#cartesian coordinates

Magic Theatre: For Madmen Only

If we describe a Cartesian ordered triad by x,y,z we can describe an analogous 6-dimensional ordered sextuplet  or 6-tuple  by x_a,y_a,z_a,x_b,y_b,z_b

The definitions that translate a 6-dimensional ordered sextuplet (hexagram in Taoist terminology) into a 3-dimensional ordered triad (trigram in Taoist terminology) are:^[1]

• (x_a + x_b) / 2 = x
• (y_a + y_b) / 2 = y
• (z_a + z_b) / 2 = z

I think the methodology will work for all scalar quantities. But as currently formulated,  mandalic geometry (MG) is a discrete geometry based entirely on unit vectors.  We are talking about the line segments between -1 and +1 in the various dimensions and only points -1, 0, and +1 in each line segment in Cartesian terms.

In essence we are not yet particularly concerned with scalars here but only with vectors :  -, +, and neutral (0).

Mathematically √−1 is important because by adding it to the real number field, as we have done, we create the algebraically complete field of complex numbers. In mathematics, a complete field is a field equipped with a metric and complete with respect to that metric. The real numbers and complex numbers are both complete fields. Cartesian coordinates- - - ordered pairs and ordered triads- - - although based on real numbers,  do not form a field. This has important implications, implications which can be ignored only at peril to the particular conceptual system involved..

The definitions above all give three possible results in Cartesian terms: -1, 0, +1.  Remember though MG hybridizes six dimensions with three dimensions and represents them superimposed. Wherever one or more zeros occurs in Cartesian coordinates we have also corresponding 6-dimensional forms,  composed of just +1s and -1s,  of which there are always two for each Cartesian zero.  A Cartesian ordered triad with one zero is associated with two such 6-dimensional forms; an ordered triad with two zeros, with four; an ordered triad with three zeros (the origin), with eight. An ordered triad without zeros will have only one associated 6-dimensional form.  This constitutes the mandalic pattern, which is an essential feature of the 6D/3D formulation of this geometric system and isomorphism naturally comes into play here as well.

Andthat is how and why all numbers in this coordinate system based on higher-dimensional extensions of the real numbers “square” to numbers which can be  either positive or negative  and then reduce or "collapse" to corresponding Cartesian forms that preserve the same sign. This is a necessary result of the fact that a primary “zero form” in
6-dimensional terms is lacking,  only +1s and -1s exist.  These can then interfere constructively and destructively as number waves, to produce a  "secondary zero"  by destructive interference  whenever linked forms differ in sign in one or more paired dimensions. Since the two linked 6-dimensional numbers are always inverse to one another, any Cartesian zero then can be substituted with two such 6-dimensional forms. This is the process that makes imaginary numbers unnecessary, replacing them with two inversely related probable numbers which behave in most ways like real numbers  and  are  distributed  throughout the entire geometric system.

“Hybridization” is probably not the best term here but will be used until I can think of a better descriptor. What I intend is not actual joining and unification,  but rather  a superposition and conceptual commingling in three-dimensional terms. Such a representational mapping substitutes for all Cartesian forms  "equivalent" forms  containing only 1s and -1s, no zeros.  In so doing, it effectively converts the Cartesian coordinate system from just a ring to a field as well, properly interpreted. Basically then, the probable numbers do for the real numbers much the same as the complex numbers do,  but with even greater and more utilitarian results which are also more easily managed.

In operational terms, complex numbers perform two rather simple binary operations: a scaling and a rotation. Scaling capability is clearly inherited through its real number lineage; rotational capacity, from its imaginary number lineage.  Together,  scaling and rotation combine to augment or diminish an axis of growth and produce vector ambulation in a circular path about a central origin point of reference.  The scaling factor  could be said to detemine the  radius of revolution;  the rotation factor, the angle of revolution. And that’s pretty much all there is to the “great mystery”  of complex numbers.  Their importance  resides in the great number of fields of endeavor where the combination of these two superpowers is necessary and/or convenient.

Nature uses this combination of scaling and rotation in many of its processes.  Atomic and subatomic proceedings  are probably not among these.  How then did it come about that  quantum mechanics  arrived at the notion that  rotation and scaling  could be applicable to modeling of discontinuos states of being?  Both refer to changes through continuous space. I think it was an accident of history. In 1925, Erwin Schrödinger, in his search for a way to explain  certain mysteries then perplexing the greatest physicists of the day,  hit upon his  eponymous equation  which appeared to do the trick.  So well,  in fact,  that quantum mechanics has been  justly considered  the single most successful description of reality ever devised. And the equation that basically accomplished this success involves the imaginary number i and complex numbers.^[2]

An important aspect of the operation of rotation, one which may have bearing on the Schrödinger equation and its huge success, has been largely overlooked. The result of a rotation can often mimic the result of inversion (reflection through a point), making the two indistinguishable by measurement alone. To someone wearing a blindfold there is no way to tell whether i has by the operations of squaring and rotation changed itself into  -1  or  -1,  the inversion element of multiplication,  has simply reflected  +1,  the identity element of multiplication,  through the origin point to  -1.  Explaining away a 90° rotation with a right angle reflection will no doubt prove more difficult but let’s not just yet deny that it might be doable.

Could there be a way to reformulate the Schrödinger equation then so it contains no imaginary or complex numbers?  Many have tried to do that very thing and failed. No one has succeeded in nearly a century. Still, we might wonder if the time is ripe now to remove the blindfold. Perhaps we might do well to inquire whether quantum physics is, in some manner we don’t quite understand, a victim of its own success.

In theory, circumventing use of complex numbers in a defining equation of quantum mechanics should be possible. On what basis do I say this?  The equation we have now relies on complex numbers.  These in turn derive an ability to produce rotation from the imaginary number √−1 .  But there are  other mathematical means  to accomplish the same. Trigonometry comes most immediately to mind. The circle and cyclicity it models have a very long and distinguished history. Complex numbers as we’ve noted can also produce scaling.  But so can real numbers.  And close examination reveals  that complex numbers inherit their ability to scale from the two real numbers they contain. The hard truth ultimately is  there is nothing all that special  about  complex numbers  or complex plane. Possibly it is their utilitarian ease of use that positions them as an attractive methodology. Other routes to ease of use exist as well. There is always more than one way to skin the proverbial cat  (even a cat residing only in the mind of a physicist named Schrödinger.)

Consider also, how great is the actual need for scaling in quantum mechanics?  The distance from  centermost part of the atom  to the outer reaches of electron orbital space is in fact quite small.  Furthermore,  the elements of this universe of discourse are quantized,  so actual distances involved are moot.  In the extreme,  the question persists  as to  whether “distance” is a concept even applicable  in this context  of quantum logic. Quantum numbers  themselves  range between 0 and 2.  I can count the allowed values on the fingers of one hand.

Regarding rotation, where exactly does that come into play in the quantum realm?  Electrons do not orbit the nucleus of the atom.  They jump from orbital to orbital by discretized changes in energy involving photon exchange. In the nucleus it seems such discretized instanteous changes take place as well,  obviating any need for rotation.  Obviously physics misguided here by labeling one of the quantum numbers “spin”. Sometimes a rose is best referred to as a rose. The problem here is that we don’t really know what it is that “spin” refers to.

The quintessential equation of quantum mechanics was formulated by a physicist, not a mathematician. It is not a simple algebraic equation, but in general a linear partial differential equation,  describing the time-evolution of the system’s wave function (“state function”). “Derivations” of the Schrödinger equation  do generally demonstrate its mathematical plausibility for describing wave-particle duality. To date, however, there are  no universally accepted derivations  of Schrödinger’s equation from appropriate axioms.  Nor is there any  general agreement  as to what the equation actually signifies.  Moreover, some authors have demonstrated that certain properties  emerging from Schrödinger’s equation  can even be deduced from symmetry principles alone.  This would appear to be a worthwhile direction of investigation to pursue.  Quantum mechanics is most fundamentally about symmetry.  Let’s make Emmy Noether proud by giving her the recognition she deserves.

Finally, it was not without considerabledifficulty that Schrödinger developed his equation.  In the end,  it almost seems  he pulled it out of a hat,  as a magician might a rabbit.^[3]   Part of the  Zeitgeist  of the physics community  in the early 1920s  revolved around  the peculiar notion  that particles  behaved as waves.  Schrödinger decided to follow this direction of thought  and  find an appropriate 3-dimensional wave equation for the electron. His equation succeeded beyond his wildest dreams.  Adopted in the canon of  the new physics,  it became the cornerstone of that radically different physics, changed forever. Physics has never looked back since.

Still, one startling and haunting fact persists: nowhere else in all of physics  has it ever been found necessary to invoke complex numbers.

Once,  quite a long time ago,  I believed  imaginary numbers  were wrong. I was the one that was wrong. Later, having grown a little more clever, I came to think that √−1 was a necessary evil- - -correct but not validly applicable to quantum physics. Wrong again. Currently it is my belief that imaginary numbers are guilty of an even worse offense: both true from the mathematical standpoint and partly applicable to physics. The worst of both worlds.  Yielding results that are in large part correct, imaginary and complex numbers have managed to lead us all down the garden path for the better part of a century. Have we then gone past the point of no return?  My contention  is  that it is possible to complete the ring that Cartesian coordinates present  and  transform it to a field over the real numbers, with appeal only to higher-dimensional analogues of the reals and no need for imaginary or complex numbers,  an approach which, if actually possible, would offer certain undeniable advantages.^[4]

Essentially the method of composite dimension does away with i and complex numbers by distributing an operation analogous to that of i throughout six dimensions or three in Cartesian terms and then working with same by means of reflections (inversions) only. So an algebra based on the system necessitates use of only the real numbers and their higher dimension extensions that I have called probable numbers.  Only simple addition and multiplication  are required.  For those in the audience who are "sufficiently mad”, there is the added bonus that a kind of division by zero becomes possible. We’ll find out soon enough whether you qualify.

A few additional explanatory remarks are in order here:

Depending on the variant,  Cartesian geometry (CG),  represents space in two or three dimensions. Points in the former are referenced to two pairwise perpendicular axes; in the latter, to three.

Because Descartes assumes as axiomatic a 1:1 correspondence of number to spatial location each of his three axes becomes a facsimile of the number line, only in different dimensions.

Mandalic geometry (MG) approaches representation of space differently, using a hybrid coordinate system which relates a higher dimension space to a lower dimension space  with a 2:1 correlation.

Itcan be represented entirely commensurate with CG, but in so doing a “glass slipper effect” occurs. Just as Cinderella’s stepsisters can manage to force a too fat foot into her glass slipper, the results leave something to be desired.  In our context here,  the  "something to be desired"  is a clear and full understanding of six-dimensional reality in its own right. We end up interpreting it in time-sharing terms of probabilities and randomness.

What Descartes refers to as an ordered pair requires two higher dimension ordered pairs to represent in MG; a Cartesian ordered triad requires three higher dimension ordered pairs to represent in MG.

In Taoist terminology the notational equivalent of a Cartesian ordered pair is a  "bigram",  a two-line symbol,  each line of which  can take one of two values. As a result there are four types of bigram. Two bigrams make up a tetragram; three, a hexagram.

