The muon and tau are second and third generation, respectively, leptons and fermions. There are a total of 6 leptons in the standard model. The electron, muon, and tau are the three which have electric charge while the others, the neutrinos, do not. Both the muon and tau are much more massive than the electron and decaydue to the weak interaction. The muon decays on average 2.2 microseconds (2.2*10^-6 s) into usually an electron and two neutrinos of different types. The tau decays much quicker in 2.9 * 10^-13 seconds into hadrons(composite particles made of two or more quarks, e.g. proton). The tau is the only lepton able to decay into hadrons because it is the only one with sufficient mass.
The muon was discovered by Carl D. Anderson and Seth Neddermeyer in 1936 by studying cosmic radiation and observed particles which deflecteddifferently than electrons in a magnetic field. The radius of deflection depends on mass and charge. Since the charge is the same the difference must be accounted through a greater mass.
The tau was theoretically predicted in 1971 by Yung-su Tsai and experimentally detected between 1974-1977 at the Stanford Linear AcceleratorandLawrence Berkeley National Laboratory.
Probably the most well known experiment that involves muons is the Muon g-2 (”g minus 2″) experiment at the Fermi National Accelerator Laboratory or Fermilab. The goal is to measure the magnetic dipole moment at a very high precision because there is a slight deviation from g=2 (hence g minus 2) known as the “anomalous” part predicted by the Standard Model theory. A large enough difference between the experimentally measured and theoretically determined values could point to the existence of more undiscovered subatomic particles. Read more about the Muon g-2 experiment below:
By drilling holes into a thin two-dimensional sheet of hexagonal boron nitride with a gallium-focused ion beam, University of Oregon scientists have created artificial atoms that generate single photons.
[…]
The artificial atoms - which work in air and at room temperature - may be a big step in efforts to develop all-optical quantum computing, said UO physicist Benjamín J. Alemán, principal investigator of a study published in the journal Nano Letters.
“Our work provides a source of single photons that could act as carriers of quantum information or as qubits. We’ve patterned these sources, creating as many as we want, where we want,” said Alemán, a member of the UO’s Material Science Institute and Center for Optical, Molecular, and Quantum Science. “We’d like to pattern these single photon emitters into circuits or networks on a microchip so they can talk to each other, or to other existing qubits, like solid-state spins or superconducting circuit qubits.”
Researchers at the U.S. Department of Energy’s Lawrence Berkeley National Laboratory (Berkeley Lab) have developed a graphene device that’s thinner than a human hair but has a depth of special traits. It easily switches from a superconducting material that conducts electricity without losing any energy, to an insulator that resists the flow of electric current, and back again to a superconductor - all with a simple flip of a switch. Their findings were reported today in the journal Nature.
[…]
“Usually, when someone wants to study how electrons interact with each other in a superconducting quantum phase versus an insulating phase, they would need to look at different materials. With our system, you can study both the superconductivity phase and the insulating phase in one place,” said Guorui Chen, the study’s lead author and a postdoctoral researcher in the lab of Feng Wang, who led the study. Wang, a faculty scientist in Berkeley Lab’s Materials Sciences Division, is also a UC Berkeley physics professor.
“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.”
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 equationandFeynman 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 andLouis 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.
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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]
Given this fresh context - one of greater dimension than the number line - it soon becomes clear with little effort that a unit number[2]ofany 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 elementandall 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.
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.
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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]
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 3.0orGFDL],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 whyandhow 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.
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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. :)
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 jandk 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.
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 3.0orGFDL],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.
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 coordinatesortrigonometry 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 theoryandsuperstring theory.
Itis 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 potentialorprobable 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.
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When a geometric interpretation of imaginary numbers was at last proposed, long after they were invented, it was as though accomplished by central committee. The upshot was easily enough understood but also simplistic. In broad brushstroke here is what seems to have gone down.
