ic/tzo: image urls upd
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<center>(1)</center>
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<center>(2)</center>
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<center>(4)</center>
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<center>(6)</center>
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@ -12,7 +12,7 @@ The essence of the problem is that Schrödinger's wave function, used to describ
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Thus, a quantum object, when viewed from the classical world, always appears as a simultaneous collection or superposition of incompatible states. Due to this fundamental ambiguity, any measurement of a quantum object's state cannot be predicted precisely and only provides probabilistic values. Yet, the wave function itself is quite deterministic — in terms of complex numbers.
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<center>(7)</center>
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@ -44,7 +44,7 @@ Overall, such a radical revision of traditional scientific views on reality was,
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But without the author himself, who was disappointed with his colleagues' reaction to his discovery. Immediately after defending his dissertation, Hugh Everett essentially parted ways with the "queen of physics" forever. [i5]
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<center>([Read more](/tbc/32/))</center>
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<center>(9)</center>
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@ -10,7 +10,7 @@ All that is known on this matter today can only be gleaned from the memoirs of p
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There are also a few brief but extremely inspired letters from Pauli, where the essence of his great insight is conveyed by words like: "**Division and reduction of symmetry, this then the kernel of the brute!**… Now we are on the trail!". [o10]
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<center>(10)</center>
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@ -20,7 +20,7 @@ To make it clearer what is being discussed here, it should be reminded that for
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To put it more poetically, the scientist was searching for ways to return to the modern scientific conception of the world the ancient notion of the "soul of matter". A notion commonly accepted in antiquity, but completely lost over the last 300 years. [i8]
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<center>(11)</center>
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<center>(12)</center>
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@ -26,7 +26,7 @@ As a juggler practitioner, he even attempted to master a peculiar juggling techn
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Another interesting work by Shannon in the same vein was a "philosophical" sculpture depicting a juggler tossing other, smaller jugglers, who, in turn, juggled even smaller jugglers…
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<center>(14)</center>
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@ -38,7 +38,7 @@ Another striking example of the same series was a specialized minicomputer const
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In short, if you look at all these unserious entertainments of Claude Shannon from a slightly different, physico-theoretical perspective, linking them with the studies of Everett and Pauli, you can discover rather interesting things.
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<center>([Read more](/tbc/41/))</center>
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<center>(15)</center>
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@ -14,7 +14,7 @@ The movement of a point generates a 1-dimensional line, specifically, a straight
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Similarly, 3-dimensional space can be generated by parallel translation of a plane or rotation of a plane around one of its lines. It is clear that this process can be developed further – towards the generation of spaces of higher dimensions.
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<center>(16)</center>
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@ -22,7 +22,7 @@ The idea of rotation is embedded in the foundations of mechanics and geometry fr
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The equation describing the motion of a point in such a system, as it turned out, is equally suitable for describing the oscillations of a weight on a spring or a pendulum on a suspension, for the sinusoidal propagation of waves, and for describing the modes of string vibrations. Due to obvious connections with music, the system is called a harmonic oscillator.
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When classical physics was replaced by quantum physics, it quickly became apparent that the harmonic oscillator plays no lesser role there. More precisely, a much greater one. Not only because the strictly discrete natural frequencies of a musical string's sound are a direct mechanical analogy for the allowed orbitals of an electron in an atom. But also because the wave equations of quantum objects are fundamentally built on the idea of oscillations and the mathematics of complex numbers. And this mathematical apparatus is essentially ideally suited for solving problems about the motion of a point in a circle (in the phase space of states).
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@ -34,7 +34,7 @@ Thus, in classical mechanics, the "Foucault's pendulum" – the rotation of the
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When – in the 19th century – the tools of multidimensional geometry began to be included in the standard arsenal of mathematics, the most curious researchers began to tackle the problems of perception. In other words, long before the appearance of the concept of 4-dimensional space-time, work began to be devoted to how an observer from, say, a 2-dimensional world would perceive 3-dimensional objects. [o16]
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One of the most characteristic examples of this kind is the passage of a 3-dimensional sphere through a flat world. To the inhabitants of the plane, it would initially appear as a tiny point, then a circle of variable – first growing, then decreasing – diameter, and finally again as a vanishingly small point. Using this analogy, it is significantly easier to imagine that a 4-dimensional sphere passing through a 3-dimensional world like ours would appear as a spherical object with a size changing from zero to a maximum diameter.
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@ -42,7 +42,7 @@ From this picture, it is logical to move on to constantly oscillating quantum pa
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Another important aspect of the quantum world is that particles of matter interact – quantum entangle – not directly but necessarily through a photon or a mediator particle. For the geometric description of this picture, it is essential that a moving photon can be represented by its plane of polarization. Typically, if a photon is reflected by a particle, the plane of the wave propagates without rotation (linear polarization). But if the photon is emitted by a particle, then the plane of polarization rotates around the axis of propagation (circular polarization).
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Looking at the picture from this perspective, it is easier to imagine the mechanism of forming quantum entanglement. When a photon departs from one particle, they are already entangled, and the photon's plane of polarization already carries information about the quantum state (spin direction) of the particle. When the photon reaches another particle, the plane of polarization makes a "slice" of its current state. If the particle’s diameter is maximum, then the interaction takes place at its maximum. And if the cross-section diameter is zero at that moment, then there is no interaction and, consequently, no entanglement at all.
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@ -72,7 +72,7 @@ If we recall that Hugh Everett built his concept on Shannon's theory of informat
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In an ultra-brief formulation, this idea is since some time expressed by the maxim "**It from bit**" . And it is unlikely a coincidence that such a beautiful aphorism was invented – at the end of his long life – by John Archibald Wheeler [i12]. The teacher of Feynman and Everett, who managed to outlive his students by several decades.
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<center>([Read more](/tbc/42/))</center>
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<center>(19)</center>
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@ -6,7 +6,7 @@ The theoretical construction of the multiverse in Hugh Everett's dissertation wa
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One notable feature of the Dirac equation is that it can be written in a unique form, sometimes referred to as the zigzag representation of a spinor [o19]. In this description, every electron (or another massive fermion with spin 1/2) appears as a particle moving along a zigzag trajectory and is in a state of continuous oscillations between the left-handed "zig" phase and the right-handed "zag" phase. Each of these alternating states, by itself, is massless, and mass arises only when the entire scenario is considered collectively.