Descartes views a point as having only two essential characteristics:

• It is dimensionless.
• It is just a location in space which can be uniquely represented
  by a single ordered pairorordered triad.

Mandalic geometry rejects both of these axioms. It regards a point, or a particle so represented, as an evanescent entity emerging from interaction of two higher dimensions expressed in our world of three dimensions in such limited manner.

Thiscan be represented in context of Cartesian space but in making mandalic coordinates commensurate with Cartesian coordinates it is no longer possible to represent every “point” in space uniquely with a single mapping of number to location.  What results instead is the probabilistic distribution pattern of the mandala, which we,  from our limited vantage in spacetime, misinterpret as something it is not.

MG is a discrete geometry. The result of the mapping formula used is a mandalic configuration in which the 3-dimensional cube composed of  unit vectors in Cartesian space  becomes a  "probability distribution"  in combined mandalic space.

I have placed the quotation marksaroundprobability distribution because this is a perspective that arises  from our inability to see all that is involved accurately. I suspect this has repercussions pertinent to a full comprehension or grokking of quantum mechanics and possibly of string theory as well.

Since the 64 discrete “points” of  the unit vector hypercube of six dimensions represented by the hexagrams cannot “fit” simultaneously in the 27 discrete points of the 3-dimensional unit vector cube  by any representational method available to our inherited bio-psychocultural mechanism, a sort of time-sharing process occurs in observations and measurements of reality which we interpret in terms of probability.

What has been described here occurs at enormous velocities close to that of light, and likely refers only to processes in the subatomic quantum realm. For MG, which is also a hybridization of mathematics and physics, context is always of the essence.

There is much more to be said in explanation of mandalic geometry. I see, though, this post has already run rather long, so we will end it here. Enough has already been said in way of introduction of basic material.

Notes

[1] Since the coordinate system is describing a cube with an n-hypercube superimposed,  there is an additional constraint placed on all coordinates in
the 6-tuples.  All scalar values must be identical for x, y and z values.  That constraint assures that all vectors though they may differ in sign (direction) maintain equal magnitudes.

When the 6-tuples are dimensionally reduced to 3-tuples by the method I’ve called “compositing of dimension”  the resulting geometric figure consists of four different dimensional amplitudes of 6-tuples collapsed.  The amplitudes of dimension correspond  in spatial terms  to the vertices,  edge centers,  face centers and cube center. The pattern that emerges is that of a mandala. This is a highly symmetric pattern though all symmetries aren’t necessarily apparent immediately, even using Taoist notation. The probability distribution of the 6-tuples allots the hexagrams in the following manner:  one to each vertex;  two to each edge center; four to each face center; one to the cube center. The result is  placement of 64 6-tuples  in 27 positions of discrete 3-tuples  in the specific mandalic distribution pattern described.

Think here of the analogy of a hydrogen atom confined within a cubic space of specified side length determined by the nuclear and atomic force fields. The single electron,  existing in such quantized energy levels that are possible,  can assume various different locations in different orbital shells,  but every location in a given orbital must be equidistant from the nuclear proton. Once reduced by dimensional compositing the 6-tuples described here fill four distinct shells that have different radii or distances from the center.  From center to periphery these distances can be described as zero;  one (or square root one);  square root 2; and square root 3. (Pythagorean theorem)

[2] Schrödinger was not entirely comfortable with the implications of quantum theory. About the probability interpretation of quantum mechanics that came out of Solvay ‘27 he wrote:  "I don’t like it,  and I’m sorry I ever had anything to do with it.“ ["A Quantum Sampler”. The New York Times. 26 December 2005.]

[3] In later years another great physicist, Richard Feynman, would remark, “Where did we get that (equation) from? Nowhere. It is not possible to derive it from anything you know. It came out of the mind of Schrödinger.”

[4] A different approach to avoiding the need for complex numbers from the one I am suggesting is described here. To my mind it offers little of value other than an interesting alternative explanation of what complex numbers are and do. A similar conclusion seems to have been reached by the author.

© 2016 Martin Hauser

Please note:  The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent).  This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added.  To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering.  To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 313-

Beyond the Enlightenment Rationalists:
From imaginary to probable numbers - IV

(continued from here)

One of the notable things the Rationalists  failed to take into account in their analysis and codification of square roots  was  the significance of context. In so doing they assured that all related concepts they developed would eventually degenerate into a series of errors of conflation.  Do  not ever underestimate the importance of context.

Mathematicians, for example, can show that for any 3-dimensional cube  there exists  a  2-dimensional square,  the area of which equals the volume of the cube.^[1] And although that is true, something has been lost in translation. This is another of the sleights of hand mathematicians are so fond of.  Physicists cannot afford to participate in such parlor tricks as these, however mathematically true they might be.^[2]

We will begin now, then, to examine how the mandalic coordinate approach stacks up against that of imaginary numbers and quaternions. The former are holistic and respective of the natural order; the latter are irresponsibly rational, simplistic and, in final analysis, wrong about how nature works.^[3] Ambitious endeavor indeed, but let’s give it a go.

We’ve already looked at how the standard geometric interpretation of imaginary numbers in context of the complex plane is based on rotations through continuous Euclidean space.  You can brush up on that aspect of the story here if necessary. The mandalic approach to mapping of space is more complicated and far more interesting.  It involves multidimensional placement of elements in a discrete space, which is to say a discontinuous space,  but one fully commensurate with both Euclidean and Cartesian 3-dimensional space. The holo-interactive manner in which these elements relate to one another leads to a  probabilistic mathematical design  which preserves commutative multiplication,  unlike quaternions which forsake it.

Transformations between these elements are based on inversion (reflection through a point) rather than rotation which cannot in any case reasonably apply to discrete spaces.  The spaces that quantum mechanics inhabits are decidedly discrete.  They cannot be accurately detailed using imaginary and complex numbers or quaternions.  To discern the various, myriad transitions which can occur among mandalic coordinates requires some patience. I think it cannot be accomplished overnight but at least in the post next up we can make a start.^[4]

(continuedhere)

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.

© 2016 Martin Hauser

Please note:  The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent).  This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added.  To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering.  To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 309-

Beyond the Enlightenment Rationalists:
From imaginary to probable numbers - III

(continued from here)

My objection to the imaginary dimension is not that we cannot see it.  Our senses cannot identify probable dimensions either, at least not in the visually compelling manner they can the three Cartesian dimensions. The question here is not whether imaginary numbers are mathematically true. How could they not be? The cards were stacked in their favor. They were defined in such a manner, – consistently and based on axioms long accepted valid, – that they are necessarily mathematically true. There’s a word for that sort of thing. –The word is  tautological.– No,  the decisive question is whether imaginary numbers apply to the real world; whether they are scientifically true, and whether physicists can truly rely on them to give empirically verifiable results with maps that accurately reproduce mechanisms actually used in nature.^[1]

The geometric interpretation of imaginary numbers was established as a belief system using the Cartesian line extending from  -1,0,0  through the origin  0,0,0 to 1,0,0  as the sole real axis left standing in the complex plane. In 1843,  William Rowan Hamilton introduced two additional axes in a quaternion coordinate system.  The new 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.

(continuedhere)

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.

[SOURCE]

A quaternion, on the other hand,  is a four-element vector composed of a single real element and three complex elements. It can be used to encode any rotation in a  3D coordinate system.  There are other ways to accomplish the same, but the quaternion approach offers some advantages over these.  For our purposes here what needs to be understood is that mandalic coordinates encode a hybrid 6D/3D discretized space. Quaternions are applicable only to continuous three-dimensional space.  Ultimately,  the two reside in different worlds and can’t be validly compared. The important point here is that each has its own appropriate domain of judicious application. Quaternions can be usefully and appropriately applied to rotations in ordinary three-dimensional space, but not to locations or changes of location in quantum space.  For description of such discrete spaces, mandalic coordinates are more appropriate, and their mechanism of action isn’t rotation but inversion (reflection through a point.) Only we’re not speaking here about inversion in Euclidean space, which is continuous, but in discrete space, a kind of quasi-Boolean space,  a higher-dimensional digital space  (grid or lattice space). In the case of an electron this would involve an instantaneous jump from one electron orbital to another.

[2] I think another laudatory feature of mandalic coordinates is the fact that they are based on a thought system that originated in human prehistory, the logic of the primal I Ching. The earliest strata of this monumental work are actually a compendium of combinatorics and a treatise on transformations,  unrivaled until modern times, one of the greatest intellectual achievements of humankind of any Age.  Yet its true significance is overlooked by most scholars, sinologists among them.  One of the very few intellectuals in the West who knew its true worth and spoke openly to the fact, likely at no small risk to his professional standing, was Carl Jung, the great 20th century psychologist and philosopher.

It is of relevance to note here that all the coordinate systems mentioned are, significantly,  belief systems of a sort.  The mandalic coordinate system  goes beyond the others though,  in that it is based on a still more extensive thought system, as the primal I Ching encompasses an entire cultural worldview.  The question of which,  if any,  of these coordinate systems actually applies to the natural order is one for science, particularly physics and chemistry, to resolve.

Meanwhile, it should be noted that neither the complex plane nor quaternions refer to any dimensions beyond the ordinary three, at least not in the manner of their current common usage.  They are simply alternative ways of viewing and manipulating the two- and three-dimensions described by Euclid and Descartes. In this sense they are little different from  polar 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.

Neo-Boolean - II: Logic Gates
Thinking Inside the Lines

(continued from here)

We have already looked briefly at three of the more important Boolean operators or logic gates:  AND, OR,andXOR.NOT just toggles  any two Boolean truth values  (true/false; on/off; yes/no).  Here we introduce two new logic gates which do not occur in Boolean algebra. Both play an important role in mandalic geometry though.

We’ll refer to the first of these new operators or logic gates as INV standing for  inversionorinvert.  This is similar to Boole’s NOT except that it produces toggling betweeen  yang/+ and yin/- instead of 1 and 0. Because it is based on binary arithmetic, Boole’s NOT has been thought of as referring to inversion also (as in ONorOFF). Although both ANDandINV act as toggling logic gates they have very different results in the greater scheme of things,  since nature has created a  prepotent disparity between a  -/+ toggle  and a  0/1 toggle  in basic parameters of geometry, spacetime, and being itself. This makes Boole’s AND just a statement of logical opposition, notinversion.

Recognition of this important difference is built into mandalic geometry structurally and functionally,  as it is also into Cartesian coordinate dynamics and the logic of the I Ching,  but lacking in  Boole’s symbolic logic. This is necessarily so, as there is no true negative domain in Boolean algebra.  The OFF state of electronics and computers, though it may sometimes be thought of in terms of a negative state, is in fact not. It relates to the  Western zero (0), not the  minus one  of the number line. Where Boolean algebra speaks of  NOT 1  it refers specifically to zero and only to zero. When mandalic geometry asserts  INV 1  it refers specifically to  -1  and only to  -1 . The inversion of yang then is yin and the inversion of yinisyang.^[1]  In the I Ching,  Taoist thought,  and mandalic geometry the two are not opposites but complements and, as such, interdependent.