The 3 dimensions of Descartes’ coordinate system-a number already deficient from the perspective of mandalic geometry-were reduced to just one. Of the real number axes then only the x-axis remained. This meant from the get-go that any geometric figure that ensued could encompass only linearity in terms of real numbers and dimensions. It was applicable only to a line segment, so the complex plane that resulted could describe just one real dimension and one imaginary dimension. It consecrated the number line in a single dimension, to exclusion of its proper habitation in two others besides. Strike one for imaginary numbers.[1]
With that as background let’s look now at the rotations described by this geometric interpretation of imaginary numbers in the context of the complex plane.
i in the complex or cartesian plane. Real numbers lie on the horizontal axis, and imaginary numbers lie on the vertical axis By Loadmaster (David R. Tribble) (Own work) [CC BY-SA 3.0orGFDL],via Wikimedia Commons
The number 1 is the multiplicative identity element for real numbers and the number -1 is the reflection inversion element for real numbers. Put another way, the number one times any number equals that number; the number -1 times any number is a negative of that number or the inverse number through a reference point, usually taken as zero. Multiplying by 1 then leaves 1, -1, i and -i all unchanged. Multiplying by -1 changes -1 to 1, 1 to -1, i to -i, and -i to i. In terms of rotations in the complex plane, these changes all involve a rotation through 180 degrees. Multiplication of the number 1 by i changes it to i; i by i changes it to -1; -1 by i to -i; and -i by i to 1. These changes all involve rotations through 90 degrees. And finally, multiplication of 1 by -i changes it to -i; -i by -i changes it to -1; -1 by -i to i; and i by -i to 1: all changes involving rotations through -90 degrees.
The figure below shows another way to interpret these rotations that amounts to the same tbing: i1 = i; i2 =-1; i3 = -i; i4 = 1. Click to enlarge.
Four numbers on the real line multiplied by integer powers of the imaginary unit, which corresponds to rotations by multiples of the right angle. By Keφr [CC0],via Wikimedia Commons
I think a committee of some sort must have come up with this resplendent plan. For certain it was an Academy of Mathematics and Sciences that endorsed and enthroned it. All bow to central authority.
I had planned to include a comparison of imaginary numbers and probable numbers in this post as well but because that is a long discussion itself, it will have to wait till the next post. I might add it should prove well worth the wait.
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 3.0orGFDL],via Wikimedia Commons
Notes
[1] Mathematician William Rowan Hamilton subsequently addressed this deficiency in 1843 with his quaternions, a number system that extends the complex numbers to three-dimensional space. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space or, in other words, as the quotient of two vectors. This complicated matters even more by introducing a non-commutative multiplication operation to the system, though to be fair the quaternion coordinate system has found some useful applications mainly for calculations involving 3-dimensional rotations, as in 3-dimensional computer graphics,computer vision, and crystallographic texture analysis. Still it becomes problematic when theoretical physics attempts use of quaternions in calculations pertaining to atomic and subatomic spaces where rotations do not actually take place. The conclusion to be drawn here is that quaternions can be usefully, if somewhat clumsily, applied to 3-dimensional macro-spaces but are inapproriate for accurate description of higher dimensional spaces. What is here unfortunate and misleading is that quaternions apparently do describe outcomes of events in the quantum realm to some partial degree, if not the mechanisms of the events themselves. Physicists would not long tolerate them were that not so.
[ADDENDUM (24 APRIL, 2016) Since writing this I’ve learned that quaternions are not currently used in quantum physics nor were they ever, to any great degree, in the past.]
In other words, sometimes the right answer can be reached by a wrong method. In the case under discussion here, we should note that it is possible for a rotation to mimic inversion (reflection through a point). A 90° rotation in two dimensions can mimic a single inversion in a single plane through an edge of a square, and a 180° rotation in two dimensions can mimic a single inversion through a diagonal of a square or two successive inversions through two perpendicular edges of a square. A 180° rotation in three dimensions can mimic three inversions through three mutually perpendicular edges of a square; a combination of one inversion through a diagonal of a square and another through an edge perpendicular to the plane of the first inversion; or a single inversion through a diagonal of the cube. Subatomic paricles exist as discrete or quantized entities and would follow such methods of transformation rather than rotations through a continuous space. Of course, transformations involving a diagonal would require more transformative energy than one involving a single edge.
Such patterns of relationship and transformation could no doubt be described in terms of quantum states and quantum numbers without too much difficulty by a knowledgeable theoretical physicist. Surely doing so could be no more difficult than using quaternions, which may give a correct answer while also misleading and limiting knowledge of the the true workings of the quantum realm by using an incorrect mechanism, one non-commutative to boot. Nature doesn’t approve of hat tricks like that.
