions and molecules such as morphogens. Gap junctions (very narrow) may also allow quantum tunneling. Therefore, we have a Gap Junction Biologic Network or syncytium (GJBN).^{16} The cells are entangled, so they can act in synchrony as a single quantum object. The neurons and glial cells also form a GJBN syncytium a kind of ‘hyper-neuron.’ The synapse and the gap junction have both classical and quantum entanglement properties. Somatic cells, lack the larger-scale phi connectome found in the brain. They are incapable of orchestrated objective reduction (Orch-OR), the ‘BING’ of a conscious moment. However, there is an objective reduction, OR, a small ‘bing’ of proto-consciousness. Nonetheless, this may be sufficient in order to interact with the larger Orch-OR system wherein one might say that the mind and the body are truly one indivisible quantum whole. Figure 10 below illustrates the possible relationship of ‘mind’/consciousness. Bohm suggests consciousness may reside in the implicate order.^{1}

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**Figure 10** [author’s original figure]

Microtubule quantum coherence scaled up by phi connectome. Collapse of the wave function Orch-OR, ‘BING’ = a moment of consciousness with objective reduction of the cortical quantum wave function to the classical, and immediate restoration of cortical quantum wave function by: **1**. Heat restoration of microtubular quantum coherence, and **2**. the phi connectome. Before Orch-OR reduction, there are many possible linear superpositions of a multitude of possible ‘moments of consciousness’ or BINGs. Which BING remains after an Orch-OR reduction is a kind of Darwinian/free will selection of the most ‘fit’ of the conscious moments. Quantum state reductions are irreversible, so that Orch-OR moments of consciousness have a direction, moving from a conscious past to a conscious future (τ -τ). The stream of consciousness gives one the impression of movement in time from a past into a future, but the ‘past’, ‘present’, and ‘future’ of conscious time are not in accord with the past, present, and future of measured time, ‘t.’^{1} By contrast, equations of physics are symmetric in ‘t’, i.e. ‘t’ can be replaced by ‘-t.’ The index ‘x-k’ is used in Figure 8A to suggest that the phi connectome exists at scales^{2} smaller than the microtubule, and the index ‘x+n’ is used to suggest that the phi connectome exists at the scale of the cosmos. Bandyopadhyay proposes a clock-within-a-clock fractal structure. These clocks behave as if they were entangled forming a macroscopic quantum coherent structure. Each clock ticks faster than the clock immediately above it, but slower than the clock just below it in the frequency chain. When the largest/slowest clock ticks once, the innermost/fastest clocks have already returned their output, so that the largest clock perceives that the computation has been instantaneous, analogous with the idea that the brain is a macroscopic coherent quantum object. The fastest clock is the basic self-synthesized ‘seed’ from which the whole frequency fractal chain—the phi connectome, emerges. Perhaps this ‘seed’ is derived from an even earlier seed at the Planck scale.

## 3- Symmetry Breaking and Self-Synthesis

The basic formulation— **Asymmetry + Entropy à Complexity**, can be expanded:

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**Figure 11** [author’s original figure] HEEMFG* = High Energy Electromagnetic Field Generator.

Figure 11 illustrates the overall scheme of self-synthesis in a random system at thermodynamic equilibrium, where space and time symmetry are broken by entropy or rapid high-frequency vibrations (randomization, Brownian motion, heat, increase in entropy).^{5},^{11},^{13},^{17} Figure 12 below illustrates symmetry breaking in Figure 11 as a loop or dynamic process.

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**Figure 12** [author’s original figure]

Illustrating that a symmetric system in thermodynamic equilibrium can be ‘driven’ into a far-from-equilibrium state of broken symmetry by an increase in random motion/noise (entropy/ΔS↑) resulting in *complexity* /edge of chaos. ^{5},^{11},^{13} Complexity entails the notion of non-local, non-algorithmic formation of quasicrystals having the golden ratio, phi (φ) = 1.618 inherent in their structure. The golden ratio in quasicrystals is scaled up as a fractal, forming the phi connectome. A symmetric substrate can, in a sense, self-synthesize to become a quantum time quasicrystal in which both space and time symmetries are broken wherein the system exhibits long-range correlations, i.e. macroscopic coherent quantum-mechanical properties.^{4}, ^{5}, ^{8},^{10},^{11},^{12},^{13},^{14},^{16}, ^{20}

