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:
The electron is a first generation fermionand a lepton. Fermions are particles with have half-integer spinthat follow Fermi-Dirac statistics and obey the Pauli exclusion principle. The Pauli exclusion principle states that two identical fermions can’t occupy the same quantum state (i.e. have the same quantum numbers within a quantum system). Leptons are a subcategory within fermions that can exist independently (without binding together) and do not interact through the strong force unlike quarks. Lastly the generations of the fermions loosely refers to the higher masses for particles in higher generations.
The existence of electrons was first discovered by J.J. Thompson in 1897 when he experimented with cathode ray tubes like the one depicted above. By applying electric and magnetic fields across the cathode ray Thompson was able to determine the mass-to-charge ratio of the particles in the cathode ray. With this he found that the particles were much smaller than any atom and by testing different sources, these negatively charged particles exist in every element.
Electrons are one of the primary charge carriers in atoms alongside protons but are the primary contributors to electric current. Electrons also have an intrinsic property known as spin which contributes to paramagnetismin certain materials.
Above is a video of an electron riding a light wave. The video was taken using a stroboscope which captures. More on it here (article)andhere (video).
Research involving electrons covers almost every corner of modern physics from high energy particle physics to condensed matter physics and even quantum computing. I have linked articles on recent research with a focus on electrons below for further reading.
University of Chicago scientists are part of an international research team that has discovered superconductivity–the ability to conduct electricity perfectly–at the highest temperatures ever recorded.
[…]
Using advanced technology at UChicago-affiliated Argonne National Laboratory, the team studied a class of materials in which they observed superconductivity at temperatures of about minus-23 degrees Celsius (minus-9 degrees Fahrenheit)–a jump of about 50 degrees compared to the previous confirmed record.
Though the superconductivity happened under extremely high pressure, the result still represents a big step toward creating superconductivity at room temperature–the ultimate goal for scientists to be able to use this phenomenon for advanced technologies. The results were published May 23 in the journal Nature; Vitali Prakapenka, a research professor at the University of Chicago, and Eran Greenberg, a postdoctoral scholar at the University of Chicago, are co-authors of the research.
Many of you may recognize this photo of the x-ray diffraction pattern of DNA found by Rosalind Franklin and her PhD student, Raymond Gosling. But, you may wonder how one could figure out from this image that DNA is structured as a double helix and even how x-ray crystallography works.
X-Ray Crystallography
X-ray crystallography is a method of determining the positions and arrangements of atoms in a crystal. Crystals are usually defined to be a highly ordered and repeating microscopic structure of a solid rather than the macroscopic crystals we know like quartz which actually tend to be “polycrystals” because at a microscopic level they do have the highly ordered structure required. Ice is also a polycrystal composed of many smaller ice crystals.
1.) X-ray beams are shot at the crystals
The x-rays interact with electrons of the atoms. This interaction or collision is typically modeled by Thomson scattering where the energy and thus frequency of the x-rays do not change after diffraction. This is similar to light going through a diffraction grating.
2.) Beam is diffracted
The x-rays are diffracted based on the crystal lattice structure of the substance. This is dependent on the characteristics of the bonds between atoms like the bond angles and bond lengths. Also the spacing between molecules also determines the diffraction.
3.) Diffraction pattern
The diffracted x-rays are light waves so they interfere both constructively and destructively. The resulting intensities of the x-rays are recorded on a screen behind the sample to create a diffraction pattern. The sample is rotated to take more data. After sufficient data is taken a model for the crystal structure for the sample can be developed. With a diffraction pattern an electron density map can be made which depicts the location and size of electron clouds in the substance.
Richard P. Feynman an astounding theoretical physicist and professor
∆ Quantum mechanics & particle physics
∆ Quantum electrodynamics (QED) for which he shared a Nobel Prize
∆ Superfluidity of liquid helium
The diagram above is of a vector boson fusion producing a Higgs boson. Feynman developed this method of representing particle interactions which have been important to the understanding of work in particle accelerators such as the Large Hadron Collider.