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Beyond the Enlightenment Rationalists: From imaginary to probable numbers - I
Imaginary numbers arose in the history of mathematics as a result of misunderstanding the dimensional character of numbers. There was a failure to acknowledge that numbers exist in a context of dimension. This has earlier been addressed at length.[1] Simply put, numbers exist always in a particular dimensional context. Square numbers pertain to a context of two dimensions and therefore to a plane, not a line. Square roots then ought justly reference a two-dimensional geometrical context rather than the linear one mathematics has maintained ever since mathematicians of the Age of Enlightenment decreed it so. Square roots contrary to the way mathematics would have it can neither exist in nor be found in any single line segment, because they do not originate in the number line but in the two-dimensional square.
Algebra, not geometry, provided the breeding ground for imaginary numbers. They were given a geometric interpretation as an afterthought only, long after the fact of their invention. Rationalist algebraists, feeling compelled to give meaning to equations of the form b2 = -4 came up with the fantastic notion of imaginary numbers. Only indirectly did these grow out of nature, by way of minds of men obsessed with reason.[2]
Descartes knew of the recently introduced square roots of negative numbers. He thought them preposterous and was first to refer to the new numbers by the mocking name imaginary, a label which stuck and which continues to inform posterity of the exact manner in which he viewed the oddities. It is one of the ironies of history that when at last a geometrical interpretation of square root of negative numbers was offered it involved swallowing up Descartes’ own y-axis. Poetic justice? Or ultimate folly?
Had the essential dimensional nature of numbers been recognized there would have been no need to inquire what the square root of -1 was. It would have been clear that there was no square root of -1 nor any need for such as +1 also has no square root. As linear numbers, neither -1 nor +1 can legitimately be said to have a square root. Both, though, have two-dimensional analogues and these do have square roots, not recognized as such unfortunately by the mathematics hegemony.[3]
In the next post we will look at a comparison between imaginary numbers, which were formulated in accordance with this misconstrual about how numbers relate to dimensions, and probable numbers which grow organically out of a consideration of how numbers and dimensions actually relate to one another in nature.[4] The first of these approaches can be thought of as rational planning by a central authority; the second, as the holistic manner in which nature attends to everything, all at once, and without rational forethought.
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 3.0orGFDL],via Wikimedia Commons
Notes
[1] See the series of about nine posts that begins here.
[2] The Rationalists missed here a golden opportunity to relate number and dimension by defining square root much too narrowly. They seem to have been so mesmerized by their algebraic equations that they failed to pursue the search into deeper significance pertaining to essential linkages between dimension and number that intuition and imagination might have bestowed.
[3] As Shakespeare correctly pointed out, a rose by any name would smell as sweet. Plus one times plus one certainly equals plus one but that has nothing to do with actual square root really, just with algebraic linear multiplication. Note has often been made in these pages of the difference between mathematical truth and scientific truth. Whereas mathematics demands only adherence to its axioms and consistency, science requires empirical proof. Mathematics defined square root in a certain manner centuries ago, and has since been devoutly consistent in its adherence to that definition. In so doing it has preserved a cherished doctrine of mathematical truth, as though in formaldehyde. It has also for many centuries contrived to be consistently scientifically incorrect. The problem lies in the fact it has converted physicists and near everyone else to its own insular worldview.
[4] For an early discussion of the probable plane, potential dimensions, and probable numbers see here.
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The scrupulous 3-dimension world we humans inhabit is in fact biological, not physical, in origin. Its limitations are determined by our specific sensory, motor and mental apparatus and abilities. It only hints at the real world, and while doing so it combines some highly erroneous observations as well. Molluscs and insects and arachnids all have a very different perspective of their environment. We would find discomfort in the world view of an octopus, as we do in the quantum world view.[1][2]
Dimension is a term laymen toss about haphazardly. Mathematicians and physicists have a more precise interpretation concerning dimension. For them, any independent parameter constitutes a separate dimension. But when it comes down to the nitty-gritty, what if anything can truly be separate and independent? Those are both relative terms. Nothing that exists is really fully isolate and independent. That is one of the substratal premises from which mandalic geometry evolves: relationships invariably exist. And relationships can always change. Mandalic geometry therefore is a geometry of process - a spacetime geometry, not one of space alone.
