An American engineer, Guido Fetta, is set to test out his version of the EM Drive, the propellant-less propulsion technology that seems to break Newton’s Third Law of Motion. A successful test could lead to more credibility for the unlikely tech.
blindmen6 reblogged
New research by a group from Hokkaido University in Japan has shown that the plastic ionic crystal, quinuclidinium perrhenate, has ferroelectric properties and can serve as a model for finding new plastic ionic crystals that demonstrate ferroelectricity with directional tunability. Their work appears in Nature Chemistry.
Ferroelectric materials are spontaneously polarizable. The polarity of ferroelectric materials can be reversed by placing the material in an inverted external electric field. This ability to switch the material’s polarity has been used for several applications including memory devices. Organic ferroelectric materials, in particular, are of interest because they are non-toxic and easier to make than their inorganic counterparts.
However, there are several obstacles to practical application of organic ferroelectric crystals. Each crystal in a substance can be polarized only along its polarization axis, which is dependent on the molecular dipole orientation within the crystal. For effective performance of ferroelectric materials, the polarization axis of the individual crystals in a substance must align in a particular direction. In contrast to the high symmetry seen in inorganic ferroelectrics, the low symmetry in organic crystals has made it difficult to fabricate ferroelectric materials with their polarization axes aligned in the desired direction.
Source: phys.org
blindmen6 reblogged
Neutrinos Hint of Matter-Antimatter Rift
An early sign that neutrinos behave differently than antineutrinos suggests an answer to one of the biggest questions in physics.
In the same underground observatory in Japan where, 18 years ago, neutrinos were first seen oscillating from one “flavor” to another — a landmark discovery that earned two physicists the 2015 Nobel Prize — a tiny anomaly has begun to surface in the neutrinos’ oscillations that could herald an answer to one of the biggest mysteries in physics: why matter dominates over antimatter in the universe.
The anomaly, detected by the T2K experiment, is not yet pronounced enough to be sure of, but it and the findings of two related experiments “are all pointing in the same direction,” said Hirohisa Tanaka of the University of Toronto, a member of the T2K team who presented the result to a packed audience in London earlier this month.
“A full proof will take more time,” said Werner Rodejohann, a neutrino specialist at the Max Planck Institute for Nuclear Physics in Heidelberg who was not involved in the experiments, “but my and many others’ feeling is that there is something real here.”
The long-standing puzzle to be solved is why we and everything we see is matter-made. More to the point, why does anything — matter or antimatter — exist at all? The reigning laws of particle physics, known as the Standard Model, treat matter and antimatter nearly equivalently, respecting (with one known exception) so-called charge-parity, or “CP,” symmetry: For every particle decay that produces, say, a negatively charged electron, the mirror-image decay yielding a positively charged antielectron occurs at the same rate. But this cannot be the whole story. If equal amounts of matter and antimatter were produced during the Big Bang, equal amounts should have existed shortly thereafter. And since matter and antimatter annihilate upon contact, such a situation would have led to the wholesale destruction of both, resulting in an empty cosmos.
Somehow, significantly more matter than antimatter must have been created, such that a matter surplus survived the annihilation and now holds sway. The question is, what CP-violating process beyond the Standard Model favored the production of matter over antimatter?
Many physicists suspect that the answer lies with neutrinos — ultra-elusive, omnipresent particles that pass unfelt through your body by the trillions each second.
To that end, starting in 2010, scientists with the T2K experiment generated beams of neutrinos or antineutrinos in Tokai, Japan, and aimed them toward the Super-Kamiokande neutrino observatory, a sensor-lined tank of 50,000 tons of pure water located nearly 200 miles away in Kamioka. Occasionally, these ghostly particles interacted with atoms inside the water tank, generating detectable flashes of radiation. Detecting a difference in the behavior of the neutrinos and antineutrinos would provide an important clue about the preponderance of matter over antimatter, perhaps opening up a route beyond the Standard Model to a more complete theory of nature. Already, the strange properties of neutrinos provide a possible outline of that fuller story.
Primordial Neutrinos
The 1998 discovery that neutrinos switch flavors on the fly “may change our most fundamental theories,” President Bill Clinton said at the time, “from the nature of the smallest subatomic particles to how the universe itself works.”
Neutrino oscillations defied the Standard Model’s prediction that the particles are massless, like photons. In order for neutrinos to oscillate, each of their three possible flavors (electron, muon and tau) must be a quantum-mechanical mixture, or “superposition,” of three possible masses. Quantum superpositions evolve over time. So a neutrino might start out with its three mass components giving it pure muon flavor, but as the components evolve at different rates, electron flavor gradually enters the mixture, and the neutrino will have some probability of being measured as an electron neutrino.
There’s no mechanism within the Standard Model by which neutrinos might acquire their tiny, nonzero masses. Also unknown is why all neutrinos are observed to be “left-handed,” spinning clockwise with respect to their direction of motion, while all antineutrinos are right-handed, spinning counterclockwise.
Experts overwhelmingly favor a double-duty explanation of neutrino mass and single-handedness called the “seesaw mechanism,” whereby the known, lightweight, left-handed neutrinos have much heavier right-handed counterparts, and the known antineutrinos likewise have superheavy left-handed counterparts (the light and heavy masses are inversely related, like two sides of a seesaw). For this seesaw explanation to work, the neutrinos and antineutrinos on each side of the seesaw must actually be the same particle, except for their opposite handedness. Numerous experiments are now hunting for an extremely rare radioactive decay that would confirm this “Majorana” nature of neutrinos, thereby shoring up the logic of the seesaw mechanism.
If the theory is correct, then the heavy neutrinos and antineutrinos would have populated the hot young universe, when there was enough energy to beget beastly particles. They would have since decayed. Physicists wonder: Might their decays have produced the matter-antimatter asymmetry? This is the question to which an answer may be emerging — and much sooner than expected.
Tilted Seesaw
There’s good reason to think that neutrinos violate CP symmetry. The one established instance of CP violation in the laws of physics arises among the quarks — the building blocks of protons and neutrons — whose flavor mixing is described by a mathematical matrix similar to the one for neutrino mixing. In the quark case, though, the value of a numerical factor in the matrix that creates a disparity between quarks and antiquarks is very small. Quarks and antiquarks behave far too symmetrically to account for the universe’s matter-antimatter imbalance.
But the neutrino mixing matrix comes equipped with its own factor by which neutrinos and antineutrinos can violate CP symmetry. (Paradoxically, they can behave differently from one another even if they are Majorana particles, identical except for their opposite handedness.) If the lightweight neutrinos and antineutrinos violate CP symmetry, then the hypothetical heavy primordial neutrinos and antineutrinos must as well, and their asymmetric decays could easily have produced the universe’s glut of matter. Discovering CP violation among the lightweight neutrinos “will boost that general framework,” said Neal Weiner, a theoretical physicist at New York University.
The question is, how large will the CP-violation factor be? “The fear was that it would be small,” said Patricia Vahle, a physicist at the College of William & Mary — so small that the current generation of experiments wouldn’t detect any difference between neutrinos’ and antineutrinos’ behavior. “But it is starting to look like maybe we will be lucky,” she said.
To search for CP violation, the T2K scientists looked for evidence that neutrinos and antineutrinos oscillated between muon and electron flavors with unequal probabilities as they traveled between Tokai and Kamioka. The amount of CP violation once again works like a seesaw, with the rate of muon-to-electron neutrino conversions on one side, and corresponding antineutrino conversions on the other. The larger the value of the factor in the matrix, the greater the seesaw’s tilt.