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In this description, there is an coupling constant, which in Dirac's theory controls the speed of shifts between the "zig" and "zag" parts of the Dirac spinor. In the later Higgs theory, which appeared in the 1960s, this constant turns into a special — Higgs — field, which enters into equations as another interaction that gives fermions mass…
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@ -30,11 +30,11 @@ The still poorly studied physics of granular media [i18] — sand, powders, susp
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Speaking more specifically about oscillons, the main feature of this variety of waves in a granular medium is their rare stability. Once arisen, this solitary wave can rise and fall, maintaining its identity indefinitely long — as long as the experiment lasts.
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Another equally important feature of oscillons is the specificity of their interaction, explicitly referring to the long-standing Bjerknes’ theory of pulsations. Being in the same phase of oscillation, oscillons repel each other, and being in opposite phases, they attract each other.
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Putting the facts slightly differently, the new discovery has revealed remarkable possibilities. By combining oscillons with Bjerknes’ theory, a clear and comprehensible explanation emerges — not only for well-known phenomena (which are explained in textbooks rather clumsily), but also for the still-mysterious secrets of electricity and magnetism.
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@ -56,7 +56,7 @@ In mathematical terminology, a situation when all orientation directions of elem
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In classical physics, a very fitting example of this phenomenon is provided by the phenomenon of antiferromagnetism. Like other substances with magnetic properties, antiferromagnets consist of molecules with a dipole moment that behave like tiny magnets. At high temperatures, all these little magnets are oriented randomly within the substance, meaning that every direction is equally probable, and the entire system as a whole is symmetrical.
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When the system's temperature drops, at a certain point, spontaneous alignment of the magnets along a single axis occurs. Symmetry of directions in the system turns out to be broken. Moreover, in antiferromagnetic substances, each magnet, during spontaneous ordering, aligns antiparallel to its neighbors. In other words, their common direction's axis is one, but the poles of neighboring molecules are pointed in opposite directions.
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@ -64,7 +64,7 @@ Juxtaposing this picture of spontaneous symmetry breaking with the phenomenon of
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Formulating more precisely, it would be more accurate to speak of the electron not as a "hilltop" but as the lower point of a conical "pit" of the oscillon. Because in conditions of a double membrane, constantly in a state of vibrations, phases of the oscillon like "hill" turn out to be less stable and perform the role of anti-particles. That is, they disappear as a result of annihilation. Such a pair of membranes vibrating in antiphase is commonly named in modern physics as a "brane-antibrane" system.
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Thus, on a pair of membranes, only the stable version of oscillons remains — in the form of a conical "pit" (the proton) and its "bottom" in the form of a point-like microvortex (the electron), synchronously jumping from one surface to the other — along the time axis [i16]
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<center>(23)</center>
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@ -18,7 +18,7 @@ Not to mention that, on a theoretical level, no one has yet succeeded in beautif
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But before delving into the peculiarities of the mechanism connecting quantum gravity and quantum entanglement in the model, it is useful to consider several important ideas and consequences that arise from the overall two-brane construction. One central idea here is the **concept of the universe as a closed one-sided surface**. The simplest example of one-sided surface is the **Möbius strip**. [i21][i22]
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Accordingly, the world as a Möbius strip is the simplest and most natural explanation for why the number of positive and negative electric charges in the universe is always equal. So the total electric charge of the universe remains invariably equal to zero.
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@ -56,7 +56,7 @@ Soon after, in 2001, an essentially similar experimental result emerged. In lase
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Studies of nonlinear optics phenomena are vital on their own and particularly interesting for sharing many similarities with the physics of quantum superfluids like Bose-Einstein Condensates. Specifically, the behavior of quantum vortices in BECs and laser optics is described by similar equations.
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As laser experiments have shown, once a spirally twisted light beam passes through a cylindrical lens, its previously round core starts to flatten into an elongated ellipse, stretching into a thin line that is nearly nonexistent. After the light passes the lens's focus — or "**compression point**" — this line reshapes into an ellipse, with energy inside circulating in the opposite direction…[i23]
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@ -64,7 +64,7 @@ As laser experiments have shown, once a spirally twisted light beam passes throu
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A notable feature in the mechanism of an optical vortex or "topological charge" inversion is the experimentally observed phase where it extends into a thin line or vortex tube.
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This result is particularly fascinating for two reasons. First, as the image of this phenomenon evidently resembles astronomical images of spiral galaxies with a bar in the core. This same concept — using the metaphor of a spinning "garden sprinkler" — often appears in popular explanations for a range of physical theories, from nuclear physics to superstrings and quantum gravity. [i28]
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<center>(27)</center>
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@ -34,11 +34,11 @@ In other words, at the moment of convergence of the branes, the tube's axis is p
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Here, to complete the picture, it remains to find in the phase of brane convergence also the fermionic partner for each quantum of light or single photon. For this, it is time to recall that unusual particles are emitted from both ends of the vortex tube during the brane convergence. One of them – the tachyon leaving the membrane – should be given a separate section further, while the second particle, eventually having the hallmark of a graviton (spin 2) and consisting of what seems like two parts – with a "longitudinally divided mode" – is exactly what is needed for SUSY. [i33]
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There is reason to believe that at the phase of maximum brane convergence, this particle appears as a pair of flat identical vortices rotating in opposite directions – like a vortex ring "in section". This configuration is known in physics as the "Kelvin Oval" [i34], has soliton properties – a stable solitary wave – and is most famous for propagating strictly in a straight line as a single entity. During the branes convergence the axes of rotation of the vortices in this pair are perpendicular to the membrane, so geometrically they should be considered fermions at this moment.
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And remarkably, the total spin of this pair of vortices with an antiparallel combination of rotation axes sums up from the values (+1/2) and (–1/2). That is, equal to zero – as with the **Higgs boson**, whose influence is presumed in the Standard Model to be responsible for **generating the inert mass of quantum particles**…
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@ -66,7 +66,7 @@ Other substantial details in Randall and Sundrum, however, look different. But h
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In the RS model, it is assumed that only the 3 known interactions of quantum physics operate on our brane, while all gravity is concentrated on the second membrane, called the "gravitybrane." Calculations of the model based on General Relativity showed that the energy of the branes in this situation curves the fifth dimension extremely strongly, giving the entire structure very specific features.