The second added logic gate that will be introduced now is the REV operator standing for reversionorrevert. This operator produces no change in what it acts upon.  It is the multiplicative identity element (also called the neutral elementorunit element),  as INV is the inverse element. In ordinary algebra the inverse element is -1, while the identity element is 1. In mandalic geometry and the I Ching the counterparts are yinandyang, respectively. If Boolean algebra lacks a dedicated identity operator, it nonetheless has its Laws of Identity which accomplish much the same in a different way:

• A = A
• NOT A = NOT A

Again, Boolean algebra has no true correlate to the INV operator. There can be no  sign inversion formulation  as it lacks negatives entirely. Although Boolean algebra may have served analog and digital electronics and digital computers quite well for decades now,  it is incapable of doing the same for any quantum logic applications in the future, if only because it lacks a negative domain.^[2]  It offers up bits readily but qubits only with extreme difficulty and those it does are like tears shed by crocodiles while feeding.

(to be continued)

Image: Boolean Search Operators. [Source]

Notes

[1] Leibniz’s binary number system, on which Boole based his logic, escapes this criticism, as Leibniz uses 0 and 1 simply as notational symbols in a modular arithmetic and not as  contrasting functional elements in an algebraic context  of either the Boolean or ordinary kind.

In the field of computers and electronics,  Boolean refers to a data type that has two possible values representing true and false.  It is generally used in context to a deductive logical system known as Boolean Algebra. Binary in mathematics and computers, refers to a base 2 numerical notation. It consists of two values 0 and 1. The digits are combined using a place value structure to generate equivalent numerical values. Thus, both are based on the same underlying concept but used in context to different systems. [Source]

[2] Moreover,  I expect physics will soon enough discover that what it now calls antimatter  is in some sense and to some degree a necessary constituent of  ordinary matter.  I can already hear  the loudly objecting voices  declaring matter  and  antimatter  in contact  necessarily annihilate one another,  but that need not invalidate the thesis just proposed.  My supposition revolves around the meaning of “contact” at Planck scale and the light speed velocity at which subatomic particles are born, interact and decay only to be revived again in an eternal dance of creation and re-creation. Material particles exist in some kind of structural and functional  homeostasis,  not all that unlike the  anabolic  and catabolic mechanisms that by means of negative feedback maintain all entities of the biological persuasion in the  steady state  we understand as life. Physics has yet to  get a full grip  on  this  aspect of reality,  though moving ever closer with introduction of quarks and gluons to its menagerie of performing particles.

© 2016 Martin Hauser

Please note:  The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent).  This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added.  To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering.  To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 304-

Mandalic geometry, Cartesian coordinates and Boolean algebra: Relationships - I

(continued from here)

In attempting to understand the logic of the I Ching it is important to know the differences between ordinary algebra  and  Boolean algebra and how Boolean algebra is related to the binary number system.^[1]

In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted
1 and 0 respectively. Instead of elementary algebra where the values of the variables are  numbers,  and the  main operations  are  addition and multiplication,  the main
operations of Boolean algebra are the conjunctionand, denoted ∧, the disjunctionor, denoted ∨, and the negationnot, denoted ¬. It is thus a formalism for describing logical relations in the same way that ordinary algebra describes
numeric relations. [Wikipedia]

Whereas in elementary algebra expressions denote mainly numbers, in Boolean algebra they denote the truth values false and true. These values are represented with the bits (or binary digits), namely 0 and 1.  They do not behave like the integers  0 and 1,  for which
1 + 1 = 2, but may be identified with the elements of the two-element field GF(2), that is, integer arithmetic modulo 2,  for which 1 + 1 = 0.  Addition and multiplication then play the  Boolean roles  of  XOR  (exclusive-or)  and  AND  (conjunction)  respectively, with disjunction  x∨y  (inclusive-or)  definable as  x + y + xy. [Wikipedia]^[2]

Mandalic logic already occurs fully in the structure and manner of divinatory practice of the I Ching,  if some of it only implicitly.  Although mandalic geometry does not originate from either Boolean algebra or the Cartesian coordinate system but from the primal I Ching which predates them by millennia, it does combine and augment aspects of both of these conceptual systems. It extends Boole’s system of symbolic logic to include an additional logic value represented by the number -1.  This necessitates modification of some of Boole’s postulates and rules,  and increases their total number through introduction of some new ones.  The hexagrams or native six-dimensional mandalic coordinates of the I Ching are related to Cartesian triads composed of the numbers -1, 0, and 1,  making these two geometric systems  commensurate  by means of composite dimension,  a 6D/3D hybridization or mandalic coordination of structure and function (or space and time).^[3]

The introduction of composite dimension produces four distinct dimensional amplitudes  and  is solely responsible for the mandalic form. For anyone reading this who might be down on sacred geometry,  itself a subject which I respect and admire, let it be known that I am talking here about genuine mathematics and symbolic logic,  and my suspicion is that there is some genuine physics involved as well.

Kalachakra Mandala

The mandalic number system, then, is a quasi-modular number system, different from Leibniz’s binary number system which is fully modular.  Boole’s rule  1 AND 1 = 1  still holds true in mandalic logic.  However we must add to this the new logic rule that  -1 AND -1 = -1.  Individually the two rules are modular,  based on a clock arithmetic using a modulo-3 number system rather than Leibniz’s modulo-2 or binary number system, but with yet another added twist.

Together the two rules prescribe a compound system, one which is not singly modular but doubly modular.  The two components, yinandyang, are complementary and are inversely related to one another in this unified system.  This  logic organization  appears based on the figure 8 or sine wave and its negative,  allowing for periodicity, for recursive periods of interminably repeating duration,  and,  perhaps most importantly,  for wave interference,  of  constructive  and  destructive  varieties. These two geometric figures also engender an unexpected decussation of dimension not recognized by Western mathematics.  This is so because 1 AND -1 = 0 and  -1 AND 1 = 0.  The surprise here  is that  there are two distinct zeros: 0_a and 0_b.^[4] In two- or three-dimensional Cartesian terms there exists no difference between these two zeros.  However,  in terms of 6-dimensional aspects of mandalic geometry  and  the hexagrams of the I Ching, the two are clearly distinct structurally and functionally.^[5]

This arithmetic system is the basis of the logic encoded in the hexagrams of the I Ching. Each hexagram uniquely references a single 6- dimensional discretized point, of which there are 64 total. These 64 6- dimensional points of the mandalic cube are distributed among the 27 discretized points  of the ordinary 3-dimensional cube  through the compositing of dimensions  in such manner  that a mandala is formed which positions  1,  2,  4  or  8 hexagrams at each 3-dimensional point according to the   dimensional amplitude  of the particular point.  This necessarily creates a concurrent probability distribution of hexagrams through each of the three Cartesian dimensions.

TheI Chinguses a dual or composite three-valued logic system.  In place of truth values,  the variables used are yin,  yang  and the two in conjunction.  These fundamentally represent vector directions.  Yin is represented by -1, yang by 1, and their conjunction, using Cartesian or Western number terminology, by zero (0). This symbol does not occur natively in the I Ching though where the representation used is simply a combination of yin and yang symbols, most often in form of a bigram containing both  and  regarded as representing a composite dimension, namely 0_[1]  or  0_[2].^[6]

The two bigrams that satisfy the requirement are

young yang

for 0_[1]

and

young yin

for 0_[2].

Although mandalic logic is in Cartesian terms a 3-valued system, in native terms it is 4-valued.  It is not a simple modulo-3  or  modulo-4 number system, but two interrelated modulo-3 systems combined.  The best way to think about this geometric arrangement is possibly to view it as a single composite dimension having four distinct vector directions: a negative direction represented by mandalic composite yin (Cartesian -1); positive direction represented by mandalic composite yang (Cartesian 1); and two decussating relatively undifferentiated directions in some sort of equilibrium, represented by mandalic 0_[1] (composite yin/yang) and 0_[2] (composite yang/yin).  both of which  devolve  to  Cartesian 0  (balanced vector direction of the origin or center).^[7]

So we’ve seen that the number system used in the I Ching is not binary as Leibniz believed but instead doubly trinary with the two halves, in simplest terms,  inversely related and intertwined.  Still, it was an easy mistake to make because the notation used is binary.  We’ve seen too that all trigrams and hexagrams in the system can be rendered commensurate with the Cartesian coordinate system:  trigrams by simple transliteration, hexagrams by dimensional compositing. What, then, of George Boole and his eponymous logic?  How do they fit in the logic scheme of the I Ching? I’m glad you asked. Stay tuned to find out.

(continuedhere)

Images: Upper: TRANSFORMATION OF THE SYMBOL OF YIN (LINE split in two) AND YANG (STRAIGHT-LINE). BLEND: 4 bigrams, THEN 8 trigrams. (MORAN, E. ET AL. 2002: 77). Found here. Lower: Modified from an animation showing how the taijitu (yin-yang diagram) may be drawn using circles, then erasing half of each of the smaller circles. O'Dea at WikiCommons [CC BY-SA 3.0orGFDL],via Wikimedia Commons

Notes

[1] Boole’s algebra predated the modern developmentsinabstract algebra and  mathematical logic  but is seen as connected to the origins of both fields. Similarly to elementary algebra, the pure equational part of the theory can be formulated without regard to explicit values for the variables.

[2] If you are new to Boolean algebra these definitions may be confusing because in some ways they seem to fly in the face of ordinary algebra.  I’ll admit, I find them somewhat daunting.  Let me see if I can clarify the three examples given in this quote. Those of you more familiar with the language of Boolean algebra might kindly correct me in the event I err.  I’m growing more comfortable with being wrong at times.  And this is after all a work in progress.

• Boolean XOR (exclusive-or) allows that a statement of the form (x XOR y) is TRUE
  if either x or y is TRUE but FALSE if both are TRUE or if both are FALSE.  Since Boolean algebra uses binary numbers and represents  TRUE by 1,  FALSE by 0,  then
            for  x = TRUE,   y = TRUE    x + y = 1 + 1 = 0 ,    so FALSE
            for  x = FALSE,  y = FALSE   x + y = 0 + 0 = 0 ,  so FALSE
            for  x = TRUE,    y = FALSE   x + y = 1 + 0 = 1 ,   so TRUE
            for  x = FALSE,   y = TRUE    x + y = 0 + 1 = 1 ,   so TRUE

• Boolean AND (conjunction) allows that a statement of the form (x AND y) is TRUE
  only if both x is TRUE and y is TRUE. If either x or y is FALSE or both are FALSE
  then x AND y is FALSE. Here algebraic multiplication of binary 1s and 0s plays the
  role of Boolean AND. (Incidentally, binary multiplication works exactly the same
  way as algebraic multiplication. There’s a gift!)
            for  x = TRUE,    y = TRUE      xy  =  1(1) = 1,    so TRUE
            for  x = FALSE,   y = FALSE     xy = 0(0) = 0,   so FALSE
            for  x = TRUE,    y = FALSE      xy = 1(0) = 0 ,  so FALSE
            for  x = FALSE,    y = TRUE      xy = 0(1) = 0 ,  so FALSE

• Boolean OR (inclusive-or) is the truth-functional operator of (inclusive) disjunction,
  also known as alternation. The OR of a set of operands is true if and only if one or
  more of its operands is true. The logical connective that represents this operator is
  generally written as ∨ or +. As stated in the Wikipedia article logical disjunction x∨y
  (inclusive-or) is definable as x + y + xy [(x OR y) OR (x AND y)] as shown below.
  [Note: x AND y is often written xy in Boolean algebra. So watch out whichalgebra
  is being referred to, ordinary or Boolean. Are we confused yet?]
            for  x = TRUE,    y = TRUE      x + y = 1 , xy = 1 ,    so TRUE
            for  x = FALSE,   y = FALSE     x + y = 0 , xy = 0 ,   so FALSE
            for  x = TRUE,     y = FALSE     x + y = 1 , xy = 0 ,   so TRUE
            for  x = FALSE,    y = TRUE      x + y = 1 , xy = 0 ,   so TRUE
  

[3] Fundamentally, though,  the  coordinates of mandalic geometry  refer to vector directions alone, rather than to both vectors and scalars (or direction and magnitude) as do Cartesian coordinates. Yin specifies actually the entire domain of negative numbers rather than just the scalar value -1. Yang similarly refers to the entire domain of positive numbers rather than the scalar value 1 alone. Their conjunction  through the compositing of dimensions,  though represented by the symbol zero (0)  in the format commensurate with Cartesian coordinates,  refers actually to a  state or condition  not found in Western thought  outside of certain forms of mysticism  and other outsider philosophies like alchemy;  equilibration of forces in physics; equilibrium reactions in chemistry; and the kindred concept of homeostasis mechanisms of living organisms found in biology.