If the seesaw is balanced, signifying perfect CP symmetry, then (accounting for differences in the production and detection rates of neutrinos and antineutrinos) the T2K scientists would have expected to detect roughly 23 electron neutrino candidates and seven electron antineutrino candidates in Kamioka, Tanaka said. Meanwhile, if CP symmetry is “maximally” violated — the seesaw tilted fully toward more neutrino oscillations and fewer antineutrino oscillations — then 27 electron neutrinos and six electron antineutrinos should have been detected. The actual numbers were even more skewed. “What we observed are 32 electron neutrino candidates and four electron antineutrino candidates,” Tanaka said.
With so few total events, it’s too soon to know whether the apparent tilt of the seesaw, signifying a large amount of CP violation, is real or a statistical aberration. Two other new hints of CP violation, however, strengthen the case. First, the newly running NOvA experiment, which generates a beam of muon neutrinos in Illinois and measures electron neutrinos in Minnesota, found a large number of these oscillations, again suggesting that the seesaw may be tilted in favor of neutrino oscillations and away from antineutrino oscillations. Second, researchers at the Super-Kamiokande observatory detected a similar enhancement of electron neutrinos coming from Earth’s atmosphere. (T2K and NOvA both plan to submit their findings for publication later this year.)
Vahle, who presented NOvA’s new results this month in London, urged caution; even when the T2K and NOvA results are combined, their statistical significance remains at a low level known as “2 sigma,” where there’s still a 5 percent chance the apparent deviation from CP symmetry is a random fluke. The results “do give me hope that finding CP violation in neutrino oscillations won’t be as hard as many feared it would be,” she said, “but we aren’t there yet.”
If CP violation among neutrinos is real and as large as it currently seems, then the evidence will slowly strengthen in the coming years. T2K’s signal could reach 3-sigma significance by the mid-2020s. “It’s a very exciting time as we look forward to a lot more data from both experiments,” said Peter Shanahan, a NOvA co-spokesperson.
It isn’t yet known exactly how CP violation in the light neutrino oscillations would translate into CP-violating decays of the heavy set. But discovering the former would point physicists in the latter’s general direction. “If we are starting to see [CP violation] in the neutrino sector, it is certainly a critical result,” Weiner said.
Source: quantamagazine.org
blindmen6 reblogged
Earlier this year, a study made headlines worldwide by bluntly demonstrating the human capacity to be misled by “pseudo-profound bullshit” from the likes of Deepak Chopra, infamous for making profound sounding yet entirely meaningless statements by abusing scientific language.
This is all well and good, but how are we supposed to know that we are being misled when we read a quote about quantum theory from someone like Chopra, if we don’t know the first thing about quantum mechanics?
In a lecture given by Richard Feynman in 1966, the influential theoretical physicist told a story about the difference between knowing the name for something and truly understanding it:
“This boy said to me, ‘See that bird standing on the stump there? What’s the name of it?’ I said, ‘I haven’t got the slightest idea.’ He said, ‘It’s a brown-throated thrush. Your father doesn’t teach you much about science.’
I smiled to myself, because my father had already taught me that [the name] doesn’t tell me anything about the bird. He taught me 'See that bird? It’s a brown-throated thrush, but in Germany it’s called a halsenflugel, and in Chinese they call it a chung ling and even if you know all those names for it, you still know nothing about the bird — you only know something about people; what they call that bird. Now that thrush sings, and teaches its young to fly, and flies so many miles away during the summer across the country, and nobody knows how it finds its way,’ and so forth. There is a difference between the name of the thing and what goes on.
The result of this is that I cannot remember anybody’s name, and when people discuss physics with me they often are exasperated when they say, 'the Fitz-Cronin effect,’ and I ask, 'What is the effect?’ and I can’t remember the name.
There is a first grade science book which, in the first lesson of the first grade, begins in an unfortunate manner to teach science, because it starts off on the wrong idea of what science is. There is a picture of a dog — a windable toy dog — and a hand comes to the winder, and then the dog is able to move. Under the last picture, it says, 'What makes it move?’ Later on, there is a picture of a real dog and the question, 'What makes it move?’ Then there is a picture of a motorbike and the question, 'What makes it move?’ and so on.
I thought at first they were getting ready to tell what science was going to be about — physics, biology, chemistry — but that wasn’t it. The answer was in the teacher’s edition of the book: The answer I was trying to learn is that 'energy makes it move.’
Now, energy is a very subtle concept. It is very, very difficult to get right. What I meant is that it is not easy to understand energy well enough to use it right, so that you can deduce something correctly using the energy idea — it is beyond the first grade. It would be equally well to say that 'God makes it move,’ or, 'Spirit makes it move,’ or, 'Movability makes it move.’ (In fact, one could equally well say, 'Energy makes it stop.’)
Look at it this way: That’s only the definition of energy; it should be reversed. We might say when something can move that it has energy in it, but not what makes it move is energy. This is a very subtle difference. It’s the same with this inertia proposition.
Perhaps I can make the difference a little clearer this way: If you ask a child what makes the toy dog move, you should think about what an ordinary human being would answer. The answer is that you wound up the spring; it tries to unwind and pushes the gear around.
What a good way to begin a science course! Take apart the toy; see how it works. See the cleverness of the gears; see the ratchets. Learn something about the toy, the way the toy is put together, the ingenuity of people devising the ratchets and other things. That’s good. The question is fine. The answer is a little unfortunate, because what they were trying to do is teach a definition of what is energy. But nothing whatever is learned.
Suppose a student would say, 'I don’t think energy makes it move.’ Where does the discussion go from there?
I finally figured out a way to test whether you have taught an idea or you have only taught a definition. Test it this way: You say, 'Without using the new word which you have just learned, try to rephrase what you have just learned in your own language. Without using the word “energy,” tell me what you know now about the dog’s motion.’ You cannot. So you learned nothing about science. That may be all right. You may not want to learn something about science right away. You have to learn definitions. But for the very first lesson, is that not possibly destructive?
I think for lesson number one, to learn a mystic formula for answering questions is very bad. The book has some others: 'gravity makes it fall;’ 'the soles of your shoes wear out because of friction.’ Shoe leather wears out because it rubs against the sidewalk and the little notches and bumps on the sidewalk grab pieces and pull them off. To simply say it is because of friction, is sad, because it’s not science.”
Feynman’s parable about the meaning of science is a valuable way of testing ourselves on whether we have really learned something, or whether we just think we have learned something, but it is equally useful for testing the claims of others. If someone cannot explain something in plain English, then we should question whether they really do themselves understand what they profess. If the person in question is communicating ostensibly to a non-specialist audience using specialist terms out of context, the first question on our lips should be: “Why?” In the words of Feyman, “It is possible to follow form and call it science, but that is pseudoscience.”
Source: BigThink |
Video: Richard Feynman on What It Means | Blank On Blank @pbsdigitalstudios
Source: youtu.be
Is Particle Physics About to Crack Wide Open?
Hints of an unexpected new particle could be confirmed within days.
If the observations are confirmed, it will be revolutionary. This could mean nothing less than the fall of the Standard Model of particle physics (SM), which has passed every experimental test thrown at it since it was first put together over four decades ago.
Read more
Source: blogs.scientificamerican.com
blindmen6 reblogged
Harvard researchers have designed a new type of foldable material that is versatile, tunable and self actuated. It can change size, volume and shape; it can fold flat to withstand the weight of an elephant without breaking, and pop right back up to prepare for the next task.