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In essence, the strongest changes in sizes, masses, and even the flow of time that occur in this two-brane configuration when shifting through the fifth dimension are very reminiscent of the powerful space-time deformations near cosmic black holes. However, if we assume that particles from the Standard Model physics are located on one of the branes, then according to calculations, they inevitably must have a small mass. This result means that, in principle, the hierarchy problem could be solved in a completely natural and automatic way…
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Each of the two branes at the edges of the RS model is similar to the world familiar to humans – flat space without any special gravitational curvature effects. Moreover, the same picture is characteristic of any layer given by a slice of space through any point along the fifth dimension axis. All layers also have flat geometry.
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However, as a whole all these 4d-layers are glued to each other so that the five-dimensional space is very strongly curved. That is, **when transitioning in the fifth dimension from one layer to another, the scales of sizes, locations, time, mass, and energy change significantly**. But although the values of masses for particles vary greatly depending on the position in the fifth dimension, all physics invariably continues to appear 4-dimensional…
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@ -122,7 +122,7 @@ And if the extremely non-trivial geometry at the core of this model and string t
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And since the compactified, in principle unattainable micro-spaces of string theory now turn out to be the same as our macrouniverse, only with a previously hidden structure, very interesting prospects open up.
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Calabi-Yau spaces are famous for extremely intricate configurations of geometry with many holes and transitions. And **in topology, any hole means an alternative shortest path from one point of space to another**. In other words, along with the new map of the universe, the perspective of fast travels to previously unthinkable distances simultaneously appears. We just need to learn how to move along frequency layers…
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<center>(31)</center>
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@ -20,7 +20,7 @@ The last of the mentioned species of well-known quasiparticles – anyons – de
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The remarkable properties of anyons are important for several reasons. Firstly, because due to relativistic effects impacting the body of a very rapidly rotating proton, there are reasons to believe that a spherical particle can take the shape of a flat disk. And for the particle-components of the proton, rotating inside this energy vortex, unclear quantum properties are characteristic. Quarks are not quite fermions, gluons – not quite bosons.
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Secondly, in the field of hydrodynamics, which often comes to theorists' aid in understanding the mysteries of nuclear physics, there is a phenomenon close in essence and called Hyde's vacillation or wavering. Its essence lies in the fact that in flat rotating systems of nature often observed a phenomenon of self-organization in the form of a specific oscillatory process. The phase of regular waves in a liquid or gas periodically alternates with a phase of turbulent vortices, which then again are replaced by regular waves. And so on. That is, stable vacillation of the system occurs between states of order and chaos. There are reasons to believe that a similar process of vacillation of the system between quark-vortices and gluon-waves occurs in the proton during its rotation. [i42]
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<center>(36)</center>
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Completely independently of these works, a renowned theorist and Nobel laureate Frank Wilczek put forward his own set of substantive ideas in early 2012 about a specific physics-mathematics framework, one that continuously generates threads of matter’s memory in the form of crystalline structures. In particular, Wilczek demonstrated that both in classical and quantum mechanical descriptions of our world, it is possible — as it turns out — to construct crystal structures in the 4th dimension, that is, in time, in a consistent and mathematically grounded way. [o36]
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These kinds of crystals turn out to be as stable as crystals in 3-dimensional space, as they are generated in cycles of oscillations of rotating systems in their most stable state of energy minimum. Particularly interesting results were achieved by Wilczek when analyzing "time crystals," as he called them, in conditions of quantum mechanical systems — where twisted elongated spiral structures form in imaginary time…
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By carefully reformulating the Riemann Hypothesis into a series of weaker statements, Matiyasevich used a computer program to calculate and plot on the complex plane the trajectories of certain characteristics-iterations, collectively painting a picture of the "hidden life of the Riemann function."
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In these graphs, two classes of objects are distinctly visible, located on different sides of the critical line-divider passing parallel to the imaginary axis through the point 1/2. The objects on the left were named "electrons" by the author since their trajectories resemble those of particles colliding and diverging. The objects on the right behave differently, resembling twisted double spirals, and were named "trains" by Matiyasevich.
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<center>(41)</center>
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A fundamentally important detail of this problem is the intentional words "appears to be" and "impression." The importance of this nuance can be illustrated by the example of the so-called "wave-particle duality" of quantum particles – perhaps the most famous natural phenomenon with a dual description of its physical properties.
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If we look more closely at the birth and establishment of this fundamental "duality" in science, it's not too difficult to notice the following. Had historical circumstances been slightly different, and if the wave (de Broglie and Schrödinger) views on quantum mechanics had gained a dominant role, the overall picture might have turned out to be far more coherent and comprehensible.
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Since the 19th century, hydrodynamics has known the so-called solitary waves (solitons), whose behavior largely corresponds to the nature of particles [i50]. Why this happened is a different long story [i51], but scientists began seriously studying the physics of wave solitons only a century later, starting in the 1960s. In other words, when quantum physics, based on an alternative concept of particles, had long been in a state of maturity and triumphant success.
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In this way, the "incomprehensible," supposedly, wave-particle duality ended up being embedded in the foundation of a grand scientific edifice. It became a kind of basis for the subsequent construction of an entire tower of new paradoxes and difficult-to-explain dual descriptions of nature. Rebuilding the entire structure on the basis of purely wave representations for the sake of conceptual integrity and harmony of the theory seemed, to say the least, irrelevant for the scientific majority…
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An aspect of holography, which is quite significant but has been practically untouched so far, is the principle of self-similarity. Due to the peculiarities of recording wave information in a hologram, any fragment of a holographic snapshot — in contrast to a photograph — reproduces the entire image as a whole, only with possibly fewer details.
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Manifestations of this principle of self-similarity can be seen everywhere: from the Mandelbrot fractal in mathematics and fractal geometry in nature to obvious analogies in the structure of the atom, solar system, and galaxy. Here, however, it is especially useful to consider a less known example of constructive analogies of nature — based on liquid crystals. [i57]
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<center>(46)</center>
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The hierarchical **structure for the general complex of human knowledge** about the world — i.e., for "philosophy" in Descartes's terminology — should, in his opinion, look as follows: "All philosophy is **like a tree**, of which Metaphysics is the root, Physics the trunk, and all other sciences the branches that grow out of this trunk, which are reduced to three principals, namely, Medicine, Mechanics, and Ethics."
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Let us ignore the obvious archaism, so to speak, of Descartes's classification of sciences, and focus only on the tree-like structure of knowledge itself. And also recall — for a complete picture — the famous legend about the circumstances under which Descartes invented his coordinate system.