For those who created the primal I Ching relationship was considered a fundamental aspect of reality. When they thought of dimension - - - and they did, in their own way - - - relationships were always involved. Flash-forward a few thousand years - quantum mechanics accomplishes much the same with its view of interacting particles in continual motion, ever-changing, and incessantly forging transient effective links with numerous other particles of similar and different type under the influence of various fields of force.
Kant thought that human concepts and categories determine our view of the world and its laws. He held that inborn features of our minds structure our experiences. Since, in his view, mind shapes and structures experience, at some level of representation all human experience shares certain essential operational features. Among these according to Kant are our concepts relating to space and time, integral to all human experience. The same might be said about our concepts of cause and effect.
Kant further asserts that we never have direct experience of things, referred to in his writings as the noumenal world. All we experience is the phenomenal world that is relayed to us by our senses. Kant views noumena as the thing-in-itself or true reality and phenomena as our experience or perception of that thing, filtered through our senses and reasoning. According to Kant science can be applied only to things that can be observed and studied. The entire world of noumena is beyond the scope and reach of science. As an heir to Enlightenment philosophy Kant respects the value of reason but believes the noumenal world to be beyond its scope and reach. So are we fated then never to experience the noumena directly? Not by a long shot. Kant claims the noumena to be accessible but only by intellectual intuition without the aid of reason.[3]
In the world of phenomena nothing is self-existent. Everything exists by virtue of dependence on something else. Point to something, anything at all, that refutes that view and I’ll tell you you’re out of your mind - and in the noumenal world. What, pray tell, are you doing there and how did you get there anyway? If you can clearly communicate the how I may give it a try myself.[4]
[1] The world view granted us by our inherited biologic capacities has been millions of years in the making. Indeed. But that makes it still not a whit truer than had we groped it only yesterday. Evolution seems to have sacrificed a full immersive sense of reality to grant a greater degree of interoperability essential to dealing with vicissitudes of a material world and confer durability within that domain. The quest after true apprehension we feel impelled to pursue is a siren not without danger.
“The search for reality is the most dangerous of all undertakings, for it destroys the world in which you live.” -Nisargadatta Maharaj
[2] Regarding the origin and transformations of the word “scrupulous”:
Scrupulous and its close relative “scruple” (“an ethical consideration”) come from the Latin noun scrupulus, the diminutive of “scrupus.” “Scrupus” refers to a sharp stone, so scrupulus means “small sharp stone.” “Scrupus” retained its literal meaning but eventually also came to be used with the metaphorical meaning “a source of anxiety or uneasiness,” the way a sharp pebble in one’s shoe would be a source of pain. When the adjective “scrupulous” entered the language in the 15th century, it meant “principled.” Now it also commonly means "painstaking" or “careful.” [Source]
Sad to say, this fascinating word that so successfully wended its way through several related incarnations in a number of different Indo-European languages prior to its appearance in English, c.15th century, appears to be passing out of usage among English speakers in modern times. We will likely be left with the occasional utterance of “scruples” but “scrupulous” itself seems destined for oblivion.
Curiously, my election of the word here was not rationally motivated. As I was framing the thought expressed in the paragraph in my mind, the word just appeared out of nowhere and seemed to insist, “I belong here though you may not yet understand why. You really need a word with my complex heritage of multiple meanings here.” And so I went with it, not fully knowing why. Funny thing about it, my rational mind is quite unable now to come up with any other single word that suits as well.
[3] Kant’s epistemology recognizes three different sources of knowlege: sensory experience, reason, and intuition. He views intuition as independent of the other two and the only one of the three with direct access to the world of noumena. This may present as suspect at first, but then how do we explain things like what Einstein did a century ago? Einstein himself has hinted in his writings at the essential role of intuition and imagination in his thinking.
Clickhere for more slides on Kant’s philosophy by William Parkhurst from Introduction to Philosophy Lecture 13, source of the above slide reproduction.
[4] Our human penchant for categorization inevitably leads to dismemberment of holistic reality into an endless number of manifest objects, many of which we no longer recognize as essentially related.
“People normally cut reality into compartments, and so are unable to see the interdependence of all phenomena. To see one in all and all in one is to break through the great barrier which narrows one’s perception of reality.” -Thích Nhất Hạnh
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So yes. This is very much a work in progress. And we have strayed now as it happens into unfamiliar territory. Terra incognita. Therewill be dragons.[1] Dragons are errors. Errors are dangerous, and we must slay them. But all in good time. First, we should scout out the terrain. That would be prudent.