[4] This is to Westerners counterintuitive. Our customary logic and arithmetic allows for but a single zero. That two different zeros might exist concurrently or consecutively is - to our minds - irrational and we wrestle mightily with the idea. To complicate matters still more,  neither of these zeros is  conveniently  like our familiar Western zero.  So which should win out here?  Rationality or reality?  In fact,  the decision is not ours.  In the end nature decides.  Nature always decides. It stuffs the ballot box  and  casts the deciding vote much to our chagrin,  leaving us powerless to contradict what we may interpret as a whim. Our votes count for bupkis.

[5] This calls to mind also the Möbius strip which involves a twist that looks very much like a decussation to me.  The decussation or  twist in space  we are talking about here though has a sort of wormhole at its center that connects two contiguous dimensional amplitudes. I can’t say more about this just now. I need to think on it still. It seems a promising subject for reflection. (1,2,3)

[6] It needs to be pointed out here that in mandalic geometry, and similarly in the primal I Ching as well,  a bigram can be formed from any two related Lines of  hexagrams,  trigrams,  and tetragrams. The two Lines need not be (and often are not) adjacent to one another. I would think such versatility might well prove useful for modeling and mapping quantum states and interactions.

[7] Note that yin and yang in composite dimension can each take the absolute values 0, 1, and 2  but when yin has absolute value 2, yang has absolute value 0; when yang has absolute value 2,  yin has absolute value 0.  This inverse relation in fact is what makes the arrangement here a superimposed, actually interwoven, dual modulo-3 number system. It also makes the center points of mandalic lines,squares,  and cubes  more protean and less differentiated  than their vertices and elicits the different amplitudes of dimension.

The composite dimension value at the origin points(centers) of all of these geometric figures is  always  zero  in  Cartesian  terms  since the values of the differing Lines  in  the  two entangled 6-dimensional hexagrams  located here add to zero. But neither of these 6-dimensional entities is in its ground state at the center.  Both  have absolute value 1  at Cartesian 0.  Let me say that again: composite dimension values at the center or origin are zero in Cartesian terms but the values of both individual constituents are non-zero.Yin is in its ground state when yang is at its maximum and vice versa. At the center, since the two are equal and opposite they interfere destructively. This results in a composite zero ground state.

So from the perspective of  Cartesian coordinate dynamics, which is after all the customary perspective in our subjective lives,  we encounter only emptiness. But it is this very emptiness that opens to a new dimension. In the hybrid 6D/3D mandalic cube  only line centers and the cube center  have direct access through change of one dimension to face centers and only the face centers have a similar direct access through a single dimension to the cube center and edge centers. All coexist in an ongoing harmony of tensegrity. There is method to all this madness then.

© 2016 Martin Hauser

Please note:  The content and/or format of this post may not be in finalized form.  Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent).  This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added.  To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering.  To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 302-

Mandalic Line Segments,
Entanglement and Quantum Gravity
Part I

(continued from here)

We are going to consider once again now geometric line segments of mandalic geometry  and  their relation to Cartesian line segments and the Western number line. Yes,  this is sort of a detour from what I stated we would look at next. But this is not unrelated and lies at the very heart of mandalic geometry, and I’m not yet ready to address what I projected in the last remark of my previous post.

I keep returning to this subject because of its extreme importance. Beyond its significance to understanding the logic encoded in mandalic geometry and the I Ching, I believe it may also hold the key to quantum entanglement and quantum gravity.  Despite the fact that mandalic line segments are really fundamentally mental constructs,  a fiction of sorts, it is still important to understand how they are composed and how their components interact.  Though they may themselves be fictions,  the line segments and the points that compose them do in fact map a number of physical entities,  realities that may be related to quantum numbers and quantum particles and states.

When Descartes invented his coordinate system, with its points and line segments,  he based his system on the number line extended to two or three dimensions. In modeling it on the number line the space he described was imagined to bear a  necessary  one to one correspondence to the real numbers.^[1]  However this  1:1 mapping  of geometric space to the real numbers was a premise implicitly assumed by Descartes.  It was in fact axiomatic,^[2]  but apparently not stated as such.^[3]  As a result, the presumed relation has become a blind spot^[4] in Western thought,  never proved nor disproved, at least not at subatomic scales.^[5]

Neither mandalic geometry nor the primal I Ching make such an assumption. In place of Descartes’ 1:1 correspondence of geometric space and the numbers on the number line, we find a mandalic arrangement in which there are different categories of spatial location which can host one or more discrete numbers in a probabilistic manner.  This creates various dimensional amplitudes and a multidimensional waveform of component entities.^[6]

To my mind these characteristics of the mandalic coordinate system in combination with others described elsewhere make it more relevant to investigation and interpretation of many quantum phenomena which are as yet poorly understood than Cartesian coordinate dynamics may be and without need for recourse to imaginary numbers and complex plane.

(continuedhere)

Image: 6 steps of the Sierpinski carpet, animated. By KarocksOrkav (Own work) [CC BY-SA 3.0],via Wikimedia Commons

Notes

[1] Real numbers are numbers that can be found on the number line. This includes both the rational and irrational numbers.

[2] That is to say, taken for granted as self-evident.

[3] See Note [4] here.

[4] We have lived with this unproved premise so long that we no longer even question it,  or imagine that there might be an alternative which conforms better to reality at certain scales, notably subatomic scales.  The I Ching also seems to suggest  that a complete true description of complex relationships that involve a large number of dimensions,  including complex societal relationships,  requires more than a simple 1:1 correspondence between the notational symbols involved and the realities they represent.

[5] And from what I can see, no one seems to have much interest in proving or disproving this assumption.

[6] When speaking about hexagrams the number of dimensions involved is six as each Line of the hexagram encodes a value for a single distinct dimension in a 6-dimensional space.  In a hybrid 6D/3D compositing of dimensions though, two such Lines in relation reference a single Cartesian dimension in 2- or 3-space.  A concept not to be missed here is that  interactions of quantum particles  may well involve such  integration of dimension,  of dimensions  we are not even aware of beyond the unsettling fact  they upset the neat applecart of customary conceptual categories.

© 2016 Martin Hauser

Please note:  The content and/or format of this post may not be in finalized form.  Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent).  This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added.  To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering.  To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 301-

Quantum Naughts and Crosses Revisited - VIII
The Cube Sliced and Diced
Transliteration Series: Section F_H(n)

(continued from here)

We come now to the  Taoist/Cartesian  transliteration sections of the three-dimensional cube.^[1] The frontal F_H section seen below is the Cartesian xy-plane we’re all familiar with from the 2-dimensional version of the Cartesian coordinate system with the third Cartesian dimension (z) added to the labeling of points.  This gives us nine distinct Cartesian triad points: four vertices, four edge centers, and one face center.  For all of the points, the third Cartesian dimension (z) is constant in this slice,  and the vector value is positive (located toward the viewer with respect to the z=0 value of the z-axis or F_HE plane which we’ll be viewing in a future post.)

The diagram shown here relates changing and unchanging trigrams of the I Ching to corresponding Cartesian ordered triads. Descartes views each of his ordered triads as referring to a single point having substantive reality in Cartesian geometric space. The I Ching and mandalic geometry, on the other hand,  regard the trigrams as evanescent composite states of being in a spacetime which is ever-changing. They are relational elements in some ways analagous to the subatomic entities of particle physics.

Accordingly, it should be further understood each “point” here, though shown as a flat “square”,  has a third dimension implied, and is therefore actually a “cube”, only one face of which is seen.^[2]  Mandalic geometry considers the point a fictional device which actually refers to a common intersection of three or more planes in a three-dimensional context, or two or more lines in a two-dimensional context.  Moreover, mandalic geometry is a discretized geometry,  and the trigram must be considered as having a distributed domain of action. This is illustrated in all the Cartesian transliteration points by distributing eight copies of trigrams with appropriate changing and unchanging lines among eight vertex-analogues of each Cartesian point.

The key to labeling of points in this section^[3]  and  all those to follow can be found here.  Additionally,  by tradition,  adding an “x” to a yin line indicates it is a changing line and adding an “o” to a yang line indicates it is a changing line.  A changing yin line is considered an old yin line which is changing to a yang line;  a changing yang line,  an old yang line that is changing to a yinline.

Vector addition of two or more yinlines yields a yin line as result. Vector addition of two or more yang lines gives a yang line as the result. Vector addition of an unequal number of yin lines and yang lines yields as result that vector (yinoryang) in excess. Vector addition of an equal number of yin lines and yang lines gives as result Cartesian zero which, in  mandalic systematics  is to be considered a vector (direction)  rather than a scalar (magnitude).  This goes far in explaining how  the I Ching and Taoism managed without an explicit zero.

Thezero was implicit or understood without using a special symbol of designation.  Moreover,  it was conceived as representative of an order of reality  entirely different from  that distinguished by  the Western zero. It is,  however,  fully commensurate with  Cartesian coordinate dynamics. It is this alternative zero,  with its extraordinary capacities,  that provides access to potential dimensions  and to different amplitudes of dimension. This will be further elaborated in a future post where we will address how Boolean logic impacts what we’ve covered here.

For now simply note that the changing yin Line and changing yang Line  in the horizontal first dimension (x)  in each “point” shown in the middle column add to zero,  not the  zero of scalar magnitude  though, but the zero of vector equilibrium.

Section F_H(n)

In this section of the cube,  as in all frontal sections,  the third Line/dimension (z) never changes; the second Line/dimension (y) changes  only in columns,  as one progresses up or down;  the first Line/dimension changes only in the rows, progressing left or right. This is just a consequence of viewing  a two-dimensional Cartesian
xy-plane in context of a section of the three-dimensional Cartesian
xyz-cube. Although not the manner in which we are accustomed to viewing the plane,  it is nonetheless fully compatible with ordinary Cartesian coordinates.  It is simply an alternative perspective,  one more suited for analysis/demonstration of trigram relationships in a Cartesian setting.