Every man’s world picture is and always remains a construct of his mind and cannot be proved to have any other existence.
Erwin Schrodinger
Source: fyp-science.com
blindmen6 reblogged
A Recap of Some Important Ideas Regarding Mandalic Geometry
- Mandalic geometry (MG) is a new kind of mathematical methodology based on a worldview having roots that predate written history.
- It is a discrete geometry which currently consists of just a coordinate system but can be extended as Descartes did his to encompass an entire analytic geometry.
- Mandalic geometry introduces and is based on a new number system, the probable number system (or probabilistic number system.)
- Just as the complex number system combines real numbers and imaginary numbers and is more robust than either, the probable number system combines real numbers and probable numbers and is more robust than either.
- The probable number system is also more robust than the complex number system. Complex numbers combine real numbers with imaginary numbers to form the single complex plane. Composite numbers combine real numbers with probable numbers to form six interdependent composite planes.
- Axiomatic to the system is the contention that numbers can exist in different dimensions and therefore can be described as being of some particular dimension. Numbers are always viewed and treated within context of a stated dimension.
- Probable numbers are an extension of the real numbers to higher dimensions and are independent of imaginary and complex numbers.
- Mandalic geometry does not admit the existence of square root of -1 in the real world other than in mathematics invented by the human mind. In place of square root of negative numbers, MG introduces the new concept of contra-square root. In brief this involves substitution of a combination form of interactive two-dimensional analogues of +1 and -1 for -1 as currently used in imaginary number contexts. This is more fully explained elsewhere in the blog.
- Put another way, in place of imaginary numbers MG posits the existence of probable numbers. These can be considered the result of what is essentially wavelike interactions of higher dimensional numbers to form the real numbers we know in the 3-dimensional world.
- Higher dimensional numbers can interact with one another through wavelike constructive and destructive interference to generate ordinary 3-dimensional numbers. Numbers are not viewed as constants to be acted upon as Descartes so views them but rather as being themselves active and changeable. They participate in process. This feature alone enables composite numbers to mediate between mathematics and physics better than either real or complex numbers can.
- The interactions of higher dimensional numbers in the process of dimensional compositing to yield 3-dimensional numbers is a function of time and therefore probabilistic from our limited ordinary point of view. From this perspective, certain probablity distributions are the result of dimensional compositing and the consequent mandalic form. MG considers the probabilistic nature of quantum mechanics likely to be based on such.
- The probabilistic nature in three dimensions of what are here called probable numbers is what gives rise to the mandalic form which can in a sense be considered the 3-dimensional evolution of 6-dimensional numbers from protean representations through progressive differentiation of form to the stage of maximal differentiation and back again to the undifferentiated state of greatest probability.
- The mandalic form has a geometric progression of its line structures in the three Euclidean/Cartesian dimensions such that series of numbers of the form 1:2:1, 2:4:2, and 4:8:4 occur throughout all of those dimensions when a hybrid 6D/3D coordinate system results from performing 2:1 compositing from six to three dimensions.
- Mandalic geometry views points and lines in three dimensions as convenient fictions that exist only as evanescent probabilistic concurrences of analogous entities in higher dimensions.
- The probabilistic nature of MG makes it ideal for investigations and descriptions of quantum mechanics.
- The exclusion of imaginary and complex numbers and substitution of probable and composite numbers which are easily reducible to ordinary algebraic/arithmetic forms and can be worked with using the same methods as those mathematical disciplines makes MG more utilitarian and appropriate to application to quantum mechanics than are complex numbers. All operations performed are based on simple inversion (reflection through a point) and on real numbers, maintaining all the usual rules and properties of ordinary arithmetic, including commutativity (which quaternions fail to preserve.)
- MG is currently based on discrete numbers and is concerned mainly with the positive and negative integers. Fractions and irrational numbers are not excluded from the system but do not currently play a significant role. Future incarnations of MG will extend it outward beyond the unit vector cube to tile the geometric universe and inward to encompass fractional entities and fractals.
- It is a hybrid geometry resulting from superposition of 6-dimensional numbers and 3-dimensional numbers and is fully commensurate with 3-dimensional Cartesian geometry.
- It describes a linear mapping of two dimensions to one dimension which forms a field of probable numbers over the field of real numbers, analogous to the field of complex numbers but constructed on a different principle and extending to the real numbers in all three Cartesian dimensions rather than just one. The two independent higher dimensions so mapped become dependent variables in the mandalic "line" that results from the compositing of the two. This is expressed, in a sense, as two sine waves 180 degrees out of phase that mutually intersect a common Cartesian axis (x,y or z) at Cartesian +1 and -1 and are maximally separated at Cartesian 0.
- This phase difference produces wave interference of both constructive and destructive varieties. So-called "points" or "particles" they represent come into existence only discretely and intermittently at Cartesian -1, +1, and 0, the locations of intersection or confluence (-1 and +1) and maximum separation, the maxima/minima of the two entangled sine waves that occur at Cartesian 0.
- As the unit vector cube corresponds to and describes only half of each of the two sine waves, two unit vector cubes are required for a full cycle. Mandalic geometry as currently formulated with a single unit cube then needs to be extended to at least two of these. Extension in both directions of all three Cartesian axes is easily accomplished by repeatedly inverting the current single unit vector cube.
- This means that mandalic coordinates alternate positive and negative on both sides of Cartesian 0. The extensions can be continued to infinity in both directions, but not, properly speaking, positive and negative infinity since the manner of extension has created what is essentially a convention-free coordinate system which consists of repeated units of consecutively inverted unit vector cubes in which positive and negative alternate ad infinitum and every Cartesian even-numbered coordinate becomes a "zero equivalent" , or better, a neo-zero in this extended mandalic coordinate system.
- The resulting geometry is a dynamic one with "points", "lines", and "planes" coming into and passing out of existence intermittently in a time-sharing of corresponding Cartesian entities. It "persists" in time and space by means of continuous creation, destruction and re-creation and is "held together" by "force fields" produced and maintained by means of tensegrity which is based ultimately on dimension and number, and by a process that.might best be described as a "weaving of reality" with warp and woof.
- The 2:1 compositing of dimension involved creates a new number system the members of which are like the real integers in all ways except that they map differently to a Cartesian geometric space. Whereas Decartes assumes that one number maps to one point, MG does not make this assumption which is just an unproved axiom that Descartes makes implicit use of.
- The method of dimensional compositing automatically results in a mandalic formation having a geometric progression through three Euclidean/Cartesian dimensions from periphery to center (origin).
- Currently MG is limited to a description of unit vectors in a composite hybrid 6D/3D geometry but can be extended to include all scalar values and any even number of dimensions.
- The notation system used is borrowed from Taoism and foreign to most Western mathematicians. It is, however, basically equivalent to Cartesian coordinate signs (yin=minus; yang=plus); ordered pairs (=bigrams); and ordered triads (=trigrams); and extends these concepts to include ordered quads (=tetragrams) and ordered sextuplets (=hexagrams).
- This notation system is used rather than the usual Cartesian notation because it is much easier for the mind to manipulate dimensional numbers using it. It takes only a little practice to become accustomed to using it. Without its use, understanding of mandalic geometry becomes extremely difficult, if not impossible.
- As MG views a point as a concurrence of various different dimensions, it interprets Cartesian ordered pairs and triads, and their extensions to higher dimensions, as tensors and treats them as such. This makes it possible to apply operations of addition and multiplication to these mathematical entities in a manner analogous to the way William Rowan Hamilton applied these operations to complex numbers by way of what he called "algebraic couples".