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In constructing *p-*adic numbers, something similar is done, but as the base, a prime number is taken — divisible only by itself and 1 (in the German language, such an object is called *Primzahl*, which suggested to Kurt Hensel naming his discovery *p-adischen Zahlen*). Hensel found that if rational numbers, i.e., fractions, are expressed in a specific mathematical way (using modular arithmetic) through the powers of a prime number, a special, entirely full-fledged world of numbers results.
|
||||
|
||||
|
||||
|
||||
|
||||
Most importantly, this world provides a convenient approach to well-known complex problems in mathematics. In particular, the *p*-adic framework has proven very useful for clarifying general questions about the solvability of algebraic equations.
|
||||
|
||||
@ -98,7 +98,7 @@ Well, the direct relevance of this entire construction to the mathematics of *p-
|
||||
|
||||
So from the very beginning and up to this day, *p-*adic numbers have been although not the only, but undoubtedly the most important example of ultrametric spaces. Or a system of infinitely nested spheres "whose center is everywhere and the circumference nowhere."
|
||||
|
||||
|
||||
|
||||
|
||||
<center>(49)</center>
|
||||
|
||||
|
||||
@ -1,4 +1,4 @@
|
||||
|
||||
|
||||
|
||||
<center>(52)</center>
|
||||
|
||||
@ -20,23 +20,23 @@ In principle, there are various ways to discuss this " structure of our space" d
|
||||
|
||||
Basically, Heinz Hopf found a way to fill all of space with circles. Generally speaking, for this task, there are simple solutions like this – take a straight line and string onto it concentric circles ad infinitum.
|
||||
|
||||
|
||||
|
||||
|
||||
However, Hopf tackled a more general task – the construction of a mapping of a three-dimensional spherical surface or 3-sphere, located in 4-dimensional space, onto the more familiar 3-dimensional Euclidean space, which is commonly referred to as flat and denoted **R<sup>3</sup>**.
|
||||
|
||||
In a sense, this task is analogous to the problem of how to display the surface of a globe – or 2-sphere – onto the surface of a flat map. It is clear that any form of projection inevitably introduces various distortions to the picture. For this purpose, Hopf applied the known stereographic projection in geography and geometry that preserves angles between lines (this is called conformal transformation), and circles are also translates into circles or straight lines (alternatively, circles of infinite radius).
|
||||
|
||||
|
||||
|
||||
|
||||
If one develops the same analogy with the globe, i.e., the more familiar 2-sphere, one of the important features of the mapping studied by Hopf is this. When the points of the 3-sphere forming the surface in 4-dimensional space are arranged on such a globe strictly along the line of "latitude," then in Euclidean space **R<sup>3</sup>**, this configuration corresponds to a shape known as a torus of revolution (and in form corresponds to a vortex ring).
|
||||
|
||||
|
||||
|
||||
|
||||
The thickness of the tube of such a torus changes depending on the location of the latitude between the plane and the projection point. As the latitude shifts from the projection point, the torus passes through all intermediate states between two extreme ones. In one extreme, becoming ever thinner, it degenerates into a circle. In the opposite case, the torus swells to such an extent that its "hole" degenerates into a straight line perpendicular to the equatorial plane.
|
||||
|
||||
In other words, Hopf filled all of space **R<sup>3</sup>** with nested tori. But the most important thing here is this. For each point on the globe, located on the latitude line, there is a corresponding circle line on the torus surface. This line captures the "donut hole" and obliquely encircles the tube. Just as a multitude of points fill the entire circumference of latitude, so a collection of such rings, linked with each other, completely covers the surface of the respective torus.
|
||||
|
||||
|
||||
|
||||
|
||||
For historical reasons, such circles on the torus are called Clifford parallels – after an English mathematician who introduced these objects in the 19th century to study the properties of curved spaces. Therefore, this whole construction is sometimes called the Clifford-Hopf fibration. Fibers here are those linked circles that form the surfaces of the tori, and hence – fill the entire volume of space.
|
||||
|
||||
@ -44,7 +44,7 @@ This initial construction gave rise to an extremely fruitful direction in topolo
|
||||
|
||||
By the end of the 1970s, however, it became clear to physicists that the Hopf fibration plays a fundamentally important role in gauge approaches to quantum field theory. It also effectively served as the core of the entire model in Roger Penrose's twistor theory, and later in several other approaches to quantum gravity theory.
|
||||
|
||||
|
||||
|
||||
|
||||
Today, the list of possible physical applications for this construction is very long – from magnetic monopoles to the polarization of transverse waves and solid-state mechanics, from the geometric properties of quantum entanglement and the organization of qubits in a quantum computer to the relativistic deformation of the celestial sphere. [o52]
|
||||
|
||||
@ -94,23 +94,23 @@ Initially, it's helpful to consider how such seemingly different configurations
|
||||
|
||||
With the help of mathematical programs and computer graphics, among other things, specialists have shown how a 2-dimensional torus through smooth topological transformations (called homotopy) transforms through compressions, pinches, and stretches into a double-layered Riemann sphere. [o55]
|
||||
|
||||
|
||||
|
||||
|
||||
The grid of cells applied to the surface of the torus and twice reproducing the well-known configuration of a soccer ball is, of course, not accidental. Firstly, it allows for a more clear view of the deformation process, demonstrating the topological equivalence of the two distinct structures.