The following is a wonderful video of Feynman talking about light
This Is Why Quantum Field Theory Is More Fundamental Than Quantum Mechanics
“But the motivation for quantizing the field is more fundamental than that the argument between those favoring perturbative or non-perturbative approaches. You need a quantum field theory to successfully describe the interactions between not merely particles and particle or particles and fields, but between fields and fields as well. With quantum field theory and further advances in their applications, everything from photon-photon scattering to the strong nuclear force was now explicable.”
What’s wrong with quantum mechanics? It might surprise you to hear that the answer is, “it isn’t quantum enough.” The enormous differences between the quantum and the non-quantum Universe are so striking, as we replace:
* continuous matter with discrete particles, * ideal points with dual-nature wave/particle quanta, * and observable properties like position and momentum with quantum mechanical operators containing an inherent uncertainty.
But it’s still not enough. For one, the original (Schroedinger) equation of quantum mechanics doesn’t play nice with relativity, but even the relativistically invariant versions don’t describe reality fully. Why not? Because only the particles are quantized in quantum mechanics. To reveal the full behavior, you need to quantize their fields and interactions, too.
Two teams of scientists from the Technion-Israel Institute of Technology have collaborated to conduct groundbreaking research leading to the development of a new and innovative scientific field: Quantum Metamaterials. The findings are presented in a new joint paper published in the journal Science.
The study was jointly conducted by Distinguished Professor Mordechai Segev, of Technion’s Physics Department and Solid State Institute and his team Tomer Stav and Dikla Oren, in collaboration with Prof. Erez Hasman of the Technion’s Faculty of Mechanical Engineering and his team Arkady Faerman, Elhanan Maguid, and Dr. Vladimir Kleiner. Both groups are also affiliated with the Russell Berrie Nanotechnology Institute (RBNI).
The researchers have demonstrated for the first time that it is possible to apply metamaterials to the field of quantum information and computing, thereby paving the way for numerous practical applications including, among others, the development of unbreakable encryptions, as well as opening the door to new possibilities for quantum information systems on a chip.
Metamaterials are artificially fabricated materials, made up of numerous artificial nanoscale structures designed to respond to light in different ways. Metasurfaces are the 2 dimensional version of metamaterials: extremely thin surfaces made up of numerous subwavelength optical nanoantennas, each designed to serve a specific function upon the interaction with light.
One-dimensional semiconductor nanowires with strong quantum confinement effect—quantum wires (QWs)—are of great interest for applications in advanced optoelectronics and photochemical conversions. Beyond the state-of-the-art Cd-containing ones, ZnSe QWs, as a representative heavy-metal-free semiconductor, have shown the utmost potential for next-generation environmental-friendly applications.
Unfortunately, ZnSe nanowires produced thus far are largely limited to the strong quantum confinement regime with near-violet-light absorption or to the bulk regime with undiscernible exciton features. Simultaneous, on-demand, and high-precision manipulations on their radial and axial sizes—that allows strong quantum confinement in the blue-light region—has so far been challenging, which substantially impedes their further applications.
In a new article published in the National Science Review, a research team led by professor YU Shuhong at University of Science and Technology of China (USTC) has reported the on-demand synthesis of high-quality, blue-light-active ZnSe QWs by developing a flexible synthetic approach—a two-step catalytic growth strategy that enables independent, high-precision, and wide-range controls over the diameter and length of ZnSe QWs. In this way, they bridge the gap between prior magic-sized ZnSe QWs and bulk-like ZnSe nanowires.
Fundamental research sheds light on new many-particle quantum physics in atomically thin semiconductors
A paper published in Nature Communications by Sufei Shi, assistant professor of chemical and biological engineering at Rensselaer, increases our understanding of how light interacts with atomically thin semiconductors and creates unique excitonic complex particles, multiple electrons, and holes strongly bound together. These particles possess a new quantum degree of freedom, called “valley spin.” The “valley spin” is similar to the spin of electrons, which has been extensively used in information storage such as hard drives and is also a promising candidate for quantum computing.