- The probabilistic mandalic form that is the hallmark of MG conveys and necessitates a new interpretation of zero(0). In MG "zero" is not the empty null that it is in Cartesian geometry and Western mathematics generally, but rather a fount of being, so to speak, and a logic gate spanning dimensions. Wherever a zero occurs in Cartesian coordinates two Cartesian-equivalent forms are found in mandalic coordinates. So in the mandalic cube based on unit vectors the twelve edge centers, having a single Cartesian zero, have two Cartesian-equivalent forms (hexagrams); the six face centers, having two Cartesian zeros, have four Cartesian-equivalent forms; and the single cube center, the Cartesian origin point with three zeros, has eight Cartesian-equivalent forms.
- This alternative zero and the mandalic structure it inhabits force the creation of four different amplitudes of dimension in the 6-dimensional unit vector cube. These are not independent but all mutually dependent and holo-interactive within the composite 6D/3D coordinate system. All of this occurs in a context reminiscent of the one inhabited by nuclear particles. The mapping proposed by MG may in fact model the elementary force fields, electromagnetism and quantum chromodynamics. It suggests a possible mechanism for formation of the state of matter known as a quark-gluon plasma. Hidden within it may even be the secret of quantum gravity.
© 2016 Martin Hauser
Please note: The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)
Scroll to bottom for links to Previous / Next pages (if existent). This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn't make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added. To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering. To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where n = x + 1 - p. :)
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blindmen6 reblogged
Beyond the Enlightenment Rationalists: From imaginary to probable numbers - VI
(continued from here)
"O Oysters, come and walk with us!" The Walrus did beseech. "A pleasant walk, a pleasant talk, Along the briny beach: We cannot do with more than four, To give a hand to each."
* * *
“The time has come,” the Walrus said, “To talk of many things: Of shoes–and ships–and sealing-wax– Of cabbages–and kings– And why the sea is boiling hot– And whether pigs have wings.”
-Lewis Carroll, The Walrus and the Carpenter
In this segment, probable numbers will be shown to grow out of a natural context inherently rather than through geometric second thought as transpired in the history of Western thought with imaginary numbers and complex plane. To continue with development of probable numbers it will be necessary to leave behind, for the time being, all preoccupation with imaginary numbers and complex plane. It will also be necessary to depart from our comfort zone of Cartesian spatial coordinate axioms and orientation.
Probable coordinates do not negate validity of Cartesian coordinates but they do relegate them to the status of a special case. In the probable coordinate system the three-dimensional coordinate system of Descartes maps only one eighth of the totality. This means then, that the Cartesian two-dimensional coordinate plane furnishes just one quarter of the total number of corresponding probable coordinate mappings projected to a two-dimensional space.[1] It suggests also that Cartesian localization in 2-space or 3-space is just a small part of the whole story regarding actual spatial and temporal locality and their accompanying physical capacities, say for instance of momentum or mass, but actually encompassing a host of other competencies as well.
Although this might seem strange it is a good thing. Why is it a good thing? First, because nature, as a self-sustaining reality, cannot favor any one coordinate scheme but must encompass all possible - if it is to realize any. Second, because both the Schrödinger equation and Feynman path integral approaches to quantum mechanics say it is so.[2] Third, because Hilbert space demands it. This may leave us disoriented and bewildered, but nature revels in this plan of probable planes. Who are we to argue?
So how do we accomplish this feat? Well, basically by reflections in all dimensions and directions. We extend the Cartesian vectors every way possible. That would give us a 3 x 3 grid or lattice of coordinate systems (the original Cartesian system and eight new grid elements surrounding it), but there are only four different types, so we require only four of the nine to demonstrate. It is best not to show all nine in any case because to do so would place our Cartesian system at direct center of this geometric probable universe and that would be misleading. Why? Because when we tile the two-dimensional universe to infinity in all directions, there is no central coordinate system. Any one of the four could be considered at the center, so none actually is. Overall orientation is nondiscriminative.[3]
LOOKING GLASS CARTESIAN COORDINATE QUARTET
The image seen immediately above shows four Looking House Cartesian coordinate systems, correlated within a mandalic plane. This mandalic plane is one of six faces of a mandalic cube, each of which is constructed to a different plan but composed of similar building blocks, the four bigrams in various positions and orientations. A 2-dimensional geometric universe can be tiled with this image, recursively repeating it in all directions throughout the two dimensions.[4] It should not be very difficult for the reader to determine which of the four mandalic moieties references our particular conventional Cartesian geometric universe.[5]
It remains only to be added here and now that potential dimensions, probable planes, and probable numbers arise immediately and directly from the remarks above. In some ways it's a little like valence in chemical reactions. We'll likely take a look at that combinatory dynamic in context of mandalic geometry at some time down the road. Next though we want to see how the addition of composite dimension impacts and modifies the basic geometry of the probable plane discussed here.[6]
(to be continued)
Top image: The four quadrants of the Cartesian plane. These are numbered in the counterclockwise direction by convention. Architectonically, two number lines are placed together, one going left-right and the other going up-down to provide context for the two-dimensional plane. This image has been modified from one found here.
Notes
[1] To clarify further: There are eight possible Cartesian-like orientation variants in mandalic space arranged around a single point at which they are all tangent to one another. If we consider just the planar aspects of mandalic space, there are four possible Cartesian-like orientation variants which are organized about a central shared point in a manner similar to how quadrants are symmetrically arranged about the Cartesian origin point (0,0) in ordinary 2D space. But here the center point determining symmetries is always one of the points showing greatest rather than least differentiation. That is to say it is formed by Cartesian vertices, ordered pairs having all 1s, no zeros. That may have confused more than clarified, but it seemed important to say. We will be expanding on these thoughts in posts to come. Don't despair. For just now the important takeaway is that the mandalic coordinate system combines two very important elements that optimize it for quantum application: it manages to be both probabilistic and convention-free (in terms of spatial orientation, which surely must relate to quantum states and numbers in some as yet undetermined manner.) At the same time, imaginary numbers and complex plane are neither.
[2] Even if physics doesn't yet (circa 2016) realize this to be true.
[3] It is an easy enough matter to extrapolate this mentally to encompass the Cartesian three-dimensional coordinate system but somewhat difficult to demonstrate in two dimensions. So we'll persevere with a two-dimensional exposition for the time being. It only needs to be clarified here that the three-dimensional realization involves a 3 x 3 x 3 grid but requires just eight cubes to demonstrate because there are only eight different coordinate system types.
[4] I am speaking here in terms of ordinary dimensions but it should be understood that the reality is that the mandalic plane is a composite 4D/2D geometric structure, and the mandalic cube is a composite 6D/3D structure. The image seen here does not fully clarify that because it does not yet take into account composite dimension nor place the bigrams in holistic context within tetragrams and hexagrams. All that is still to come. Greater context will make clear how composite dimension works and why it makes eminent good sense for a self-organizing universe to invoke it. Hint: it has to do with quantum interference phenomena and is what makes all process possible. ADDENDUM (12 APRIL, 2016) The mandalic plane I am referring to here corresponds to the Cartesian 2-dimensional plane and is based on four extraordinary dimensions that are composited to the ordinary two dimensions, hence hybrid 4D/2D. It should be understood though that any number of extra dimensions could potentially be composited to two or three ordinary dimensions. The probable plane described in this post is not such a mandalic plane as no compositing of dimensions has yet been performed. What is illustrated here is an ordinary 2-dimensional plane that has undergone reflections in x- and y-dimensions of first and second order to form a noncomposited probable plane. The distinction is an important one.