Descartes in constructing his system of coordinates built upon the bedrockofelementary algebra and the number line. We’vepreviously called attention to the important but mostly overlooked issue of the 1:1 congruence between number and geometric/spatial position he incorporated implicitly in the logic of his coordinates and questioned the validity of such correspondence, at least with respect to subatomic scales.
Working two centuries later but very much under the influence of Descartes’ thought, George Boole introduced his own unique brand of algebra. A second major influence on the development of his symbolic logic was the binary number system of Leibniz, himself influenced to a large degree by Descartes. We need to carefully follow and connect the dots here. Great advances in human cognition rarely, if ever, occur in isolation and seclusion. There is a fine line to tread though. If progress requires the shoulders of giants to stand on, it is still difficult at times not to be overly influenced by those who came before.
Boole’s new logic, constructed in the wake of what by his time were firmly entrenched systematizations of thought by two of the most highly regarded philosopher mathematicians, was devised in such a manner as to conform to both of these conventions of system design. Significant to our purposes here are the facts that first, Boolean logic echoes Cartesian convention of attributing to each and every location in geometric space a single unique number, and second, it adheres to Leibniz’s convention of using a modulo-2 number system based on binary elements 1 and 0.[2]
The symbolic logic systems of mandalic geometry and the I Ching do not abide by either of these conventions. Instead they are based on what is best described as composite dimensions with four unique truth values (or vector directions) each, ranging from -1 through two distinctive zeros (0a; 0b) to +1, and assignment of numbers to spatial locations through all dimensions by means of probability distributions in place of a simple and simplistic 1:1 distribution. To accommodate these alternative conceptual concepts, we will need to expand and modify traditional Boolean logic as we have already done as regards Cartesian coordinate theory.
For starters here we should doubtless add, the mandalic form is the probability distribution through all dimensions, and the probability distributions are the mandalas. And movement through either or both can only be accomplished by discretized stepwise maneuvers between different amplitudes of dimension separated by obscure quantum leaps of endless being and becoming and being and unbecoming, toward and away from the centers and subcenters of holistic systems, the parts of which are always aiming towards some kind of equilibrium never quite within reach. Which then makes error also a necessary aspect of reality and not simply the fearful monster we imagined. It is error that makes achievement possible.[3][4]
Image:Here Be Dragons Map. Detail of he Carta marina (Latin “map of the sea” or “sea map”), drawn by Olaus Magnus in 1527-39. This is the earliest map of the Nordic countries that gives details and place names, by Olaus Magnus [Public domain], via Wikimedia Commons. The map was in production for 12 years. The first copies were printed in 1539 in Venice. [Wikipedia]
Notes
[1] Mapmakers during the Age of Exploration sometimes placed the phrase “here be dragons” at the edges of their known world, presumably to warn of the dangers lying in wait for sailing vessels and travelers by land who strayed too far from well-traveled routes. Here is a list of all known historical maps on which these words appear.
[2] Or in Boole’s case, we might say, attributing to each proposition in concept space a single truth value: TRUE or FALSE (var YES or NO; or, in electronics applications, ON or OFF.) What we have here, I believe, is in many instances a false dilemma or the old Aristotelian dichotomy of either/or. Quantum physics demands and deserves better. OK, true enough, Boole gets around to extending possibilities by means of multi-term propositions, which his system can readily handle. The question here, though, is whether nature can or does handle such similarly. I think not. I think it approaches the question at a more fundamental level of reasoning and reality: at the most basic level of spacetime itself.
[3] This echoes the view of cybernetics, a transdisciplinary approach for exploring regulatory systems, their structures, constraints, and possibilities.