(continuedhere)

Notes

[1] This should be viewed as a work in progress. I’m still feeling my way with this so the content and/or format may change in the future. What is demonstrated here does not yet take into account  the manner in which Boolean logic relates to the distribution of changing and unchanging trigrams nor does this series of cube sections include the all-important geometric method of composite dimension. As described,  this is simply a Taoist notation transliteration of Cartesian coordinate structure.  The meat and potatoes of the matter is yet to come.  Of particular note here, though,  is the fact that even at this early stage of translation to a version of mandalic geometry that can be considered comprehensive,  what is possibly best described as a decussationbetweenyinandyang lines is already evident at every Cartesian triad point containing a “Cartesian zero”.  Worth mentioning here, this will be a key feature addressed in future posts.

[2]Point,  square,  and cube,  have all been placed in quotation marks to indicate that what is being referred to here is actually a different category of objects or elements which should in some sense be understood as relating to fractals or fractal structure and of a different dimensionality entirely than are those ordinary geometric objects. The admittedly deficient terminology used here is necessitated by the fact that sufficiently appropriate vocabulary terms to describe the reality intended do not currently exist,  or if they do are not as yet known to me.  Since we are representing a Cartesian point (ordered triad) as a quasi-cubic structure here,  it must have  a near face (n) and a far face (f) in each section with respect to the viewer. The chart displayed details the near face (n) of Section F_H.

[3] This is the frontal section through the cube nearest a viewer. It is Descartes’ xy-plane with label of the third dimension (z) added so each point label shown is a Cartesian ordered triad rather than an ordered pair as textbooks generally show the plane. Why the difference?  Because the geometry texts are interested only in demonstrating the two-dimensional plane in isolation,  whereas we want to see it as it exists in the context of three or more dimensions. Cartesian triads are shown by convention as (x,y,z),  so the xy-plane  emerges from the first two coordinates of the points in this section, and all the z-coordinates seen here are positive (+1). The F_E plane has all of its x and y coordinates identical to those seen here but its z-coordinates are all negative (-1). The F_HE plane has all the x and y coordinates identical to those seen here but its z-coordinates are all zero (0).

© 2015 Martin Hauser

Please note:  The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent).  This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added.  To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering.  To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 296-

Beyond Boole - Part 1
Symbolic Logic for the 21st Century

Boolean Algebra:
Fundamental Operations

(continued from here)

Looking back on how we arrived at this stage of reconstruction of Western thought,  I see the difficulty arose in attempting to explain the “missing zero” of Taoism. Blame our troubles on Leibniz. It was he who introduced binary numbers to the West,  and made the fateful choice of using zero(0) instead of -1 to counter with +1.  Leibniz knew full well of the I Ching, but did not understand it well. He missed the point, seeing in it only a resemblance to his own newly devised system of numbers.

By Leibniz’s time negative numbers were firmly entrenched in the European mind.  Why did  Leibniz  ignore them completely?  In doing so he blazed a new trail that led eventually to the digital revolution of recent times. It also led to a dead end in the history of Western thought, one the West has not yet come fully face to face with. It will, though. Give it a few more years.^[1]

George Boole, the inventor of what we know today as Boolean logic or Boolean algebra, was one of the thinkers who followed in the footsteps of Leibniz, building on the trail he blazed.^[2]  When he came to devise his truth tables,  he also chose zero(0) as the counterpart to one(1).  This led to certain resounding successes.  And ultimately,  to certain failures  that introduced yet another layer to the  blind spot  of Western symbolic logic. Here we are, almost two centuries later,^[3] saddled with and hampered by the unfortunate fallout of that eventful decision still.^[4]

Most arguments in elementary algebra denote numbers. However, in Boolean algebra, they denote  truth values  falseandtrue.  Convention has decreed these values are represented with the  bits (or binary digits), namely 0 and 1.  They do not behave like the integers 0 and 1 though, for which 1 + 1 = 2,  but are identified with the elements of the  two-element field GF(2), that is, integer arithmetic modulo 2, for which 1 + 1 = 0. (1,2) This causes a substantial problem when we attempt correlation of Taoist logic and Boolean logic. As we will soon discover, Taoist logic is a hybrid logic that is based on both vector inversion and arithmetic modulo 2.  As such,  it ought prove relatable to both Cartesian coordinates and Boolean algebra, though it may necessitate “forcing a larger foot in a smaller glass slipper.”

Taoism chose ages ago to use ‘yin’ and 'yang’ as its logical symbols. Although this appears, at first, to be a binary system, like those of Leibniz and Boole, on closer inspection it proves not to be.  It is one of far greater logical complexity, alternatively binary or ternary with intermediate third element understood. This implied third element is able to bestow balance and equilibrium throughout all of the Taoist logical system.  This is where the 'missing zero’ of Taoism went.  Only it is a very different zero than the 'zero’ of Western thought.  It is a zero of infinite potential rather than one of absolute emptiness.  It is a  zero  of  continual beginnings and endings, not of finality. It is one of the things that make the I Ching totally unique in the history of human cognition.  All these hidden zeros are wormholes between dimensions and between different amplitudes of dimension.

So where does this all lead to, then? We’ve seen that the Taoist 'yin’ can readily be made commensurate with 'minus 1’ of Western arithmetic, the number line,  and  Cartesian coordinates.^[5]  But if it is to remain true to Taoist logic,  it cannot be made commensurate with the Western 'zero’. We’ve found the Taoist number system and geometry to be Cartesian-like but not Cartesian. Now we discover them to be Boolean-like, not Boolean. Sorry, Leibniz,  they are not so much as remotely like your binary system. You were far too quick to disesteem the unique qualities of the I Ching.^[6]

This all has far-reaching consequences for Western thought in general. Especially though, for symbolic logic, mathematics, and physics. More specifically for our purposes here it means that when we create our Taoist notation transliteration of Cartesian coordinates, we will need also to translate Boolean logic into terms compatible with Taoist thought, that is to say, from a two-value system based on '1s’ and '0s’ into a three-value system based on '1s’, ’-1s’, and the ever-elusive invisible balancing-act '0s’ of Taoism.^[7] We turn to that undertaking next.

(continuedhere)

Image: Fundamental operations of Boolean algebra.  Symbolic Logic, Boolean Algebra and the Design of Digital Systems. By the Technical Staff of Computer Control Company, Inc.  Other logical operations exist and are found useful by non-engineer logicians.  However, these can always be derived from the three shown. These three are most readily implementable by electronic means. The digital engineer, therefore,  is usually concerned only with these fundamental operations of conjunction, disjunction, and negation.

Notes

[1] It is at times like this that I am thankful I am not a member of Academia. Were I so, I could not afford, from a practical standpoint, to make claims such as this. Tenure notwithstanding.

[2] A knowledge of the binary number system is an important adjunct to an understanding of the fundamentals of Symbolic Logic.

[3] If we look back far enough in time, it was the introduction of “zero” as a number and a philosophical concept that led us down this tangled garden path, though the history of human thought is nothing if not interesting.

[4] Far out speculative thought here:  Were binary numbers and Boolean logic based on +1s and -1s instead of +1s and 0s,  might it not be possible to construct today a software-based quantum computer requiring no fancy juxtapositions and superpositions of subatomic particles?  Think on it for a while before dismissing the thought as irrational folly.

[5] More correctly expressed, it can be made commensurate with the domain of negative numbers, since it is a vector symbol, properly speaking, concerned only with direction, not magnitude.

[6] Unfortunately there is still little understanding of the true nature of the symbolic logic encoded in the I Ching, as exemplified by this quote:

The I Ching dates from the 9th century BC in China. The binary notation in the
I Ching is used to interpret its quaternary divination technique.

It is based on taoistic duality of yin and yang.Eight trigrams (Bagua) and a set of 64 hexagrams (“sixty-four” gua), analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou Dynasty of ancient China.

The contemporary scholar Shao Yong rearranged the hexagrams in a format that resembles modern binary numbers, although he did not intend his arrangement to be used mathematically. Viewing the least significant bit on top of single hexagrams in Shao Yong’s square and reading along rows either from bottom right to top left with solid lines as 0 and broken lines as 1 or from top left to bottom right with solid lines
as 1 and broken lines as 0 hexagrams can be interpreted as sequence from 0 to 63.

[Wikipedia]

It was this Shao Yong sequence of hexagrams (Before Heaven sequence) that Leibniz viewed six centuries after the Chinese scholar created it, so maybe he can be forgiven his error after all.

The more significant point here might be that an important  Neo-Confucian philosopher, cosmologist, poet, and historian of the 11th century either was no longer able to access the original logic and meaning of the I Ching or, at the very least, was hellbent on reinterpreting it in a manner contradictory to its original intent.  The latter is a distinct possibility,  as Neo-Confucianism was an attempt to create a more rationalist secular form of Confucianism by rejecting superstitious and mystical elements of  Taoism and Buddhism that had influenced Confucianism since the Han Dynasty (206 BC–220 AD).

[7] Taoist logic and mandalic geometry share some of the characteristics of both Cartesian coordinates and Boolean logic,  but not all of either.  Descartes’ system is indeed a ternary one when viewed in terms of vector direction rather than scalar magnitude. That fits with the requirements of Taoist logic.  It is, on the other hand, dimension-poor,  as Taoist logic and geometry require a full six independent dimensions for execution.  Boolean logic lacks the necessary third logical element -1, which causes inversion through a central point of mediation. But we shall see, it does bestow the ability to enter and exit a greater number of dimensional levels by means of its logical gates. Used together in an appropriate manner, these two can provide a key to understanding Taoist logic and geometry. Speculating even further, Taoist thought might provide a key to interpretation of quantum mechanics, the same quantum mechanics devised in the early twentieth century that no one can yet explain. Well,  I mean, actually,  Taoist thought in the formulation given it by mandalic geometry.  Why feign modesty, when this work will likely linger in near-total obscurity for the next hundred years gathering dust or whatever it is that pixels gather in darkness undisturbed.

© 2015 Martin Hauser

Please note:  The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent).  This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added.  To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering.  To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 294-

Beyond Descartes - Part 10
Taoism Meets Boolean Logic: Introduction

Logic gate symbols

(continued from here)

Before we can hope to comprehend Taoist arithmetic and geometry we need to take a short detour through Boolean logic. First and foremost, we need to see how Boolean logic^[1] relates to Cartesian coordinates. That will provide what may be the best foundation available for understanding the Taoist approach to mapping of spacetime and the methodology which mandalic geometry derived from it.^[2]

For Descartes, his coordinate system is one thing,  his coordinate geometry another.  For Taoism, the coordinate system is the geometry.^[3] Boolean logic helps to explain how the two perspectives are similar,  how different. Cartesian coordinates are static and passive. Taoist coordinates and the derivative mandalic coordinates are active and dynamic.  In brief, the latter are changeable and self-changeable, a feat carried out by means of a brand of Boolean logic intrinsic to the system. Although it is true that Descartes’ coordinates do encode much the same information,  that is not where their focus of interest lies. Accordingly they turn our own attention elsewhere and we overlook those inherent possibilities.^[4]

Descartes’ geometric system is one based on vectors, that is, on both  magnitude and direction.  But in the scheme of things,  the former has somehow eclipsed the preeminence of the latter in the Western hive mind.  The opposite is true of Taoist thought and of mandalic geometry. Direction is uniformly revered as primary and prepotent. Magnitude, or scale,  is viewed as secondary and subordinate.  This mindset allows the Boolean nuances inherent in the system to come to the fore, where they are more easily recognized and deployed.