|
||||
|
||||
|
||||
|
||||
|
||||
Secondly, as shown by fullerene molecules in nature, such a configuration is optimal in terms of minimizing energy on a spherical surface. Thirdly, most importantly, science today has substantial evidence that this very configuration underlies the structure of the universe — as a network of cells formed by superclusters of galaxies. [i71]
|
||||
|
||||
|
||||
|
||||
|
||||
More precisely, when carefully articulated, the observational data available to science allows for the supposition of a cosmic form known as Poincaré's dodecahedral space. A simplified model of such a configuration is a ball sewn from 12 pieces in the shape of regular pentagons. Or, otherwise, a regular polyhedron-dodecahedron inflated to a sphere. [i72]
|
||||
|
||||
|
||||
|
||||
|
||||
Distinct signs of this form of the cosmos were detected by J. Luminet's group in the cosmic microwave background radiation maps from the WMAP satellite [i73]. Polish researchers from Torun University identified on the same map 6 pairs of matching circles, even more definitively indicating the features of a closed-universe space in the shape of a dodecahedron. [i74]
|
||||
|
||||
|
||||
|
||||
|
||||
To clarify that this significant (but for some reason hushed-up) scientific discovery of the 21st century does not contradict the "classic" form of a soccer ball made out of 12 pentagons and 20 hexagons, one should note that even in the 32-cell variant, there are the same 6 pairs of "circles" as in the dodecahedron. Also, recall the dual topology of the cosmic sphere and the physics of forming convective cells in superfluid liquids. [i75]
|
||||
|
||||
@ -120,7 +120,7 @@ And if on one side of the sphere 12 dodecahedral structure cells form, on the ot
|
||||
|
||||
In the end, when this entire configuration stabilizes to a minimum-energy state, on each side of the sphere — inside and outside — there are identical grids of 32 soccer ball cells, shifted relative to each other by convective processes. These vortex processes effectively "cut off" the energy-costly vertices of the dodecahedron and the icosahedron, overlaying both structures shifted onto each other and ultimately generating a symmetrical, energetically optimal construction.
|
||||
|
||||
|
||||
|
||||
|
||||
If the key role of (topological) vortex effects in the picture of the formation of the cosmic cellular structure has become more or less clear, it is time to remind of the following. All the discussions here are certainly not proofs and, even more so, not the absolute truth.
|
||||
|
||||
@ -132,19 +132,19 @@ Continuing in the same direction, it is time to more thoroughly examine why the
|
||||
|
||||
As an elementary example illustrating what comprises a trivial fibration, the form of a cylinder is typically referenced, formed from segments stemming from multiple points on a circle. The surface of such a cylinder is called a circle fibration, with its constituent segments dubbed fibers.
|
||||
|
||||
|
||||
|
||||
|
||||
Accordingly, a fibration is called nontrivial if the surface formed by the fibers demonstrates one-sidedness rather than being ordinary. The simplest example of such a surface — a nontrivial fibration — is the Möbius strip.
|
||||
|
||||
To make it clearer how in the nontrivial Hopf fibration, where the fibers are circles, a crucial structure like the Möbius strip is present, it's helpful to introduce two seemingly different figures that are, in fact, topologically equivalent. One in the form of a ladder graph represents the traditional essence of the Möbius strip as a one-sided surface, while the other, employing the rubbery properties of topology, elongates the same graph into a circle, with opposite points connected by segment-fibers.
|
||||
|
||||
|
||||
|
||||
|
||||
The extent to which the nontrivial topology of a Möbius strip can be significant from the perspective of physics was recently beautifully demonstrated once more by a group of Chinese researchers from Beijing's Institute of Theoretical Physics [o56]. In 2009, they published a theoretical paper on the electronic properties of a sheet made of a new material, graphene, and taking the shape of a Möbius strip.
|
||||
|
||||
In this research, calculations demonstrated that a graphene Möbius strip behaves as a "topological insulator with a robust metallic surface" [i76]. That is, electron movements occur losslessly along the edge of the strip, while the rest (bulk) of the strip doesn't conduct electricity, showcasing insulating properties. In other words, the topology of the shape engenders the material's unusual properties.
|
||||
|
||||
|
||||
|
||||
|
||||
Three years later, in May 2012, theoretical work from the Institute for Nuclear Theory in Seattle, USA, showed that if the known physical properties of a topological insulator are assumed for the space-time of the entire universe, then it is possible to discern a completely natural topological mechanism that generates precisely three generations of fermion particles. [o57]
|
||||
|
||||
@ -156,7 +156,7 @@ To spectacularly enhance the same "layered" theme, one can revisit the smooth ho
|
||||
|
||||
One of transformations shown by Trott through animated clips illustrates the morphing process between the already known two-layered soccer ball and the trefoil knot — another noteworthy form rich in its topological properties.
|
||||
|
||||
|
||||
|
||||
|
||||
<center></center>
|
||||
|
||||
@ -182,21 +182,21 @@ Among the key achievements of the holographic approach in modern theoretical phy
|
||||
|
||||
To make the direct connection between AdS/CFT and Hopf fibration more clear and illustrative, it is useful to describe two different but equivalent approaches to filling the volume with curved surfaces. One of these methods, (a), is already familiar to us and represents Clifford parallels in the form of circles forming tori. The second method, (b), linearizes the first, so that Clifford parallels indeed become line segments, but at the same time form a curved "ruled surface" or hyperboloid of revolution with negative curvature. The boundary of such a surface is a circle or 1-sphere.
|
||||
|
||||
|
||||
|
||||
|
||||
Based on these images, showing the connections with AdS/CFT becomes simpler. Because the outer part of a torus with positive curvature can be likened to the world of the spherical boundary system of CFT (here dimension 2D). And the inner space of the "hole," bounded by a hyperboloid of negative curvature, can be considered as the world of AdS (dimension 3D, respectively).
|
||||
|
||||
In this modeling approach, AdS space-time looks like a stack of flat (2D) circles, each having a hyperbolic geometry of space, and all stacked on top of one another along the vertical time axis (forming 3D).
|
||||
|
||||
|
||||
|
||||
|
||||
If one makes a cross-section of the torus at any given moment in time, then every circle in the AdS world corresponds to a latitude circle on the outer shell — a snapshot of “our” CFT world. A 1D world that, along the same time axis, moves from the past (the bottom of the torus) to the future (the top of the torus).
|
||||
|
||||
|
||||
|
||||
|
||||
Since the "AdS world" geometrically resides in the "hole" of the torus, and every circle in the Hopf fibration forming the surface of the torus necessarily contains this "hole" inside, an intriguing possibility is visible for the point inhabitants of the "CFT world" living at the latitude. If considering the inclined circle of the Hopf fiber as their "memory," i.e., the basis of consciousness, the space inside that circle falling within the "hole" of the torus can be thought of as the 2-dimensional "hologram of consciousness." Moreover, due to the geometric features of the inclined section, this hologram allows the inhabitants of the "CFT world" to travel within their consciousness both in space and time.
|
||||
|
||||
|
||||
|
||||
|
||||
As everyone knows, in a similar way — by the "power of thought" — people in our world can travel through space-time in their dreams, reveries, and in memories of near-death experience linked to staying in the other world of spirits and souls of the deceased. In other words, there are reasons to name this space — geometrically inseparably connected with ours — as the space of the other world.