Cybernetics is relevant to the study of systems, such as mechanical, physical, biological, cognitive, and social systems. Cybernetics is applicable when a system being analyzed incorporates a closed signaling loop; that is, where action by the system generates some change in its environment and that change is reflected in that system in some manner (feedback) that triggers a system change, originally referred to as a “circular causal” relationship. [Wikipedia]
[4] This entire blog and its predecessor are in some sense the chronology of a journey from the familiar shoreline into largely uncharted waters. Hesitant at first, increasingly more daring as time has gone on and I’ve come to see errors to be stepping stones along the way. And there have beenmanyerrors along the way. Some I am not yet cognizant of. But of those I am aware, I have left most intact in spite of since being superseded by ideas superior, more correct or better formulated. I’ve done this because I think it important to map the course of a conceptual journey, how the ideas evolved from A to B to C to D. It also allows readers to participate, to a degree, in the thrill of an exciting adventure of mind, should they so choose. Happy travels.
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. :)
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 jandk 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.
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 3.0orGFDL],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.
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 coordinatesortrigonometry 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 theoryandsuperstring theory.
Itis 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 potentialorprobable 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.
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. :)
When a geometric interpretation of imaginary numbers was at last proposed, long after they were invented, it was as though accomplished by central committee. The upshot was easily enough understood but also simplistic. In broad brushstroke here is what seems to have gone down.
The 3 dimensions of Descartes’ coordinate system-a number already deficient from the perspective of mandalic geometry-were reduced to just one. Of the real number axes then only the x-axis remained. This meant from the get-go that any geometric figure that ensued could encompass only linearity in terms of real numbers and dimensions. It was applicable only to a line segment, so the complex plane that resulted could describe just one real dimension and one imaginary dimension. It consecrated the number line in a single dimension, to exclusion of its proper habitation in two others besides. Strike one for imaginary numbers.[1]
With that as background let’s look now at the rotations described by this geometric interpretation of imaginary numbers in the context of the complex plane.
i in the complex or cartesian plane. Real numbers lie on the horizontal axis, and imaginary numbers lie on the vertical axis By Loadmaster (David R. Tribble) (Own work) [CC BY-SA 3.0orGFDL],via Wikimedia Commons
The number 1 is the multiplicative identity element for real numbers and the number -1 is the reflection inversion element for real numbers. Put another way, the number one times any number equals that number; the number -1 times any number is a negative of that number or the inverse number through a reference point, usually taken as zero. Multiplying by 1 then leaves 1, -1, i and -i all unchanged. Multiplying by -1 changes -1 to 1, 1 to -1, i to -i, and -i to i. In terms of rotations in the complex plane, these changes all involve a rotation through 180 degrees. Multiplication of the number 1 by i changes it to i; i by i changes it to -1; -1 by i to -i; and -i by i to 1. These changes all involve rotations through 90 degrees. And finally, multiplication of 1 by -i changes it to -i; -i by -i changes it to -1; -1 by -i to i; and i by -i to 1: all changes involving rotations through -90 degrees.
The figure below shows another way to interpret these rotations that amounts to the same tbing: i1 = i; i2 =-1; i3 = -i; i4 = 1. Click to enlarge.
Four numbers on the real line multiplied by integer powers of the imaginary unit, which corresponds to rotations by multiples of the right angle. By Keφr [CC0],via Wikimedia Commons
I think a committee of some sort must have come up with this resplendent plan. For certain it was an Academy of Mathematics and Sciences that endorsed and enthroned it. All bow to central authority.
I had planned to include a comparison of imaginary numbers and probable numbers in this post as well but because that is a long discussion itself, it will have to wait till the next post. I might add it should prove well worth the wait.
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 3.0orGFDL],via Wikimedia Commons
Notes
[1] Mathematician William Rowan Hamilton subsequently addressed this deficiency in 1843 with his quaternions, a number system that extends the complex numbers to three-dimensional space. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space or, in other words, as the quotient of two vectors. This complicated matters even more by introducing a non-commutative multiplication operation to the system, though to be fair the quaternion coordinate system has found some useful applications mainly for calculations involving 3-dimensional rotations, as in 3-dimensional computer graphics,computer vision, and crystallographic texture analysis. Still it becomes problematic when theoretical physics attempts use of quaternions in calculations pertaining to atomic and subatomic spaces where rotations do not actually take place. The conclusion to be drawn here is that quaternions can be usefully, if somewhat clumsily, applied to 3-dimensional macro-spaces but are inapproriate for accurate description of higher dimensional spaces. What is here unfortunate and misleading is that quaternions apparently do describe outcomes of events in the quantum realm to some partial degree, if not the mechanisms of the events themselves. Physicists would not long tolerate them were that not so.