From such small and seemingly insignificant differences ensue entirely disparate worldviews.

(continuedhere)

Notes

[1] George Boole’s monumental contribution to symbolic logic was published in 1854 but was viewed as only an interesting academic novelty until the second decade of the twentieth century,  when it was at last exhumed as a mathematical masterpiece by Whitehead and Russell in their Principia Mathematica.

[2] In Boolean logic (Boolean algebra) logical propositions are represented by algebraic equations in which  multiplication  and  addition  (and negation) are replaced with ‘and’ and 'or’ (and 'not’),  and where the numbers  '0’ and '1’ represent 'false’ and 'true’ respectively. Boolean logic has played a significant role in the development of computer programming and continues to do so.

[3] This is true also of mandalic geometry in its current formulation.

[4] This might be a proper place to proclaim that nature has little use for Descartes’ breed of coordinates,  finding them far too stagnant and limiting for her purposes. Fortuitously, she devised her own choice coordinate stock long before Descartes thought to invent his.

© 2015 Martin Hauser

Please note:  The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent).  This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added.  To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering.  To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 293-

Quantum Naughts and Crosses Revisited - VII
The Cube Sliced and Diced
Cartesian Series: Sections S_H, S_HE, S_E

(continued from here)

Below are the three sagittal sections of the Cartesian 3-cube. All three have nine distinct ordered triads,  one located at each discretized Cartesian spatial locus. S_H and S_E sections have four vertices, four edge centers and one face center as did the F_H,  F_E,  T_H, and T_E sections seen earlier. The S_HE section contains four edge centers and four face centers and also, as its central point, the single cube center,  as did the F_HE and T_HEsections.^[1]

These are all Cartesian yz-planes, seen in three-dimensional context at different x-values.  For S_H,  x = +1.  For S_HE,  x = 0.  For S_E,  x = -1.

The key to labeling of points in these sections and all those to follow can be found here.

Section S_H

Section S_HE

Section S_E

Having completed our survey of sections of the Cartesian 3- cube, we are now ready to view the Taoist transliteration equivalents.  Almost. But first …

(continuedhere)

Notes

[1] See here for comment regarding the spurious appearance of edge and face centers in section F_HE. That comment applies to the S_HE and the T_HE sections as well. The deceptive appearance can be described as being an initially misleading artifact of the sectioning methodology.

© 2015 Martin Hauser

Please note:  The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent).  This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added.  To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering.  To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 292-

Quantum Naughts and Crosses Revisited - VI
The Cube Sliced and Diced
Cartesian Series: Sections T_H, T_HE, T_E

(continued from here)

We see grouped together below the three transverse sections of the Cartesian 3-cube. All three have nine distinct ordered triads, one located at each discretized Cartesian spatial locus.  Both T_H and T_E sections have four vertices, four edge centers,  and one face center as did the F_H and F_E sections seen earlier. The T_HE section contains four edge centers and four face centers, and also, as its central point, the single cube center.^[1]  These are all Cartesian xz-planes, seen in three-dimensional context at different y-values.  For T_H,  y = +1.  For T_HE,  y = 0.  For T_E,  y = -1.

The key to labeling of points in these sections and all those to follow can be found here.

Section T_H

Section T_HE

Section T_E

In the next post we’ll look at the three sagittal sections of the Cartesian 3-cube.

(continuedhere)

Notes

[1] See here for comment regarding the spurious appearance of edge and face centers in section F_HE. That comment applies to the T_HE section as well and also to the S_HE section we will view in the next post.

© 2015 Martin Hauser

Please note:  The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent).  This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added.  To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering.  To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 291-

Quantum Naughts and Crosses Revisited - V
The Cube Sliced and Diced
Cartesian Series: Section F_E

(continued from here)

We see below the third and final frontal section of the 3-cube. The  F_E section is composed of nine distinct Cartesian ordered triads, with four vertices,  four edge centers,  and one face center like the F_H section seen earlier.  All x- and y-coordinates are identical to those in the F_H section and in identical relative position. All the z-coordinates here have a vector direction of -1 instead of +1. In other words, this is a Cartesian xy-plane placed in three-dimensional context with z value of -1.

The key to labeling of points in this section and all those to follow can be found here.

Section F_E

Next up, we begin a survey of transverse sections of the Cartesian 3- cube.

(continuedhere)

© 2015 Martin Hauser

Please note:  The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent).  This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added.  To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering.  To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 290-

Quantum Naughts and Crosses Revisited - IV
The Cube Sliced and Diced
Cartesian Series: Section F_HE

(continued from here)

Below we have the second of three frontal sections through the 3-cube, labeled with the Cartesian coordinates of each point. This “slice” is through a plane that lies between an identity face, which contains the trigram  HEAVEN,  and an inversion face with the trigram  EARTH.  As such it does not belong fully to either the one or the other,  but it shares some characteristics of both. It is a plane, then, of mediation.  Again we see here nine Cartesian ordered triads. Due to an artifact of the “slicing” procedure,  the four edge centers deceptively appear as though vertices, and the four face centers could be taken as edge centers. Make note that these appearances are illusory.  At the center of this section we have the origin point of the cube, Cartesian (0,0,0).^[1]

The key to labeling of points in this section^[2] and all those to follow can be found here.

Section F_HE

(continuedhere)

Notes

[1] It might be well to note here that the origin point of the coordinate system never appears in either an identity plane or an inversion plane of any of the three section types.  All of the planes in which it appears are mediation planes of three dimensions in the case of the Cartesian 3-cube,  or of six dimensions in the case of the hybrid mandalic 6D/3D hypercube.  This is likely the rationale for why in the  I Ching  a change involving passage through this central point  is referred to as  "crossing the Great Water.“  There must be more than coincidence in the fact that Western thought refers to this point as the "origin” and Taoist thought views it as the source and beginning of all things. It’s not that something important was lost in translation.  The two notions arose independently, from two very different worldviews. Somehow in the scheme of things, the West came to equate “origin” with  "zero"  whereas the East came to equate  "origin"  with “the beginning and end of all things.”  Taoism, in particular, sees in this a focus of both creation and dissolution. As we shall soon enough discover,  this alternative perspective leads to a different species of arithmetic,  one of great antiquity though long lost in the sands of time.  Mandalic geometry has unearthed it and will reveal it here, in this blog, for the first time in millennia.  As a teaser,  it involves a different treatment of what the West calls “zero”. It is an arithmetic more in line with Boolean logic.

[2] The 2-dimensional version of this section is obtained from the  x and y coordinates, which by convention are the first and second, respectively, in the Cartesian ordered triads seen here. So the only difference between this section and the F_Hsectionpreviously viewed is the fact that the z-coordinates here are all zero (0) instead of +1.  In our next section, F_E,  the x and y coordinates will again be as seen here but all z-coordinates will be -1.  I believe I detect a trend developing here.

© 2015 Martin Hauser

Please note:  The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent).  This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added.  To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering.  To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 289-

Quantum Naughts and Crosses Revisited - III
The Cube Sliced and Diced
Cartesian Series: Section F_H

(continued from here)

The first slice through the cube, shown below, the F_H section,  is the Cartesian xy-plane we’re all familiar with from the 2-dimensional version of the Cartesian coordinate system with the third Cartesian dimension (z) added to the labeling of points.  This gives us nine distinct Cartesian triad points: four vertices, four edge centers, and one face center.  For all of the points, the third Cartesian dimension (z) is constant in this slice,  and the vector value is positive (located toward the viewer with respect to the z=0 value of the z-axis or F_HE plane which we’ll be viewing in the next post.)

The key to labeling of points in this section^[1] and all those to follow can be found here.

Section F_H

(continuedhere)

Notes

[1] This is the frontal section through the cube nearest a viewer. It is Descartes’ xy-plane with label of the third dimension (z) added so each point label shown is a Cartesian ordered triad rather than an ordered pair as textbooks generally show the plane. Why the difference?  Because the geometry texts are interested only in demonstrating the two-dimensional plane in isolation,  whereas we want to see it as it exists in the context of three or more dimensions. Cartesian triads are shown by convention as (x,y,z),  so the xy-plane  emerges from the first two coordinates of the points in this section, and all the z-coordinates seen here are positive (+1). The F_E plane, which we’ll be viewing in the post after next, has all of its x and y coordinates identical to those seen here but its z-coordinates are all negative (-1).

© 2015 Martin Hauser

Please note:  The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent).  This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added.  To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering.  To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 288-

Beyond Descartes - Part 8:
A Good Convention Gone Bad,
An Opportunity Missed

Composite Dimension and
Amplitudes of Potentiality
Episode 2

(continued from here)

We cannot blame Descartes for imaginary numbers. It was he, after all,  who christened these numbers “imaginary” due to his disdain for them.  We can,  however,  fault him  for his lack of insight  into how his coordinate system could be extended to create a viable substitute to show that imaginary numbers and the complex plane were nonsensical and make them unnecessary. Alas, that was not to be. Certain powerful forces of history decreed that imaginary numbers were here to stay and we seem stuck with them still, nearly five centuries later.

Not all would agree that imaginary numbers are a bad convention. We should all,  however,  be able to agree that they are  a convention and nothing more. They were invented by humanity.^[1]  Mathematics may not have taken to them at first - but did eventually welcome them into its fold for better or worse. The real damage was done when physics did the same without first subjecting the mathematical concepts involved to the kind of scrutiny and empirical review it demands of its own theories.

Where is the proof that imaginary numbers and complex plane in fact apply to the real world and particularly to the subatomic realm?  It is lacking in the main, and though the geometric concepts have indeed been successfully applied to a number of branches of physics  and explanations of  a variety of physical phenomena,  the reconciliation is incomplete,  the fit an uncomfortable one, and too many mysteries remain unexplained.

The term imaginary unit refers to a solution to the equation  x² = -1. By convention, the solution is usually denoted i. As no real number exists with this property,  the imaginary number i extends the real numbers and creates an entirely new and different category of numbers.  And crucially, at this point an assumption is made,  a rather sweeping assumption.  It is assumed that the properties of addition and multiplication we’re familiar with - (closure, associativity, commutativity and distributivity) - continue to hold true for this new species of number, or I should say, for this newly derived artificial species of number.  That may fly in the ivory tower^[2]  of pure mathematics,  but it lacks the wings and propelling force required to maneuver effectively in the real world that physics investigates.  Still,  the complex plane,  generated by mathematically motivated minds,  was soon adopted by physicists the world over.^[3]

Mandalic geometry offers an alternative solution in the effective combination of  dimensional numbers,  composite dimension,  and plane of potentiality. We’ll take a close look at potential numbers first. Let’s see how they stack up against  the imaginary numbers,  how  and where  they differ. Distinctions between complex plane and potential plane are subtle but they make for a world - a universe, actually - of difference. When next we meet, kindly check all preconceptions at the door.  Entirely untrodden paths await.