[5] This is perhaps a good place to mention that the six planar faces of the mandalic cube fit together seamlessly in 3-space, all mediated by the common shared central point, in Cartesian terms the origin at ordered triad (0.0.0) where eight hexagrams coexist in mandalic space. Moreover the six planes fit together mutually by means of a nuclear particle-and-force equivalent of the mortise and tenon joint but in six dimensions rather than two or three, and both positive and negative directions for each.
[6] It should also be avowed that tessellation of a geometric universe with a nondiscriminative, convention-free coordinate system need not exclude use of Cartesian coordinates entirely in all contextual usages. Where useful they can still be applied in combination with mandalic coordinates since the two can be made commensurate, irrespective of specific Cartesian coordinate orientation locally operative. Whatever the Cartesian orientation might be it can always be overlaid with our conventional version of the same. More concretely, hexagram Lines can be annotated with an ordinal numerical subscript specifying Cartesian location in terms of our local convention should it prove necessary or desirable to do so for whatever reason. On the other hand, before prematurely throwing out the baby with the bath water, we might do well to ask ourselves whether these strange juxtapositions of coordinates might not in fact encode the long sought-after hidden variables that could transform quantum mechanics into a complete theory. In mandalic coordinates of the reflexive nature described, these so-called hidden variables could be hiding in plain sight. Were that to prove the case, David Bohm and Louis de Broglie would be immediately and hugely vindicated in advancing their pilot-wave theory of quantum mechanics. We could finally consign the Copenhagen Interpretation to the scrapheap where it belongs, along with both imaginary numbers and the complex plane. ADDENDUM (24 APRIL, 2016) Since writing this I've learned that de Broglie disavowed Bohm's pilot wave theory upon learning of it in 1952. Bohm had derived his interpretation of QM from de Broglie's original interpretation but de Broglie himself subsequently converted to Niels Bohr's prevailing Copenhagen interpretation.
© 2016 Martin Hauser
Please note: The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)
Scroll to bottom for links to Previous / Next pages (if existent). This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn't make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added. To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering. To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where n = x + 1 - p. :)
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blindmen6 reblogged
Beyond the Enlightenment Rationalists: From imaginary to probable numbers - V
(continued from here)
The four Cartesian quadrants provide the two-dimensional analogue of the number line and its graphic representation in Cartesian coordinate space. This is the true native habitat of the square and, by implication, of square root. Because Enlightenment mathematicians found fit to define square root in a different context inadvertently -that of the number line- we will find it necessary to devise a different name for what ought rightly to have been called square root, but wasn't. I propose that we retain the existent definition of tradition and refer to the new relationship between opposite numbers in the square, that is to say, opposite vertices through two dimensions or antipodal numbers, as contra-square root.[1]
Modified from image found here.
Given this fresh context - one of greater dimension than the number line - it soon becomes clear with little effort that a unit number[2] of any dimension multiplied by itself gives as result the identity element of that express dimension. For the native two-dimensional context of the square the identity element is OLD YANG, the bigram composed of two stacked yang (+) Lines, which corresponds to yang (+1), the identity element in the one-dimensional context of the number line. In a three-dimensional context, the identity element is the trigram HEAVEN which is composed of three stacked yang (+) Lines. The crucial idea here is that the identity element differs for each dimensional context, and whatever that context might be, it produces no change when in the operation of multiplication it acts as operator on any operand within the stated dimension.[3]
As a corollary it can be stated that any number in any dimension n composed of any combination of yang Lines (+1) and yin Lines (-1) if multiplied by itself (i.e., squared) produces the identity element for that dimension. In concrete terms this means, for example, that any bigram multiplied by itself equals the bigram OLD YANG; any of eight trigrams multiplied by itself equals the trigram HEAVEN; and any of the sixty-four hexagrams multiplied by itself equals the hexagram HEAVEN; etc. (valid for any and all dimensions without exception). Consequently, the number of roots the identity element has in any dimension n is equal to the number 2n, these all being real roots in that particular dimension.
Similar contextual analysis would show that the inversion element of any dimension n has 2n roots of the kind we have agreed to refer to as contra-square roots in deference to the Mathematics Establishment.[4]
That leads us to the possibly startling conclusion that in every dimension n there is an inversion element that has the same number of roots as the identity element and all of them are real roots. For two dimensions the two pairs that satisfy the requirement are bigram pairs
For one dimension there is only a single pair that satisfies. That is (surprise, surprise) yin (-1)/yang (+1). What it comes down to is this:
If we are going to continue to insist on referring to square root in terms of the one-dimensional number line, then
- +1 has two real roots of the traditional variety, +1 and -1
- -1 has two real roots of the newly defined contra variety, +1/-1 and -1/+1
So where do imaginary numbers and quaternions fit in all this? The short answer is they don't. Imaginary numbers entered the annals of human thought through error. There was a pivotal moment[5] in the history of mathematics and science, an opportunity to see that there are in every dimension two different kinds of roots - - - what has been called square root and what we are calling contra-square roots. Enlightenment mathematicians and philosophers essentially allowed the opportunity to slip through their fingers unnoticed.[6]
Descartes at least saw through the veil. He called the whole matter of imaginary numbers 'preposterous'. It seems his venerable opinion was overruled though. Isaac Newton had his say in the matter too. He claimed that roots of imaginary numbers "had to occur in pairs." And yet another great mathematician, philosopher opined. Gottfried Wilhelm Leibniz, in 1702 characterized √−1 as “that amphibian between being and non-being which we call the imaginary root of negative unity.” Had he but preserved such augury conspicuously in mind he might have elaborated the concept of probable numbers in the 18th century. If only he had truly understood the I Ching, instead of dismissing it as a primitive articulation of his own binary number system.
(continued here)
Image: The four quadrants of the Cartesian plane. By convention the quadrants are numbered in a counterclockwise direction. It is as though two number lines were placed together, one going left-right, and the other going up-down to provide context for the two-dimensional plane. Sourced from Math Is Fun.
Notes
[1] My preference might be for square root to be redefined from the bottom up, but I don't see that happening in our lifetimes. Then too this way could be better.
[2] By the term unit number, I intend any number of a given dimension that consists entirely of variant elements of the number one (1) in either its positive or negative manifestation. Stated differently, these are vectors having various different directions within the dimension, but all of scalar value -1 (yin) or +1 (yang). All emblems of I Ching symbolic logic satisfy this requirement. These include the Line, bigram, trigram, tetragram, and hexagram. In any dimension n there exist 2n such emblems. In sum, for our purposes here, a unit number is any of the set of numbers, within any dimension n, which when self-multiplied (squared) produces the multiplicative identity of that dimension which is itself, of course, a member of the set. ADDENDUM (01 MAY 2016): I've since learned that mathematics has a much simpler way of describing this. It calls all these unit vectors. Simple, yes?
[3] I think it fair to presume that this might well have physical correlates in terms of quantum mechanical states or numbers. Here's a thought: why would it be necessary that all subatomic particles exist in the same dimension at all times given that they have a playing field of multiple dimensions, - some of them near certainly beyond the three with which we are familiar? And why would it not be possible for two different particles to be stable and unchanging in their different dimensions, yet become reactive and interact with one another when both enter the same dimension or same amplitude of dimension?