[ADDENDUM (24 APRIL, 2016) Since writing this I’ve learned that quaternions are not currently used in quantum physics nor were they ever, to any great degree, in the past.]
In other words, sometimes the right answer can be reached by a wrong method. In the case under discussion here, we should note that it is possible for a rotation to mimic inversion (reflection through a point). A 90° rotation in two dimensions can mimic a single inversion in a single plane through an edge of a square, and a 180° rotation in two dimensions can mimic a single inversion through a diagonal of a square or two successive inversions through two perpendicular edges of a square. A 180° rotation in three dimensions can mimic three inversions through three mutually perpendicular edges of a square; a combination of one inversion through a diagonal of a square and another through an edge perpendicular to the plane of the first inversion; or a single inversion through a diagonal of the cube. Subatomic paricles exist as discrete or quantized entities and would follow such methods of transformation rather than rotations through a continuous space. Of course, transformations involving a diagonal would require more transformative energy than one involving a single edge.
Such patterns of relationship and transformation could no doubt be described in terms of quantum states and quantum numbers without too much difficulty by a knowledgeable theoretical physicist. Surely doing so could be no more difficult than using quaternions, which may give a correct answer while also misleading and limiting knowledge of the the true workings of the quantum realm by using an incorrect mechanism, one non-commutative to boot. Nature doesn’t approve of hat tricks like that.
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. :)
How Quantum Physics Allows Us To See Back Through Space And Time
“If it weren’t for this rare transition, from higher energy spherical orbitals to lower energy spherical orbitals, our Universe would look incredibly different in detail. We would have different numbers and magnitudes of acoustic peaks in the cosmic microwave background, and hence a different set of seed fluctuations for our Universe to build its large-scale structure out of. The ionization history of our Universe would be different; it would take longer for the first stars to form; and the light from the leftover glow of the Big Bang would only take us back to 790,000 years after the Big Bang, rather than the 380,000 years we get today.
In a very real sense, there are a myriad of ways that our view into the distant Universe — to the farthest reaches of deep space where we detect the earliest signals arising after the Big Bang — that would be fundamentally less powerful if not for this one quantum mechanical transition. If we want to understand how the Universe came to be the way it is today, even on cosmic scales, it’s remarkable how subtly dependent the outcomes are on the subatomic rules of quantum physics. Without it, the sights we see looking back across space and time would be far less rich and spectacular.”
What gives the Universe the properties we see today? Is it gravity, working on the largest of cosmic scales? It plays a role, but perhaps ironically, the subatomic physics that governs electron transitions within atoms is maybe even more important.
so a couple of days ago a had a lecture and it was… quite intense. the lightbulb moments that made sense simultaneously didn’t make sense, lmao. I later got to reviewing the content to better understand it, but here’s a…
protip! - find a new place to study or work. for the past year, i’ve been working at the same table in the same area. but studying (and eating lunch) at two different places within two days (with earbuds in to focus) made me feel more refreshed and energized. you should try it :)
The ether is considered a universal medium consisting of a primary substance, attenuated beyond conception, which fills all space and connects all matter. This medium, or field of force, is responsible for action at a distance—a concept where an object can interact with other objects even though they are separated in space. This idea still baffles today’s physicists, but was understood by Nikola Tesla long before Albert Einstein coined his “spooky action at a distance”.
Before I get into Tesla’s explanation of the ether, I must first recall the famous 1887 Michelson-Morley experiment, because I know some readers will immediately bring it up. The experiment was intended to detect the ether using light beams and mirrors to record the speed of light through the ether relative to the Earth’s movement around the Sun; however, the two scientists failed to detect the ether and it became one of the most famous failed experiments in history. Surprisingly though, the experimenters did not account for the fact that the speed of light was relative to the observer moving with the apparatus, which led to the null effect. What it did, rather, was prove that the average velocity of light for a round trip between a beam splitter and a mirror was independent of motion through space. Either way, physicists agreed that by its nature, the ether cannot be detected and it is unnecessary for explaining how light travels through space.