(continuedhere)

Image: (lower left) Imaginary unit i in the complex or Cartesian plane. Real numbers lie on horizontal axis, imaginary numbers on the vertical axis.  By Loadmaster  (David R. Tribble), (Own work) [CC BY-SA 3.0orGFDL], via Wikimedia Commons; (lower right) A diagram of the complex plane. The imaginary numbers are on the vertical axis, the real numbers on the horizontal axis. By Oleg Alexandrov [GFDLorCC-BY-SA-3.0],via Wikimedia Commons

Notes

[1] Let those who suppose differently, who believe them to be an indelible part of nature itself, prove their case. Until they do, I will see fit to call such numbers manmade inventions.

[2] I use the term ivory tower without malice of any kind in this context, rather judiciously, because mathematics demands no more than internal consistency for its particular brand of truth. It is not much interested in examining its definitions and axioms to determine how they shape up against hard reality. Mathematicians leave that  "sordid work"  to physicists and philosophers, both of whom are more willing to dig in  the mire of nature,  seeking its actual relics.  Enthusiastically to persist in such a real world-oblivious manner as pure mathematicians do, I think, requires a very special type of mind, one I don’t fully understand myself.

[3] In some circles this would be considered no less than a monumental leap of faith, particularly in view of the many unproved assumptions made in creation of imaginary and complex numbers. This was, in fact,  the New Faith  promulgated by Descartes’ contemporaries, the rationalists of the Age of Reason,  to supplant the Old Faiths of Religion and Scholasticism.

© 2015 Martin Hauser

Please note:  The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent).  This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added.  To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering.  To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 284-

Beyond Descartes - Part 7

Composite Dimension and
Amplitudes of Potentiality
Episode 1

(continued from here)

Having frightened away all the cognitive wusses with my remark in that last post about the complexity of composite dimension and of the mandalic coordinate system  based on it,  I have a confession to make to those followers who remain. Although understanding the ideas involved requires a step back and viewing them from a different perspective alien to our Western modes of thought, composite dimension and the plane of potentiality are at once  more natural  and  far less complicated  than are imaginary numbers and the complex plane. Stay with me here. There is a light at the end of the tunnel growing ever brighter.

The 6D/3D mandalic cube is a hybrid structure having four levels of amplitude potentiality represented geometrically by 27 3D points which correspond to Cartesian points centered about Cartesian (0,0,0) and 64 6D points,  corresponding to the 64 hexagrams,  similarly centered and distributed among the 27 Cartesian points  in such a way  as to create a probability distribution through all three Cartesian dimensions,  that is with geometric progression of the number of hexagrams resident in the different amplitudes or orbitals. This gives rise to the mandalic form of the coordinate system. There are  four well-defined orbitals or shells  in this unique geometric arrangement of hexagrams and,  parenthetically, whatever it is they represent in physical terms.^[1]

We can conceptually abstract and decompose the 3D moiety of this concept entity, the part corresponding to Cartesian space. In doing so we identify a cube having a single center and eight vertices, all points by Euclidean/Cartesian reckoning, twelve edges (lines), each having an edge center (points), and six faces (planes), each having a center (point) equidistant from its four vertices. Each vertex point is shared equally by three faces or planes of the cube and each edge, by two adjacent faces or planes. We have  previously analyzed in detail  how the six planes of the 3D cube dovetail with one another and the repercussions involved. (See hereandhere.) One of the most important consequences we find is that each face center coordinates in a special way all four vertices of the face. This becomes particularly significant  in consideration of the composite dimension-derived hypercube faces of mandalic geometry.

The 6D moiety follows an analogous but more complex plan and has been formulated so as to be commensurate with the convention of the Cartesian coordinate system.  It also introduces measurement of a discretized time  to the coordinates,  thus rendering the geometry one of spacetime.  The hybrid 6D/3D configuration introduces probability as well through its bell curve/normal distribution (1, 2) of hexagrams; and also,  the two new directions,  manifestation (differentiation) and potentialization (dedifferentiation).^[2] These unfamiliar directions are unique to mandalic geometry and the I Ching upon which it is based.

In the lower diagram above, the figure on the right represents the skeletal structure of the hybrid 6D/3D coordinate system;  the figure on the left, the skeletal structure of the corresponding 3D Cartesian moiety. The  27 discretized points  of the cube on the left have become 64 points of the 6D hypercube on the right.  In the next post we will begin to flesh these two skeletons out.^[3] The end results are nothing short of amazing.

(continuedhere)

Notes

[1] With this remark I am avowing that mandalic geometry is intended not just as an abstract pure mathematical formulation,  but rather as a logical/geometrical mapping of energetic relationships that exist at some scale of subatomic physics, Planck scale or other. I maintain the possibility that this is so despite the obvious and unfortunate truth  that we cannot now ascertain just what it is the hexagrams represent, and may, in fact, never be able to.

[2] Manifestation/differentiation corresponds to the direction of divergence; potentialization/dedifferentiation, to the direction of convergence. The former is motion away from a center; the latter, motion toward a center. Convergenceanddivergence are the two directions found in every Taoist line that do not occur in Cartesian space, at least not explicitly as such.  There are functions in Cartesian geometry that converge toward zero as a limit. To reach zero in Cartesian space however is to become ineffective. That is quite different from gaining increased potential, potential which can then be used subsequently in new differentiations. (See also the series of posts beginning here.)  Both the terms differentiationanddedifferentiation  were  brazenly borrowed  from the field of biology,  while the designations manifestandunmanifest  have been shamelessly appropriated from Kantian philosophy, though similar concepts also occur in different terminology in deBroglie-Bohmian pilot-wave theoryasexplicitandimplicit.

[3] In the figure of the cube on the lower left above there is a single Cartesian triad (point) identifying each vertex (V),  edge center (E),  face center (F),  and cube center C.  In the right figure, the  hybrid 6D/3D hypercube  at each vertex has one resident hexagram identifying it,  two resident hexagrams at each edge center, four resident hexagrams identifying each face center, and eight resident hexagrams identifying the hypercube center. This brings the total of hexagrams to 64, the number found in the I Ching and the total possible number (2⁶ = 64). This geometric progression of hexagram distribution,  through three Cartesian dimensions constitutes the mandalic form. It is entirely the result of composite dimension.

© 2015 Martin Hauser

Please note:  The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent).  This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added.  To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering.  To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 283-

Beyond Descartes - Part 5

Reciprocation, Alternation, Decussation
Imaginary and Complex Numbers

(continued from here)

Previously in this blog a number of attempts have been made to explicate the Taoist number line and contrast it with the Western version of the same.  It is essential to do this and to do it flawlessly,  first because different systems of arithmetic result from the two, and secondly because the mandalic coordinate system is based on the former perspective while the Cartesian coordinate system is based on the latter.^[1]

What has been offered earlier has been accurate to a degree, a good first approximation. Here we intend to present a more definitive account of the Taoist number line,  describing both how it is similar to and how it differs from the  Western number line  used by Descartes in formation of his coordinate system.  This will inevitably transport us  well beyond that comfort zone offered by the more accessible three-dimensional cubic box that has heretofore engaged us.

Both Taoist and Western number lines observe directional locative division of their single dimension into two major partitions:  positive and negative for the West;  yinandyang for Taoism.^[2]  There the similarities essentially end.  From its earliest beginnings Taoism recognized a second directional divisioning in its number line, that of manifest/unmanifestorbeingandbecoming.^[3]  The West never did such.  As a result, some time later the West found it necessary to invent imaginary numbers.^[4][5]

Animaginary number is a complex number that can be written as a real numbermultiplied by theimaginary uniti, which is defined by its property i² = −1. [Wikipedia]

Descartes knew of these numbers but was not particularly fond of them.  It was he, in fact, who first used the term “imaginary” describing them in a derogatory sense. [Wikipedia]  The term “imaginary number” now just denotes a complex number with a real part equal to 0,  that is, a number of the form bi. A complex number where the real part is other than 0 is represented by the form a + bi.

In place of the complex plane, Taoism has (and always has had from time immemorial)  a plane of potentiality.  An explanation of this alternative plane was attempted earlier in this blog,  but it can likely be improved. This post has simply been a broad brushstrokes overview. In the following posts we will look more closely at the specifics involved.^[6]

(continuedhere)

Image (lower): A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram representing the complex plane. “Re” is the real axis, “Im” is the imaginary axis, and i is the imaginary unit which satisfies i² = −1. Wolfkeeper at English Wikipedia [GFDL (http://www.gnu.org/copyleft/fdl.html) or CC BY-SA 3.0 (http://creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons

Notes

[1] The arithmetic system derived from the Taoist number line can perhaps best be understood as a  noumenal  one. It applies to the world of ideas rather than to our phenomenal world of the physical senses, but it may also apply to the real world, that is, the real real world which we can never fully access.

Much of modern philosophy has generally been skeptical of the possibility of knowledge independent of the physical senses, and Immanuel Kant gave this point of view its canonical expression: that the noumenal world may exist, but it is completely unknowable to humans. In Kantian philosophy, the unknowable noumenon is often linked to the unknowable “thing-in-itself” (Ding an sich, which could also be rendered as “thing as such” or “thing per se”), although how to characterize the nature of the relationship is a question yet open to some controversy. [Wikipedia]

[2] From the perspective of physics this involves a division into two major quanta of charge, negative and positive, which like yinandyang can be either complementary or opposing.  Like forces repel one another and unlike attract. This is the basis of electromagnetism, one of four forces of nature recognized by modern physics. But it is likely also the basis, though not fully recognized as such, of the strong and weak nuclear forces, possibly of the force of gravity as well. I would suspect that to be the case. The significant differences among the forces  (or force fields, the term physics now prefers to use)  lie mainly, as we shall see, in intricate twistings and turnings through various dimensions or directions that negative and positive charges undergo in particle interactions.

[3] It is this additional axis of probabilistic directional location, along with composite dimensioning, both of which are unique to mandalic geometry, that make it a geometry of spacetime,  in contrast to Descartes’ geometry which, in and of itself, is one of space alone. The inherent spatiotemporal dynamism that is characteristic of  mandalic coordinates  makes them altogether more relevant for descriptions of particle interactions than Cartesian coordinates, which often demand complicated external mathematical mechanisms to sufficiently enliven them to play even a partial descriptive role, however inadequate.

[4] In addition to their use in mathematics, complex numbers, once thought to be  "fictitious" and useless,  have found practical applications in many fields, including chemistry, biology, electrical engineering, statistics, economics,  and, most importantly perhaps, physics..