The paper, titled “Revealing the biexciton and trion-exciton complexes in BN encapsulated WSe2,” was published in the Sept. 13, 2018, edition of Nature Communications. Results of this research could lead to novel applications in electronic and optoelectronic devices, such as solar energy harvesting, new types of lasers, and quantum sensing.
Shi’s research focuses on low dimensional quantum materials and their quantum effects, with a particular interest in materials with strong light-matter interactions. These materials include graphene, transitional metal dichacogenides (TMDs), such as tungsten diselenide (WSe2), and topological insulators.
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.
Scientists pursuing better performance in a well-known type of iron-based magnet also discovered wide-gap semiconducting behavior and a quantum state useful for quantum information processing—all in a single low-cost material that has been in existence for decades.
Scientists at the U.S. Department of Energy’s Critical Materials Institute, or CMI, study ways to make lower-cost, easier-to-obtain materials used as ingredients in technologies that are in demand now or are developing for the future. In this case, the researchers were investigating ways to create a stronger iron-based permanent magnet, something referred to as a “gap” magnet.
Permanent magnets fall into two broad categories. The strongest-performing permanent magnetscontainrare-earth metals like samarium, neodymium, and dysprosium—their properties make them the best and often only choice for applications like computer hard disk drives and motors in hybrid and electric vehicles. These magnets are typically expensive, and their rare-earth components can be difficult to obtain. The second, iron-based permanent magnets, are inexpensive and made of readily available materials, but their performance is often too poor for many advanced applications. In between the high performing rare-earth magnets and low-performing iron-based magnets is a “gap,” where there is a great need for permanent magnets that perform in the mid-range of desirable properties. Filling that gap reduces the need for rare-earth magnets, and in turn hard to source rare-earth materials.
Researchers have shown how to shuttle lithium ions back and forth into the crystal structure of a quantum material, representing a new avenue for research and potential applications in batteries, “smart windows” and brain-inspired computers containing artificial synapses.
The research centers on a material called samarium nickelate, which is a quantum material, meaning its performance taps into quantum mechanical interactions. Samarium nickelate is in a class of quantum materials called strongly correlated electron systems, which have exotic electronic and magnetic properties.
The researchers “doped” the material with lithium ions, meaning the ions were added to the material’s crystal structure.
The addition of lithium ions causes the crystal to expand and increases the material’s conduction of the ions. The researchers also learned that the effect works with other types of ions, particularly sodium ions, pointing to potential applications in energy storage.
The work advances the possibility of applying quantum mechanical principles to existing optical fiber networks for secure communications and geographically distributed quantum computation. Prof. David Awschalom and his 13 co-authors announced their discovery in the June 23 issue of Physical Review X.
“Silicon carbide is currently used to build a wide variety of classical electronic devices today,” said Awschalom, the Liew Family Professor in Molecular Engineering at UChicago and a senior scientist at Argonne National Laboratory. “All of the processing protocols are in place to fabricate small quantum devices out of this material. These results offer a pathway for bringing quantum physics into the technological world.”
The findings are partly based on theoretical models of the materials performed by Awschalom’s co-authors at the Hungarian Academy of Sciences in Budapest. Another research group in Sweden’s Linköping University grew much of the silicon carbide material that Awschalom’s team tested in experiments at UChicago. And another team at the National Institutes for Quantum and Radiological Science and Technology in Japan helped the UChicago researchers make quantum defects in the materials by irradiating them with electron beams.
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.
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. :)
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.
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.
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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. :)
Taoism and the primordial I Chingare in agreement that temporal changes have two different aspects: sequent and cyclic. Western thought in general follows suit. The I Ching differs from the other two in asserting that the direction of change - for both sequent and cyclic change - is fully reversible, with the proviso that sufficiently small units of measurement are involved.[1] The probability that reversal can be achieved diminishes proportionately to the magnitude of change that has taken place.[2]
Taoist appropriation of bigrams and trigrams of the I Ching to model such phenomena as change of seasons and phases of the moon is plausible if not quite legitimate. The natural phenomena so modeled are macroscopic and vary continuouslyandinexorably throughout an ever-repeating cyclic spectrum. And there’s the rub.