Pais’ patents^{11} utilize the same process, namely, accelerated vibrations cause symmetry breaking producing ‘far-from-equilibrium’ states: **quantum vacuum plasma + HEEMFG à local polarized vacuum**. We can relate these ideas to: 1. Glotzer’s work involving Monte Carlo simulations with polyhedra in a confined space, and 2. transformation of the *random* asymmetry graph #30 to the *complex* asymmetry graph #110 (Figure 13 below). Microtubule quantum coherence is maintained by *noise*, i.e. metabolic heat.^{4},^{5}

Brownian motion is, in itself, a non-local process, inasmuch as it acts on *all* parts of a substrate confined to a small volume and a small time interval. In the Pais patents, symmetry breaking occurs when external vibrations (e.g. a high energy electromagnetic field generator HEEMFG) of 100 MHz *or more* are applied to a symmetric (in thermodynamic equilibrium) substrate. If the applied external vibrations are *faster* (acceleration) than the vibrations of the particles in the substrate, i.e. faster than the relaxation time^{17} for vibrations in the substrate, then the substrate particles are ‘captured’ such that their randomly distributed vibrations become coherent, e.g. Cooper pairs and lattice distortion (Bipolaronic superconductivity).^{17} That is, the substrate exhibits broken symmetry, is pushed away from thermodynamic equilibrium, and becomes capable of doing work. This same process is seen in the Pais patents for room temperature superconductivity, and as a way of reducing inertial mass in a craft.^{11}

Figure 13 below illustrates how a random network becomes a complex small-world network.

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**Figure 13** [author’s original figure]

Figure 13 illustrates how a random network (asymmetry graph, A-graph #30) can be transformed into a complex small-world network (asymmetry graph #110) by increasing connections between nodes at short and long distances while decreasing connections between nodes at intermediate distances (PàV) *at all scales*. The formula S = k ln N applies to rule #110 (a small-world network), and a similar formula is found in Monte Carlo simulations with various polyhedra in confined spaces in which entropic bonds are formed when entropy is maximized, where W is the number of ways (microstates) of arranging the confined polyhedra, i.e., S = k ln W, where the *number of ways of being ordered is greater than the number of ways of being disordered*.^{4},^{20}

Note that A-graph #110 is scale free, i.e., a fractal. In A-graph #110, the area under H1 = the area under H2, indicating that the total information flow under A-graph #30 is merely ‘divided in two,’ a partition at all scales (fractal split) of information flow, from the symmetry of A-graph #30 to asymmetry of A-graph #110. In the phi connectome, we have a *spatiotemporal fractal structure* scaled up from the microtubule to the whole brain. At every scale the system is quantum coherent and exhibits bi-directional information flow such that the entire brain/body operates as a macroscopic coherent quantum object, a single oscillator.^{12} See Figure 19 which illustrates information movement in capacitors (tubulins) arranged in a spiral or helix. Coordination of these capacitors via the associated electromagnetic field of the Hinductor^{12} is analogous to quantum coherence in the microtubule caused by the effect of quasicrystal formation in vicinal water illustrated in figures 3, 4, and 5 above. Moreover, patterns of information transmission through the Hinductor shown in figure 5a of the Sahu patent^{8} ^{12} suggest bi-directional information flow in two interpenetrating golden helices/spirals having adjacent Fibonacci numbers 5 and 8. The microtubule in most mammals has 13 columns made up of two interpenetrating helices running in opposite directions, one having a pitch of 5 and the other a pitch of 8. The EEG may not reveal frequency phase shifts, but when the EEG is expressed in terms of frequencies of a complex equation, such as z2 = z + c, where z is a complex number (z = x + iy) and c is a constant, then the EEG can make sense. Plots of this equation define a frequency fractal or resonance chain. If the function diverges at each iteration, it is called an escape-time fractal (e.g., Mandelbrot, and Julia).

Vibrations from each layer or scale connect with vibrations of adjacent layers such that the entire system is interconnected, forming one coherent quantum object. Since the structure is not only a spacial fractal, but also a temporal fractal, then time has to be included which means that from the microtubule to the neuron to the whole brain and body, a chain or resonance band is formed, extending from A to H, giving a complex function AJ = F(x, y) + iG(x, y), a resonance chain or frequency fractal. The system has the structure of a clock within a clock within a clock, etc. where all the innermost clocks are imaginary. Each clock has its own resonance band and shares a common region in its resonance band with adjacent inner and upper layers forming a chain of resonance bands, a frequency fractal. The brain and somatic tissues are *a whole*.

**[...]**

- Quote paper
- Dr. Marshall Goldberg (Author), 2020, Heat-Induced Quasicrystal Formation in Vicinal Water. Quantum Coherence in Microtubules and the Phi Connectome, Munich, GRIN Verlag, https://www.grin.com/document/961537

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