[5] The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers. He called them “fictitious” during his attempts to find solutions to cubic equations in the 16th century.  At the time, such numbers were poorly understood,  consequently regarded by many as fictitious or useless as negative numbers and zero once were. Many other mathematicians were slow to adopt use of imaginary numbers, including Descartes, who referred to them in his La Géométrie, in which he introduced the term imaginary,  that was intended to be derogatory. Imaginary numbers were not widely accepted until the work of Leonhard Euler (1707–1783) and Carl Friedrich Gauss (1777–1855).  Geometric interpretation of  complex numbers as points in a complex plane  was first stated by mathematician and cartographer Caspar Wessel in 1799. [Wikipedia]

[6] What I have called here the plane of potentiality occurs only implicitly in the Taoist I Ching but is fully developed in mandalic geometry. It may be related to  bicomplex numbers  or tessarines in abstract algebra, the existence of which I only just discovered. Unlike the quaternions first described by Hamilton in 1843, which extended the complex plane to three dimensions, but unfortunately are not commutative,  tesserines or bicomplex numbers  are hypercomplex numbers in a commutative,  associative  algebra over real numbers,  with two imaginary units (designated i and k). Reading further, I find the following fascinating remark,

The tessarines are now best known for their subalgebra of real tessarines t = w + y j, also called split-complex numbers, which express the parametrization of the unit hyperbola. [Wikipedia]

The rectangular hyperbola x2-y2 and its conjugate, having the same asymptotes. The Unit Hyperbola is blue, its conjugate is green, and the asymptotes are red. By Own work (Based on File:Drini-conjugatehyperbolas.png) [CC BY-SA 2.5],via Wikimedia Commons

Note to self:  Also investigate Cayley–Dickson constructionandzero divisor. Remember,  this is a work still in progress,  and if a  bona fide mathematician  believes division by zero is possible in some circumstances,  (as is avowed by mandalic geometry), I want to find out more about it.

© 2015 Martin Hauser

Please note:  The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent).  This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added.  To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering.  To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 281-

Beyond Descartes - Part 4
Directional Locatives

(continued from here)

Descartes derives his directional locatives from considerations of human anatomy, as does most of Western culture. The descriptive terms generally used for orientation purposes include left/right;up/down; and forward/backward.^[1] The first two sets have been extended also to refer to the cardinal directions, North/South and East/West.

To the degree that they conform to Cartesian coordinates, mandalic coordinates adhere to this schema as well.  However, mandalic geometry and the Taoist I Ching upon which it is largely based constitute a system of combinatorial relationships that is rooted mainly in  radial symmetry rather than bilateral symmetry. For mandalic coordinates, the principal directional locatives can be characterized as  divergentandconvergent, and the principal movements or changes in position, as centrifugalandcentripetal.^[2]

One of the important consequences of this alternative geometric perspective is that the frame of reference as well as the complex pattern produced are more integrative than in the method of Descartes. Looked at another way, Descartes is most enamored by specification of location of individual points whereas mandalic geometry is more concerned with relationships of parts - and the overall unification of the entire complex holistic system.^[3]

From this one seemingly small difference an enormous disparity grows in a manner reminiscent of chaos theory.^[4] Cartesian coordinates and mandalic coordinates can be made commensurate, but remain after all two exclusive systems of spatial awareness,  leading to very disparate results arising out of what seem small initial differences.^[5]

(continuedhere)

Image (bottom): Animation of a double compound pendulum showing chaotic behaviour. By Catslash (Own work). [Public domain], via Wikimedia Commons.^[6]

Notes

[1] Such terminology is of little use, despite its biological origins, to an amoeba or octopus,  not to mention those  extraterrestrials  who have been blessed with a second set of eyes at the back of their heads. (We wuz cheated.)

[2] To be more correct, the radial symmetry involved is of a special type. It is not simple planar radial symmetry, nor even the three-dimensional symmetry of a cube and its circumscribed and inscribed spheres. It is all of those but also the symmetry involved in all the different faces of a six-dimensional hypercube and the many relationships among them.

[3] To be fair, Descartes eventually gets around to relating his points in a systematic whole we now know as analytic geometry (1,2).  But as great an achievement though it might be,  Cartesian geometry  lacks the overarching cosmographical implications which characterize mandalic geometry and the I Ching. Descartes’ system is purposed differently, arising as it does out of a very different world view. To paraphrase George Orwell,

“All geometries are sacred, but some geometries are more sacred than others.”

[4] Chaos theory was summarizedbyEdward Lorenzas:

“When the present determines the future, but the approximate present does not approximately determine the future.”

[5] An example of one unique result of mandalic coordination of space is the generation of a geometric/logical probability wave of all combinatorial elements that occur in the 6D/3D hybrid composite dimension specification of the system. I envision this as offering a possible model at least,  if not an actual explanation, of the  probabilistic nature  of quantum mechanics.  Extrapolating this thought to its uttermost conclusion, it is not entirely inconceivable, to my mind at least, that probability itself might be the result of composite dimensioning. (And for such a brash remark I would almost surely be excommunicated from the fold were I but a member.)

[6] Starting the pendulum from a slightly different initial condition would result in a completely different trajectory.  The double rod pendulum is one of the simplest dynamical systems that has chaotic solutions. [Wikipedia]

© 2015 Martin Hauser

Please note:  The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent).  This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added.  To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering.  To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 280-

Beyond the Enlightenment Rationalists:
From imaginary to probable numbers - VI

(continued from here)

“O Oysters, come and walk with us!” The Walrus did beseech. “A pleasant walk, a pleasant talk, Along the briny beach: We cannot do with more than four, To give a hand to each.”

* * *

“The time has come,” the Walrus said, “To talk of many things: Of shoes–and ships–and sealing-wax– Of cabbages–and kings– And why the sea is boiling hot– And whether pigs have wings.”

-Lewis Carroll, The Walrus and the Carpenter

In this segment, probable numbers will be shown to grow out of a natural context inherently rather than through geometric second thought as transpired  in the history of Western thought  with imaginary numbers and complex plane.  To continue  with development of probable numbers it will be necessary to leave behind,  for the time being,  all preoccupation with imaginary numbers and complex plane.  It will also be necessary  to depart from our comfort zone of Cartesian spatial coordinate axioms and orientation.

Probable coordinates do not negate validity of Cartesian coordinates but they do relegate them to the status of a special case.  In the probable coordinate system the three-dimensional coordinate system of Descartes maps only one eighth of the totality. This means then, that the Cartesian two-dimensional coordinate plane furnishes just one quarter of the total number of  corresponding probable coordinate mappings  projected to a two-dimensional space.^[1]  It suggests also that  Cartesian localization  in 2-space or 3-space is just a small part of the whole story regarding actual spatial and temporal locality and their accompanying physical capacities, say for instance of momentum or mass, but actually encompassing a host of other competencies as well.

Although this might seem strange it is a good thing. Why is it a good thing?  First, because nature, as a self-sustaining reality, cannot favor any one coordinate scheme but must encompass all possible - if it is to realize any.  Second,  because both the Schrödinger 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.

© 2016 Martin Hauser

Please note:  The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent).  This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added.  To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering.  To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 311-

Neo-Boolean - II: Logic Gates
Thinking Inside the Lines

(continued from here)

We have already looked briefly at three of the more important Boolean operators or logic gates:  AND, OR,andXOR.NOT just toggles  any two Boolean truth values  (true/false; on/off; yes/no).  Here we introduce two new logic gates which do not occur in Boolean algebra. Both play an important role in mandalic geometry though.

We’ll refer to the first of these new operators or logic gates as INV standing for  inversionorinvert.  This is similar to Boole’s NOT except that it produces toggling betweeen  yang/+ and yin/- instead of 1 and 0. Because it is based on binary arithmetic, Boole’s NOT has been thought of as referring to inversion also (as in ONorOFF). Although both ANDandINV act as toggling logic gates they have very different results in the greater scheme of things,  since nature has created a  prepotent disparity between a  -/+ toggle  and a  0/1 toggle  in basic parameters of geometry, spacetime, and being itself. This makes Boole’s AND just a statement of logical opposition, notinversion.

Recognition of this important difference is built into mandalic geometry structurally and functionally,  as it is also into Cartesian coordinate dynamics and the logic of the I Ching,  but lacking in  Boole’s symbolic logic. This is necessarily so, as there is no true negative domain in Boolean algebra.  The OFF state of electronics and computers, though it may sometimes be thought of in terms of a negative state, is in fact not. It relates to the  Western zero (0), not the  minus one  of the number line. Where Boolean algebra speaks of  NOT 1  it refers specifically to zero and only to zero. When mandalic geometry asserts  INV 1  it refers specifically to  -1  and only to  -1 . The inversion of yang then is yin and the inversion of yinisyang.^[1]  In the I Ching,  Taoist thought,  and mandalic geometry the two are not opposites but complements and, as such, interdependent.

The second added logic gate that will be introduced now is the REV operator standing for reversionorrevert. This operator produces no change in what it acts upon.  It is the multiplicative identity element (also called the neutral elementorunit element),  as INV is the inverse element. In ordinary algebra the inverse element is -1, while the identity element is 1. In mandalic geometry and the I Ching the counterparts are yinandyang, respectively. If Boolean algebra lacks a dedicated identity operator, it nonetheless has its Laws of Identity which accomplish much the same in a different way:

• A = A
• NOT A = NOT A

Again, Boolean algebra has no true correlate to the INV operator. There can be no  sign inversion formulation  as it lacks negatives entirely. Although Boolean algebra may have served analog and digital electronics and digital computers quite well for decades now,  it is incapable of doing the same for any quantum logic applications in the future, if only because it lacks a negative domain.^[2]  It offers up bits readily but qubits only with extreme difficulty and those it does are like tears shed by crocodiles while feeding.

(to be continued)

Image: Boolean Search Operators. [Source]

Notes

[1] Leibniz’s binary number system, on which Boole based his logic, escapes this criticism, as Leibniz uses 0 and 1 simply as notational symbols in a modular arithmetic and not as  contrasting functional elements in an algebraic context  of either the Boolean or ordinary kind.

In the field of computers and electronics,  Boolean refers to a data type that has two possible values representing true and false.  It is generally used in context to a deductive logical system known as Boolean Algebra. Binary in mathematics and computers, refers to a base 2 numerical notation. It consists of two values 0 and 1. The digits are combined using a place value structure to generate equivalent numerical values. Thus, both are based on the same underlying concept but used in context to different systems. [Source]

[2] Moreover,  I expect physics will soon enough discover that what it now calls antimatter  is in some sense and to some degree a necessary constituent of  ordinary matter.  I can already hear  the loudly objecting voices  declaring matter and  antimatter  in contact  necessarily annihilate one another,  but that need not invalidate the thesis just proposed.  My supposition revolves around the meaning of “contact” at Planck scale and the light speed velocity at which subatomic particles are born, interact and decay only to be revived again in an eternal dance of creation and re-creation. Material particles exist in some kind of structural and functional  homeostasis,  not all that unlike the  anabolic  and catabolic mechanisms that by means of negative feedback maintain all entities of the biological persuasion in the  steady state  we understand as life. Physics has yet to  get a full grip  on  this  aspect of reality,  though moving ever closer with introduction of quarks and gluons to its menagerie of performing particles.

© 2016 Martin Hauser

Please note:  The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent).  This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added.  To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering.  To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 304-