As they occur and function in the I Ching bigrams and trigrams are dicontinuous discrete elements, formed by other similarly discontinuous discretized entities, and they follow evolutionary courses which are most often nonrepetitive. So the Taoist usage is misleading at best, annihilative at worst. Unfortunately, as the I Ching itself evolved through centuries of commentaries and reinterpretations, it became ever more contaminated and tainted by these Taoist corruptions of meaning, at the same time that it was being inundated by Confucian sociological and ethical reworkings. What we have today is an amalgam, the various parts of which do not sit well with one another.[3]
Though it may in part be hyperbole to prove a point, the stark difference between the two approaches, that of Taoism and that of the I Ching, is epitomized by comparison of the Taoist diagram of the cycle of seasons with diagrams at the top and bottom of the page, which are based on the number, logic, and coordinate systems of The Book of Changes.[4] The increased complexity of the latter diagrams should not prove a stumbling block, as they can be readily understood in time with focus and attention to detail. The important take-away for now is that in the I Ching bigrams exist within a larger dimensional context than the Taoist diagram avows, and this context makes all their interactions more variable, conditional, and complex. As well, the same can be said of trigrams and hexagrams.
One of the more important aspects of these differences has to do with the notion of equipotentiality. As bigrams and trigrams function within higher dimensional contexts in the I Ching, this introduces a possibility of multiple alternative paths of movement and directions of change. Put another way, primordial I Ching logic encompasses many more degrees of freedom than does the logic of Taoism.[5] There is no one direction or path invariably decreed or favored. An all-important element of conditionality prevails. And that might be the origin of what quantum mechanics has interpreted as indeterminism or chance.
Next up, a closer look at equipotentiality and its further implications.
[1] There are exceptions. Taoist alchemy describes existence of certain changes that admit reversibility under special circumstances. Other than the Second Law of Thermodynamics (which is macroscopic in origin, not result of any internally irreversible microscopic properties of the bodies), the laws of physics neglect all distinction between forward-moving timeandbackward-moving time. Chemistry recognizes existence of certain states of equilibrium in which the rates of change in both directions are equal. Other exceptions likely occur as well.
[2] Since change is quantized in the I Ching, which is to say, it is divided into small discretized units, which Line changes model, the magnitude of change is determined by the number of Line changes that have occurred between Point A and Point B in spacetime. Reversal is far easier to achieve if only a single Line change has occurred than if three or four Lines have changed for example.
[3] Ironically, Taoism itself has pointed out the perils of popularity. Had the I Ching been less popular, less appealing to members of all strata of society, it would have traveled through time more intact. Unless, of course, it ended up buried or burned. What is fortunate here is that much of the primordial logic of the I Ching can be reconstructed by focusing our attention on the diagrammatic figures and ignoring most of the attached commentary.
[4] These diagrams do not occur explicitly in the I Ching. The logic they are based on, though, is fully present implicitly in the diagramatic structural forms of hexagrams, trigrams, and bigrams and the manner of their usage in I Ching divinatory practices.
[5] Or, for that matter, than does the logic of Cartesian coordinate space if we take into account the degrees of freedom of six dimensional hexagrams mapped by composite dimensional methodology to model mandalic space. (See Note [4] here for important related remarks.)
[6] This is the closest frontal section to the viewer through the 3-dimensional cube using Taoist notation. See here for further explanation. Keep in mind this graph barely hints at the complexity of relationships found in the 6-dimensional hypercube which has in total 4096 distinct changing and unchanging hexagrams in contrast to the 16 changing and unchanging trigrams we see here. Though this model may be simple by comparison, it will nevertheless serve us well as a key to deciphering the number system on which I Ching logic is based as well as the structure and context of the geometric line that can be derived by application of reductionist thought to the associated mandalic coordinate system of the I Ching hexagrams. We will refer back to this figure for that purpose in the near future.
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. :)