It was Heinrich Hertz, who during the same time as the Michelson-Morley experiment, demonstrated the notion of action at a distance proving the existence of electromagnetic waves first predicted by James Clerk Maxwell in 1864. Since these waves travel across space, there must be a medium carrying the waves. Like Maxwell, Hertz postulated that ether was structureless beyond conception, and yet solid and possessed a rigidity incomparably greater than the hardest steel. Electromagnetic waves were then believed to be transverse waves (waves that vibrate at ninety degrees angles).
In the early 1890s, Nikola Tesla repeated Hertz’s experiments with a much improved and a far more powerful apparatus, coming to the conclusion that what Hertz observed were longitudinal waves in a gaseous medium propagated by alternate compression and expansion. After discovering these results, Tesla declared that light, and other electromagnetic waves, are not transverse waves (a theory still believed today in conventional physics), but instead are a longitudinal disturbance in the ether involving alternate compressions and rarefactions. In his own words, “light can be nothing else than a sound wave in the ether. Since light has such a constancy of velocity, light can only be explained by assuming that it is dependent solely on the physical properties of the medium, especially density and elastic force.” It wasn’t until after Nikola Tesla met with Hertz and explained his results that Hertz then changed his views on the ether and accepted that it was a gaseous medium rather than a stationary one.
Believing that the ether was one of the most important results of modern scientific research, Tesla refused to abandon it because in his mind the ether was an important key to understanding how electrical energy could travel through space without wires. He displayed this phenomenon in numerous experiments and lectures throughout the 1890s.
It wasn’t until 1896 when Tesla finally obtained experimental proof of the ether. He invented a new form of vacuum tube which could be charged to any high potential and operated with pressures up to 4,000,000 volts. In 1929, Tesla spoke of these vacuum tubes saying, “One of the first striking observations made with my tubes was that a purplish glow for several feet around the end of the tube was formed, and I readily ascertained that it was due to the escape of the charges of the particles as soon as they passed out into the air; for it was only in a nearly perfect vacuum that these charges could be confined to them. The coronal discharge proved that there must be a medium besides air in the space, composed of particles immeasurably smaller than those of air, as otherwise such a discharge would not be possible. On further investigation I found that this gas was so light that a volume equal to that of the earth would weigh only about one-twentieth of a pound.”
To explain the density of the ether, Tesla referred to William Thomson’s equations. In 1932, Tesla said, “Its density has been first estimated by Lord Kelvin and conformably to his finding a column of one square centimeter cross section and of a length such that light, traveling at a rate of three hundred thousand kilometers per second, would require one year to transverse it, should weigh 4.8 grams. This is just about the weight of a prism of ordinary glass of the same cross section and two centimeters length which, therefore, may be assumed as the equivalent of the ether column in absorption. A column of the ether one thousand times longer would thus absorb as much light as twenty meters of glass. However, there are suns at distances of many thousands of light years and it is evident that virtually no light from them can reach the earth. But if these suns emit rays immensely more penetrative than those of light they will be slightly dimmed and so the aggregate amount of radiations pouring upon the earth from all sides will be overwhelmingly greater than that supplied to it by our luminary. If light and heat rays would be as penetrative as the cosmic, so fierce would be the perpetual glare and so scorching the heat that life on this and other planets could not exist.”
According to Nikola Tesla’s ether theory, all matter in the universe is metamorphous from the ether. When the ether is set in motion, it becomes gross matter. All matter, then, is merely ether in motion. In 1900, Tesla said, “By being set in movement, ether becomes matter perceptible to our senses; the movement arrested, the primary substance reverts to its normal state and becomes imperceptible. If this theory of the constitution of matter is not merely a beautiful conception, which in its essence is contained in the old philosophy of the Vedas, but a physical truth, then if the ether whirl or atom be shattered by impact or slowed down and arrested by cold, any material, whatever it be, would vanish into seeming nothingness, and, conversely, if the ether be set in movement by some force, matter would again form. Thus, by the help of a refrigerating machine or other means for arresting ether movement and an electrical or other force of great intensity for forming ether whirls, it appears possible for man to annihilate or to create at his will all we are able to perceive by our tactile sense.”
In summary, Tesla experimented, and proved his theories using the scientific method. His methods were far more superior to other physicists of his time, because he had the motors and transformers invented by himself to help with his experiments. These include the induction motor, his Telsa coil, and many more apparatuses. In the future the ether may be referred to as dark matter, the force etc., but Nikola Tesla’s ether theory will surely be proven true in years to come.