[4] Since in any contra-pair (antipodal opposites) of any dimension, either member of the pair must be regarded once as operator and once as operand. So for the two-dimensional square, for example, there are two antipodal pairs (diagonals) and either vertex of each can be either operator or operand. So in this case, 2 x 2 = 4. For trigrams there are four antipodal pairs, and 2 x 4 = 8. For hexagrams there are thirty-two antipodal pairs and 2 x 32 = 64. In general, for any dimension n there are 2 x 2n/2 = 2n antipodal pairs or contra-roots.
[5] Actually lasting several centuries, from about the 16th to the 19th century. Long enough, assuredly, for the error to have been discovered and corrected. Instead, the 20th century dawned with error still in place, and physicists eager to explain the newly discovered bewildering quantum phenomena compounded the error by latching onto √−1 and quaternions to assuage their confusion and discomfiture. This probably took place in the early days of quantum mechanics when the Bohr model of the atom still featured electrons as traveling in circular orbits around the nucleus or soon thereafter, visions of minuscule solar systems still fresh in the mind. At that time rotations detailed by imaginary numbers and quaternions may have still made some sense. Such are the vagaries of history.
[6] I think an important point to consider is that imaginary and complex numbers were, -to mathematicians and physicists alike,- new toys of a sort that enabled them to accomplish certain things they could not otherwise. They were basically tools of empowerment which allowed manipulation of numbers and points on a graph more easily or conveniently. They provided their controllers a longed for power over symbols, if not over the real world itself. In the modern world ever more of what we humans do and want to do involves manipulation of symbols. Herein, I think, lies the rationale for our continued fascination with and dependence on these tools of the trade. They don't need to actually apply to the world of nature, the noumenal world, so long as they satisfy human desire for domination over the world of symbols it has created for itself and in which it increasingly dwells, to a considerable degree apart from the natural world's sometimes seemingly too harsh laws.
© 2016 Martin Hauser
Please note: The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)
Scroll to bottom for links to Previous / Next pages (if existent). This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn't make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added. To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering. To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where n = x + 1 - p. :)
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blindmen6 reblogged
Beyond the Enlightenment Rationalists: From imaginary to probable numbers - IV
(continued from here)
One of the notable things the Rationalists failed to take into account in their analysis and codification of square roots was the significance of context. In so doing they assured that all related concepts they developed would eventually degenerate into a series of errors of conflation. Do not ever underestimate the importance of context.
Mathematicians, for example, can show that for any 3-dimensional cube there exists a 2-dimensional square, the area of which equals the volume of the cube.[1] And although that is true, something has been lost in translation. This is another of the sleights of hand mathematicians are so fond of. Physicists cannot afford to participate in such parlor tricks as these, however mathematically true they might be.[2]
We will begin now, then, to examine how the mandalic coordinate approach stacks up against that of imaginary numbers and quaternions. The former are holistic and respective of the natural order; the latter are irresponsibly rational, simplistic and, in final analysis, wrong about how nature works.[3] Ambitious endeavor indeed, but let's give it a go.
We've already looked at how the standard geometric interpretation of imaginary numbers in context of the complex plane is based on rotations through continuous Euclidean space. You can brush up on that aspect of the story here if necessary. The mandalic approach to mapping of space is more complicated and far more interesting. It involves multidimensional placement of elements in a discrete space, which is to say a discontinuous space, but one fully commensurate with both Euclidean and Cartesian 3-dimensional space. The holo-interactive manner in which these elements relate to one another leads to a probabilistic mathematical design which preserves commutative multiplication, unlike quaternions which forsake it.
Transformations between these elements are based on inversion (reflection through a point) rather than rotation which cannot in any case reasonably apply to discrete spaces. The spaces that quantum mechanics inhabits are decidedly discrete. They cannot be accurately detailed using imaginary and complex numbers or quaternions. To discern the various, myriad transitions which can occur among mandalic coordinates requires some patience. I think it cannot be accomplished overnight but at least in the post next up we can make a start.[4]
(continued here)
Image: A drawing of the first four dimensions. On the left is zero dimensions (a point) and on the right is four dimensions (A tesseract). There is an axis and labels on the right and which level of dimensions it is on the bottom. The arrows alongside the shapes indicate the direction of extrusion. By NerdBoy1392 (Own work) [CC BY-SA 3.0 or GFDL], via Wikimedia Commons
Notes
[1] If only in terms of scalar magnitude. Lost in translation are all the details relating to vectors and dimensions in the original. Conflation does not itself in every case involve what might be termed 'error' but because it always involves loss or distortion of information, it is nearly always guaranteed to eventuate in error somewhere down the line of argument. The point of all this in our context here is that, in the history of mathematics, something of this sort occurred when the Rationalists of the Enlightenment invented imaginary and complex numbers and again when quaternions were invented in 1843. These involved a disruption of vectors and dimensions as treated by nature. The loss of information involved goes a long way in explaining why no one has been able to explain why and how quantum mechanics works in a century or more. These misconstrued theses of mathematics behave like a demon or ghost in the machine that misdirects, albeit unintentionally, all related thought processes. What we end up with is a plethora of confusion. The fault is not in quantum mechanics but in ourselves, that we are such unrelentingly rational creatures, that so persistently pursue an unsound path that leads to reiterative error.
[2] Because physicists actually care about the real world; mathematicians, not so much.
[3] It must be admitted though that it was not the mathematicians who ever claimed imaginary numbers had anything to do with nature and the real world. Why would they? Reality is not their concern or interest. No, it was physicists themselves who made the mistake. The lesson to be learned by physicists here I expect is to be careful whose petticoat they latch onto. Not all are fabricated substantially enough to sustain their thoughts about reality, though deceptively appearing to do just that for protracted periods of time.
[4] My apologies for not continuing with this here as originally intended. To do so would make this post too long and complicated. Not that transformations among mandalic coordinates are difficult to understand, just that they are very convoluted. This is not a one-point-encodes-one-resident-number plan like that of Descartes we're talking about here. This is mandala country.
© 2016 Martin Hauser
Please note: The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)
Scroll to bottom for links to Previous / Next pages (if existent). This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn't make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added. To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering. To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where n = x + 1 - p. :)
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Davide Castelvecchi 20 April 2016
Hints of a new subatomic particle at the world’s most powerful atom smasher have inspired theoretical physicists to write more than 300 papers in the past four months. Now, a journal has published four of them, forming a condensed guide to what has become a zoo of possible explanations for a mysterious anomaly in the data collected by the Large Hadron Collider (LHC). . .