|
||||
|
||||
|
||||
@ -1,4 +1,4 @@
|
||||
|
||||
|
||||
|
||||
<center>(58)</center>
|
||||
|
||||
@ -24,7 +24,7 @@ Particularly noteworthy in this picture are the following points: First, in anci
|
||||
|
||||
In particular, in Plato's work "The Republic," the functioning of the world order is explained as the result of the action of the cosmic spindle of Ananke. In the image of the goddess Ananke, we should remind, in Greek mythology, represents Necessity or Inevitability.
|
||||
|
||||
|
||||
|
||||
|
||||
Ananke was the mother of the three Moirai — executors of human fates. She herself, according to Plato, governs the entire spinning of the fabric of the cosmos. Between Ananke's knees rotates the spindle, whose axis is the axis of the universe. And the daughters-Moirai — as best as they can — assist this universal rotation…
|
||||
|
||||
@ -56,7 +56,7 @@ Far more significant for the physic-mathematical science are the 12 indicated su
|
||||
|
||||
Why and to what extent this is important for the modern scientific worldview will become clearer as we delve into the mythological ideas of the ancients, the religious philosophy of the East, and the revelations of mystics of the West.
|
||||
|
||||
|
||||
|
||||
|
||||
<center>(59)</center>
|
||||
|
||||
@ -66,7 +66,7 @@ Under the number 59, it should be clarified, this remarkable dream appears in th
|
||||
|
||||
That is, Pauli understood quite definitely that from the depths of his own consciousness he received through this image some extremely important information about the workings of the universe. The scientist remembered the picture perfectly and in details, but — despite his undeniable talents in mathematics and physics — was unable to comprehend what exactly this picture meant.
|
||||
|
||||
|
||||
|
||||
|
||||
The compact format of this book is not very suitable for detailed decoding of the entire construction of the "World Clock," however, to explain the key parameters of the structure — two mutually perpendicular dials with 32 sectors— there is quite enough space. But for the result of the decryption to appear as a natural and convincing explanation, it's necessary to recall some facts from the history of clocks.
|
||||
|
||||
@ -96,7 +96,7 @@ An unquestionably important point is that here we have an absolutely precise coi
|
||||
|
||||
Equally significant seems another — hitherto unmentioned — feature of the "clocks of the world," referring to the process of overall evolution of the cosmos and humankind. In Pauli's dream, this is conveyed as a change of color in the outer ring of the dial: "A ring borders the circle, which was once dark, but now is golden."
|
||||
|
||||
|
||||
|
||||
|
||||
<center></center>
|
||||
|
||||
@ -162,7 +162,7 @@ If we attempt to translate the Buddhist worldview into the language of contempor
|
||||
|
||||
Besides offering a beautiful explanation for the striking difference in the forces of electromagnetism and gravity through the use of two branes, the "RS model" also provides mathematical justification for a multi-layer structure of multiverse space, located between two boundary membranes.
|
||||
|
||||
|
||||
|
||||
|
||||
<center></center>
|
||||
|
||||
@ -186,7 +186,7 @@ Intuitively, it's probably not difficult to comprehend that by combining these t
|
||||
|
||||
To make the general dynamics of the development of this geometric structure in time and space more comprehensible, it's useful to recall two significantly different ways through which the space is filled with circles: (a) a trivial way and (b) a nontrivial Hopf fibration.
|
||||
|
||||
|
||||
|
||||
|
||||
By a non-accidental, naturally occurring coincidence, these two images in the simplified (a) and more intricate — but also more adequate — view (b) reflect the process of the evolution of the world and humanity as envisioned by ancient Buddhists and Empedocles. To put it differently, we can say that these are two different projections that display a unified multi-dimensional structure as an easier-to-understand three-dimensional image for us.
|
||||
|
||||
@ -228,11 +228,11 @@ One of the most remarkable features of the Hopf fibration is that important deta
|
||||
|
||||
Speaking specifically about mathematics, the inclined circles forming the torus surface were historically discovered and described by the French astronomer Yvon Villarceau back in 1848. That is, almost a century before Hopf's discovery of topological fibrations, Villarceau showed that when a torus is bisected by a plane touching its surface twice, a characteristic figure is formed, consisting of two circles whose centers lie on each other's perimeters.
|
||||
|
||||
|
||||
|
||||
|
||||
A direct outcome derived from the discovery of Villarceau's circles is this fact. Through any randomly chosen point on the torus surface, it turns out that four different circles always pass. Two are obvious and relate to the torus as a rotating figure's circular generatrices. Thus, one circle lies in a plane parallel to the equatorial plane of the torus, and the other — perpendicular to it. As for the other two, less-obvious circles, these are two "oblique" Villarceau circles, generally lying in different planes corresponding to different bisecting planes.
|
||||
|
||||
|
||||
|
||||
|
||||
For the constructed correspondences picture, these 4 circles intrinsic to each point on the torus matter for the following reasons. The first two circles — horizontal and vertical —indicate the universe's closure on space and time axes. The skew circle of Villarceau, clearly, signifies the "world line" trajectory of any object in the universe's evolution. But the fact that TWO such skew circles actually pass through any point — that fact requires particular attention and consideration.
|
||||
|
||||
@ -244,7 +244,7 @@ Yet, geometry allows for situations where both circles end up in one plane or be
|
||||
|
||||
Naturally, it's no accident that such a configuration was known in the so-called "sacred geometry" of various Eastern and Western cultures for many centuries and even millennia before Yvon Villarceau's discovery. This figure appears under different names but is most frequently called Vesica Piscis in mystical literature of the European tradition of the last 3-4 centuries, which translates as "fish bladder" from Latin.
|
||||
|
||||
|
||||
|
||||
|
||||
For a clearer understanding of the natural interpretation of this "sacred" geometry from the perspective of a person's psychological self-realization, two points ought to be emphasized. Firstly, the well-known formula that "nature is a sphere whose center is everywhere and circumference nowhere."
|
||||
|
||||
@ -258,7 +258,7 @@ Such profound ideas of universal unity, in fact, can be discovered in the simple
|
||||
|
||||
The essence of these ideas, however, people have always been able to grasp without any science. Undoubtedly curious and worthy of mention is the fact that the central element of the Vesica Piscis, having a special name Mandorla (or "almond" in Italian), was quite popular in the religious visual art of medieval Europe.
|
||||
|
||||
|
||||
|
||||
|
||||
The characteristic form of the Mandorla was often used to depict a glowing (or conversely dark) halo around Christian saints, the Virgin Mary, and her son Jesus. What exactly the ancient masters intended in this image is hard to say. But it is impossible to deny that for this characteristic form, at least three comparisons are valid.
|
||||
|
||||
@ -272,13 +272,13 @@ In other words, concerning the model of the universe, the life line and consciou
|
||||
|
||||
A stronger image for the universal unity of the world is probably impossible to conceive.
|
||||
|
||||
|
||||
|
||||
|
||||
<center>(61)</center>
|
||||
|
||||
Another very important — or archetypal — image that has consistently persisted in human culture from ancient times to the present is the symbol of the butterfly.
|
||||
|
||||
|
||||
|
||||
|
||||
A rather unexpected and original illustration demonstrating the truly universal character of this archetype (ranging from the mysteries of the universe to human evolution) can be found in the following fragment from the poems of Vladimir Nabokov:
|
||||
|
||||
@ -321,7 +321,7 @@ How close this concept might have been to both the writer and the physicist is e
|
||||
|
||||
As a visual illustration of this idea applied to the field of physics, it would be appropriate, perhaps, to provide the following comparison picture, presupposing for the viewers not only the ability to analyze facts but also at least a minimum of imagination.
|
||||
|
||||
|
||||
|
||||
|
||||
In these pictures, one can see three essentially simple images reflecting fundamentally important ideas for modern physics. (a) The construct of a [graviton as a pair of photons](/tbc/44/), providing unity to a world split into two identical parts. (b) A strange attractor, as a graph of the "bifurcated" behavior of a nonlinear dynamic system on the brink of order and chaos. (c) A cross-section of a torus as Villarceau circles, as a simplest representation of the holographic idea of [AdS/CFT correspondence](/tbc/62/) and the inseparable unity of two substantially different worlds, forming a whole.
|
||||
|
||||
@ -531,7 +531,7 @@ As scientists themselves explain this heightened interest, butterfly wings — n
|
||||
|
||||
The specificity of the structure of butterfly wings is such that there is an array formed by light-scattering elements in a regular structure. Thanks to such a regular grid-like nanostructure, a butterfly wing always reflects approximately the same light wave regardless of the angle at which the light falls.
|
||||
|
||||
|
||||
|
||||
|
||||
Using scientific terminology, such butterfly wings demonstrate the characteristics of so-called "structural coloration," not the usual "chemical" kind. Much more familiar to everyone, chemical coloration — or the intrinsic color of a specific chemical substance — is explained as the scattering of light in one range of wavelengths and the absorption of waves from other parts of the spectrum. All dyes used have precisely this kind of innate chemical color.
|
||||
|
||||
@ -561,7 +561,7 @@ In the collective consciousness of humanity, for conveying and comprehending thi
|
||||
|
||||
<center>(62)</center>
|
||||
|
||||
|
||||
|
||||
|
||||
In the memorable dream of Wolfgang Pauli, known as the "Great Vision of the World Clock" [i80], there is one important detail which, for some reasons, has not been mentioned much here before.
|
||||
|
||||
@ -583,13 +583,13 @@ In the system of quite intricate notions of the ancient Egyptians about the stru
|
||||
|
||||
To delve into the details of this intriguing construct here is clearly not the place, but it is absolutely necessary to emphasize two most important components of the Egyptian soul, in the aggregate unity ensuring its immortal life forever. One part is called Ba, the second Akh, and both are depicted as birds.
|
||||
|
||||
|
||||
|
||||
|
||||
The Ba, as a part of the soul, closer to the affairs of the Egyptian’s life, was interpreted as the center of one’s human personality. It was depicted as a bird with a human head and left the physical body after death, in the hope of reuniting with the Akh.
|
||||
|
||||
As for Akh, this highest part of the soul was interpreted as a shining "pure mind" or a personal divine essence that lives by itself and fills every life with meaning. The name of the essence, "Akh," originated from the concept of "sparkling," "shining," "glowing."
|
||||
|
||||
|
||||
|
||||
|
||||
A Ba that had lived righteously could happily reunite with the cosmic wisdom of the Akh, which was depicted as “just a bird” — an ibis. However, such an end to life was, as far as one can judge, always considered an extremely rare event. As for the souls of ordinary people, after leaving the body and passing the “judgment of the gods,” they would settle across various levels of the afterlife, the Duat, which was described in great detail in the sacred texts of the Egyptians.
|
||||
|
||||
@ -615,7 +615,7 @@ But regardless of this, the birds manage to come to an agreement and set off on
|
||||
|
||||
In the end, finally, thirty of the most persistent reach Mount Qaf, where Simurgh resides. Finally, they see their king in person. And here is where the birds suddenly grasp that they themselves are Simurgh. Each individually and all together…
|
||||
|
||||
|
||||
|
||||
|
||||
<center>#</center>
|
||||
|
||||
@ -679,7 +679,7 @@ Research confirmed that on any scale – from the very large to the microscopica
|
||||
|
||||
Having constructed with the help of a computer the final fractal drawing – the limit set of one of Klein's symmetric iterative procedures – the researchers, captivated by the beauty of the result, could not resist and gave it the lofty-poetic name "The Glowing Limit."
|
||||
|
||||
|
||||
|
||||
|
||||
Every part of any such fractal structure contains the essence of the whole. Based on Mumford’s popularly written book, any literate computer user today can write a program that allows for increasingly magnified views of any chosen fragment of an image, and observe the same lace-like structure repeating itself on ever finer levels — revealing worlds within worlds within worlds, and so on…
|
||||
|
||||
@ -729,7 +729,7 @@ Now, Capra felt all this directly: he "saw" cascades of energy coming down from
|
||||
|
||||
<center>#</center>
|
||||
|
||||
|
||||
|
||||
|
||||
The canonical depiction of Shiva Nataraja, which clearly took shape in the bronze sculptures of the 10th–11th centuries, is now considered the most expressive and comprehensive symbol of Hinduism, portraying Shiva as the energy and life of all existence.
|
||||
|
||||
@ -757,7 +757,7 @@ For any attentive reader, it's probably already clear that the image of Shiva's
|
||||
|
||||
If the idea of a direct correspondence of the images still seems murky and not at all obvious to the reader, it is beneficial to examine some of the main details of the Nataraja image, which are crucially important from a physics point of view. Primarily, the damaru drum is of interest, widely used in the religious music of Hinduism and Tibetan Buddhism.