Source: nature.com
The memristor (/ˈmɛmrᵻstər/; a portmanteau of memory resistor) was a term coined in 1971 by circuit theorist Leon Chua as a missing non-linear passive two-terminal electrical component relating electric charge and magnetic flux linkage.[1] The operation of RRAM devices was recently connected to the memristor concept.[2] According to the characterizing mathematical relations, the memristor would hypothetically operate in the following way. The memristor's electrical resistance is not constant but depends on the history of current that had previously flowed through the device, i.e., its present resistance depends on how much electric charge has flowed in what direction through it in the past. The device remembers its history—the so-called non-volatility property.[3] When the electric power supply is turned off, the memristor remembers its most recent resistance until it is turned on again.[4][5]
In 2008, a team at HP Labs claimed to have found Chua's missing memristor based on an analysis of a thin film of titanium dioxide;[8] the HP result was published in Nature.[4] The memristor is currently under development by various teams including Hewlett-Packard, SK Hynix and HRL Laboratories.[citation needed]
These devices are intended for applications in nanoelectronic memories, computer logic and neuromorphic/neuromemristive computer architectures.[9]Commercial availability of memristor memory has been estimated as 2018.[10]In March 2012, a team of researchers from HRL Laboratories and theUniversity of Michigan announced the first functioning memristor array built on a CMOS chip.[11]
Image: Relations between the four fundamental electronic variables and devices that implement these relations. Vectorised version of File:Memristor.png by User:Linear77. By Parcly Taxel (Own work) [CC BY-SA 3.0 (http://creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons
See also
Source: Wikipedia
blindmen6 reblogged
Beyond the Enlightenment Rationalists: From imaginary to probable numbers - III
(continued from here)
My objection to the imaginary dimension is not that we cannot see it. Our senses cannot identify probable dimensions either, at least not in the visually compelling manner they can the three Cartesian dimensions. The question here is not whether imaginary numbers are mathematically true. How could they not be? The cards were stacked in their favor. They were defined in such a manner, -- consistently and based on axioms long accepted valid, -- that they are necessarily mathematically true. There's a word for that sort of thing. --The word is tautological.-- No, the decisive question is whether imaginary numbers apply to the real world; whether they are scientifically true, and whether physicists can truly rely on them to give empirically verifiable results with maps that accurately reproduce mechanisms actually used in nature.[1]
The geometric interpretation of imaginary numbers was established as a belief system using the Cartesian line extending from -1,0,0 through the origin 0,0,0 to 1,0,0 as the sole real axis left standing in the complex plane. In 1843, William Rowan Hamilton introduced two additional axes in a quaternion coordinate system. The new j and k axes, similar to the i axis, encode coordinates of imaginary dimensions. So the complex plane has one real axis, one imaginary; the quaternion system, three imaginary axes, one real, to accomplish which though involved loss of commutative multiplication. The mandalic coordinate system has three real axes upon which are superimposed six probable axes. It is both fully commensurate with the Cartesian system of real numbers and fully commutative for all operations throughout all dimensions as well.[2]
All of these coordinate systems have a central origin point which all other points use as a locus of reference to allow clarity and consistency in determination of location. The mandalic coordinate system is unique in that this point of origin is not a null point of emptiness as in all the other locative systems, but a point of effulgence. In that location where occur Descartes' triple zero triad (0.0.0) and the complex plane's real zero plus imaginary zero (ax=0, bi=0), we find eight related hexagrams, all having neutral charge density, each of these consisting of inverse trigrams with corresponding Lines of opposite charge, canceling one another out. These eight hexagrams are the only hexagrams out of sixty-four total possessing both of these characteristics.[3]
So let's begin now to plot the points of the mandalic coordinate system with the view of comparing its dimensions and points with those of the complex plane.[4] The eight centrally located hexagrams all refer to and are commensurate with the Cartesian triad (0,0,0). In a sense they can be considered eight alternative possible states which can exist in this locale at different times. These are hybrid forms of the four complementary pair of hexagrams found at antipodal vertices of the mandalic cube. The eight vertex hexagrams are those with upper and lower trigrams identical. This can occur nowhere else in the mandalic cube because there are only eight trigrams.[5]
From the origin multiple probability waves of dimension radiate out toward the central points of the faces of the cube, where these divergent force fields rendezvous and interact with reciprocal forces returning from the eight vertices at the periphery. converging toward the origin. Each of these points at the six face centers are common intersections of another eight particulate states or force fields analogous to the origin point except that four originate within this basic mandalic module and four without in an adjacent tangential module. Each of the six face centers then is host to four internal resident hexagrams which share the point in some manner, time-sharing or other. The end result is the same regardless, probabilistic expression of characteristic form and function. There is a possibility that this distribution of points and vectors could be or give rise to a geometric interpretation of the Schrödinger equation, the fundamental equation of physics for describing quantum mechanical behavior. Okay, that's clearly a wild claim, but in the event you were dozing off you should now be fully awake and paying attention.
The vectors connecting centers of opposite faces of an ordinary cube through the cube center or origin of the Cartesian coordinate system are at 180° to each other forming the three axes of the system corresponding to the number of dimensions. The mandalic cube has 24 such axes, eight of which accompany each Cartesian axis thereby shaping a hybrid 6D/3D coordinate system. Each face center then hosts internally four hexagrams formed by hybridization of trigrams in opposite vertices of diagonals of that cube face, taking one trigram (upper or lower) from one vertex and the other trigram (lower or upper) from the other vertex. This means that a face of the mandalic cube has eight diagonals, all intersecting at the face center, whereas a face of the ordinary cube has only two.[6]
The circle in the center of this figure is intended to indicate that the two pairs of antipodal hexagrams at this central point of the cube face rotate through 90° four times consecutively to complete a 360° revolution. But I am describing the situation here in terms of revolution only to show an analogy to imaginary numbers. The actual mechanisms involved can be better characterized as inversions (reflections through a point), and the bottom line here is that for each diagonal of a square, the corresponding mandalic square has a possibility of 4 diagonals; for each diagonal of a cube, the corresponding mandalic cube has a possibility of 8 diagonals. For computer science, such a multiplicity of possibilities offers a greater number of logic gates in the same computing space and the prospect of achieving quantum computing sooner than would be otherwise likely.[7]
Similarly, the twelve edge centers of the ordinary cube host a single Cartesian point, but the superposed mandalic cube hosts two hexagrams at the same point. These two hexagrams are always inverse hybrids of the two vertex hexagrams of the particular edge. For example, the edge with vertices WIND over WIND and HEAVEN over HEAVEN has as the two hybrid hexagrams at the center point of the edge WIND over HEAVEN and HEAVEN over WIND. Since the two vertices of concern here connect with one another via the horizontal x-dimension, the two hybrids differ from the parents and one another only in Lines 1 and 4 which correspond to this dimension. The other four Lines encode the y- amd z-dimensions, therefore remain unchanged during all transformations undergone in the case illustrated here.[8]
This post began as a description of the structure of the mandalic coordinate system and how it differs from those of the complex plane and quaternions. In the composition, it became also a passable introduction to the method of composite dimension. Additional references to the way composite dimension works can be found scattered throughout this blog and Hexagramium Organum. Basically the resulting construction can be thought of as a tensegrity structure, the integrity of which is maintained by opposing forces in equilibrium throughout, which operate continually and never fail, a feat only nature is capable of. We are though permitted to map the process if we can manage to get past our obsession with and addiction to the imaginary and complex numbers and quaternions.[9]
In our next session we'll flesh out probable dimension a bit more with some illustrative examples. And possibly try putting some lipstick on that PIG (Presumably Imaginary Garbage) to see if it helps any.
(continued here)
Image: A drawing of the first four dimensions. On the left is zero dimensions (a point) and on the right is four dimensions (A tesseract). There is an axis and labels on the right and which level of dimensions it is on the bottom. The arrows alongside the shapes indicate the direction of extrusion. By NerdBoy1392 (Own work) [CC BY-SA 3.0 or GFDL], via Wikimedia Commons
Notes
[1] For more on this theme, regarding quaternions, see Footnote [1] here. My own view is that imaginary numbers, complex plane and quaternions are artificial devices, invented by rational man, and not found in nature. Though having limited practical use in representation of rotations in ordinary space they have no legitimate application to quantum spaces, nor do they have any substantive or requisite relation to square root, beyond their fortuitous origin in the Rationalists' dissection and codification of square root historically, but that part of the saga was thoroughly misguided. We wuz bamboozled. Why persist in this folly? Look carefully without preconception and you'll see this emperor's finery is wanting. It is not imperative to use imaginary numbers to represent rotation in a plane. There are other, better ways to achieve the same. One would be to use sin and cos functions of trigonometry which periodically repeat every 360°. (Read more about trigonometric functions here.) Another approach would be to use polar coordinates.