|
||||
|
||||
|
||||
|
||||
|
||||
The damaru is a small two-membrane percussion instrument played with just one hand. The body of the drum is made of wood in the shape of an hourglass — two cones with converging tips and bases covered with leather.
|
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|
||||
@ -843,7 +843,7 @@ The most well-known of these gods, Quetzalcoatl or "Feathered Serpent" in litera
|
||||
|
||||
The image of another god, named Tezcatlipoca or "Smoking Mirror," is directly related to the creation of the world, in its functions most closely akin to the image of Shiva in Hinduism, thus can appear both in wrathful and benevolent guises — like nature itself.
|
||||
|
||||
|
||||
|
||||
|
||||
One of the most important features of the image of Tezcatlipoca is his pronounced asymmetry: the god has only a left leg, while in place of the second, right leg, he is depicted either with a mirror or a serpent. The significant cosmic role of the mirror and the general notion of the universe as a “hall of mirrors” are, it is worth noting, characteristic of the Hua-yen school of Buddhism [i88], and in modern science — of the dodecahedral model by Jean-Pierre Luminet and Jeffrey Weeks. [i89]
|
||||
|
||||
@ -869,7 +869,7 @@ To finish the chapter on a positive note, it's much nicer to talk about a signif
|
||||
|
||||
In the summer of 2004, on the grounds of CERN, the European Organization for Nuclear Research near Geneva, a two-meter statue of Shiva Nataraja was ceremoniously installed as a gift from the Indian government to commemorate many years of fruitful cooperation with European scientists.
|
||||
|
||||
|
||||
|
||||
|
||||
Inscribed on the pedestal of the statue are words from Fritjof Capra’s book The Tao of Physics — about Shiva’s cosmic dance as a metaphor of universal significance:
|
||||
|
||||
@ -883,7 +883,7 @@ Our visible part of the universe on the sculpture is just the circle of light fr
|
||||
|
||||
For this reason, perhaps the most important element of the sculpture today should be considered Shiva's calming and encouraging gesture – "Abhaya Hasta." That is, "do not fear, friends," those who possess true knowledge have none and nothing to fear – in literally all respects…
|
||||
|
||||
|
||||
|
||||
|
||||
<center>(64)</center>
|
||||
|
||||
@ -921,7 +921,7 @@ Particularly of historical interest in this context are the so-called "cultural
|
||||
|
||||
As a particularly impressive example of such enlightening activities, it is worth recalling a character called the god Thoth or Djehuty – revered as the founder of all, essentially, intellectual life of Ancient Egypt. It is worth mentioning that he was depicted as a man with the head of an ibis bird (a symbol of the shining pure mind Akh).
|
||||
|
||||
|
||||
|
||||
|
||||
The cult of the wisest god Thoth, according to historical evidence, was one of the most ancient on the entire territory of the Egyptian state. Since in each region – or nome – of Egypt it was customary to particularly revere one of the deities of their extensive pantheon, the center of worship for Thoth was the capital of the 15th nome, the city of Hermopolis, or Khemenu, famous throughout the country for its richest library.
|
||||
|
||||
@ -1041,7 +1041,7 @@ In other words, no new "colonizer invasion" threatens us anymore. While we were
|
||||
|
||||
There is only one thing that inspires hope. Most normal representatives of humanity do clearly acknowledge what madness war is. And only we ourselves can and must stop this madness. In connection with which it seems more than appropriate to choose such a wonderful symbol as "The World in Our Hands" as the finale.
|
||||
|
||||
|
||||
|
||||
|
||||
The author of this monument, where the shadow from two folded hands forms a giant peace bird on the ground, is the Mexican artist and sculptor Glenda Hecksher. It is also worth mentioning this sculpture, adorning one of Mexico City's squares, for one highly significant reason.
|
||||
|
||||
@ -1057,7 +1057,7 @@ There are quite a few ways to wake up. One of the simplest recipes has been form
|
||||
|
||||
<center>THE END</center>
|
||||
|
||||
|
||||
|
||||
|
||||
### Inside links
|
||||
|
||||
|
||||
@ -37,22 +37,20 @@ class TZO
|
||||
// images
|
||||
$md = preg_replace_callback(
|
||||
'#!\[]\('
|
||||
// 1=url, 2=year, 3=month, 4=filename
|
||||
. '(https://kniganews\.org/wp-content/uploads/(\d{4})/(\d{2})/([^/\s]+))'
|
||||
// 5=caption: either a backslash+any-char, or anything but " or \
|
||||
// 1 = filename
|
||||
. '(/images/tzo/[\w\d\-.]+)'
|
||||
// 2 = caption
|
||||
. '(?:\s+"((?:\\\\.|[^"\\\\])*)")?'
|
||||
. '\)#',
|
||||
function($m) {
|
||||
$year = $m[2];
|
||||
$month = $m[3];
|
||||
$filename = $m[4];
|
||||
$caption = isset($m[5]) ? stripcslashes($m[5]) : '';
|
||||
$new_file = $year.$month.'_'. $filename;
|
||||
$path = APP_ROOT.'/public/ic/images/tzo/'.$new_file;
|
||||
$file = $m[1];
|
||||
$name = basename($file);
|
||||
$caption = isset($m[2]) ? stripcslashes($m[2]) : '';
|
||||
$path = APP_ROOT.'/public/ic/images/tzo/'.$file;
|
||||
list($w, $h) = $this->getImageSize($path); // TODO cache image size
|
||||
return (
|
||||
'<figure>'
|
||||
.'<div class="img-wrapper"><img src="'.htmlescape('/images/tzo/'.$new_file).'" width="'.$w.'" height="'.$h.'" alt="'.substr($filename, strrpos($filename, '.')+1).'"></div>'
|
||||
.'<div class="img-wrapper"><img src="'.htmlescape($file).'" width="'.$w.'" height="'.$h.'" alt="'.substr($name, strrpos($name, '.')+1).'"></div>'
|
||||
.($caption != '' ? '<figcaption>'.htmlescape($caption).'</figcaption>' : '')
|
||||
.'</figure>'
|
||||
);
|
||||
|
||||
Loading…
x
Reference in New Issue
Block a user