[SOURCE]
A quaternion, on the other hand, is a four-element vector composed of a single real element and three complex elements. It can be used to encode any rotation in a 3D coordinate system. There are other ways to accomplish the same, but the quaternion approach offers some advantages over these. For our purposes here what needs to be understood is that mandalic coordinates encode a hybrid 6D/3D discretized space. Quaternions are applicable only to continuous three-dimensional space. Ultimately, the two reside in different worlds and can't be validly compared. The important point here is that each has its own appropriate domain of judicious application. Quaternions can be usefully and appropriately applied to rotations in ordinary three-dimensional space, but not to locations or changes of location in quantum space. For description of such discrete spaces, mandalic coordinates are more appropriate, and their mechanism of action isn't rotation but inversion (reflection through a point.) Only we're not speaking here about inversion in Euclidean space, which is continuous, but in discrete space, a kind of quasi-Boolean space, a higher-dimensional digital space (grid or lattice space). In the case of an electron this would involve an instantaneous jump from one electron orbital to another.
[2] I think another laudatory feature of mandalic coordinates is the fact that they are based on a thought system that originated in human prehistory, the logic of the primal I Ching. The earliest strata of this monumental work are actually a compendium of combinatorics and a treatise on transformations, unrivaled until modern times, one of the greatest intellectual achievements of humankind of any Age. Yet its true significance is overlooked by most scholars, sinologists among them. One of the very few intellectuals in the West who knew its true worth and spoke openly to the fact, likely at no small risk to his professional standing, was Carl Jung, the great 20th century psychologist and philosopher.
It is of relevance to note here that all the coordinate systems mentioned are, significantly, belief systems of a sort. The mandalic coordinate system goes beyond the others though, in that it is based on a still more extensive thought system, as the primal I Ching encompasses an entire cultural worldview. The question of which, if any, of these coordinate systems actually applies to the natural order is one for science, particularly physics and chemistry, to resolve.
Meanwhile, it should be noted that neither the complex plane nor quaternions refer to any dimensions beyond the ordinary three, at least not in the manner of their current common usage. They are simply alternative ways of viewing and manipulating the two- and three-dimensions described by Euclid and Descartes. In this sense they are little different from polar coordinates or trigonometry in what they are attempting to depict. Yes, quaternions apply to three dimensions, while polar coordinates and trigonometry deal with only two. But then there is the method of Euler angles which describes orientation of a rigid body in three dimensions and can substitute for quaternions in practical applications.
A mandalic coordinate system, on the other hand, uniquely introduces entirely new features in its composite potential dimensions and probable numbers which I think have not been encountered heretofore. These innovations do in fact bring with them true extra dimensions beyond the customary three and also the novel concept of dimensional amplitudes. Of additional importance is the fact that the mandalic method relates not to rotation of rigid bodies, but to interchangeability and holomalleability of parts by means of inversions through all the dimensions encompassed, a feature likely to make it useful for explorations and descriptions of particle interactions of quantum mechanics. Because the six extra dimensions of mandalic geometry may, in some manner, relate to the six extra dimensions of the 6-dimensional Calabi–Yau manifold, mandalic geometry might equally be of value in string theory and superstring theory.
It is possible to use mandalic coordinates to describe rotations of rigid bodies in three dimensions, certainly, as inversions can mimic rotations, but this is not their most appropriate usage. It is overkill of a sort. They are capable of so much more and this particular use is a degenerate one in the larger scheme of things.
[3] This can be likened to a quark/gluon soup. It is a unique and very special state of affairs that occurs here. Physicists take note. Don't let any small-minded pure mathematicians dissuade you from the truth. They will likely write all this off as "sacred geometry." Which it is, of course, but also much more. Hexagram superpositions and stepwise dimensional transitions of the mandalic coordinate system could hold critical clues to quantum entanglement and quantum gravity. My apologies to those mathematicians able to see beyond the tip of their noses. I was not at all referring to you here.
[4] Hopefully also with dimensions and points of the quaternion coordinate system once I understand the concepts involved better than I do currently. It should meanwhile be underscored that full comprehension of quaternions is not required to be able to identify some of their more glaring inadequacies.
[5] In speaking of "existing at the same locale at different times" I need to remind the reader and myself as well that we are talking here about particles or other subatomic entities that are moving at or near the speed of light,- - -so very fast indeed. If we possessed an instrument that allowed us direct observation of these events, our biologic visual equipment would not permit us to distinguish the various changes taking place. Remember that thirty frames a second of film produces the illusion of motion. Now consider what thirty thousand frames a second of repetitive action would do. I think it would produce the illusion of continuity or standing still with no changes apparent to our antediluvian senses.
[6] Each antipodal pair has four different possible ways of traversing the face center. Similarly, the mandalic cube has thirty-two diagonals because there are eight alternative paths by which an antipodal pair might traverse the cube center. This just begins to hint at the tremendous number of transformational paths the mandalic cube is able to represent, and it also explains why I refer to dimensions involved as potential or probable dimensions and planes so formed as probable planes. All of this is related to quantum field theory (QFT), but that is a topic of considerable complexity which we will reserve for another day.
[7] One advantageous way of looking at this is to see that the probabilistic nature of the mandalic coordinate system in a sense exchanges bits for qubits and super-qubits through creation of different levels of logic gates that I have referred to elsewhere as different amplitudes of dimension.
[8] Recall that the Lines of a hexagram are numbered 1 to 6, bottom to top. Lines 1 and 4 correspond to, and together encode, the Cartesian x-dimension. When both are yang (+), application of the method of composite dimension results in the Cartesian value +1; when both yin (-), the Cartesian value -1. When either Line 1 or Line 4 is yang (+) but not both (Boole's exclusive OR) the result is one of two possible zero formations by destructive interference. Both of these correspond to (and either encodes) the single Cartesian zero (0). Similarly hexagram Lines 2 and 5 correspond to and encode the Cartesian y-dimension; Lines 3 ane 6, the Cartesian z-dimension. This outline includes all 9 dimensions of the hybrid 6D/3D coordinate system: 3 real dimensions and the 6 corresponding probable dimensions. No imaginary dimensions are used; no complex plane; no quaternions. And no rotations. This coordinate system is based entirely on inversion (reflection through a point) and on constructive or destructive interference. Those are the two principal mechanisms of composite dimension.
[9] The process as mapped here is an ideal one. In the real world errors do occur from time to time. Such errors are an essential and necessary aspect of evolutionary process. Without error, no change. And by implication, likely no continuity for long either, due to external damaging and incapacitating factors that a natural world devoid of error never learned to overcome. Errors are the stepping stones of evolution, of both biological and physical varieties.
© 2016 Martin Hauser
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#mandalic geometry#physics#quantum physics#philosophy#symbolic logic#combinatorics#transitions#imaginary numbers#quaternions#coordinate systems#mandalic coordinates#dimensions#dimension#space#continuous space#discrete space#rotation#inversion#I Ching#Carl Jung#complexity science#interdisciplinary#transdisciplinary#potential dimension#probable dimension#probable plane#quantum field theory
blindmen6 reblogged
6-as-above-so-below-9-deactivat
Rotation or illusion of rotation? I suppose it depends on what dimension one’s mind is moving in.
blindmen6 reblogged
Stepwise transformations and entanglement.
Escher understood transformation and entanglement before the physicists did.