Rolling dimensions

So apparently tumblr has changed the dashboard image dimensions from 500 width to 540 width. This particularly affects me because I generally use a mspaint/oekaki style of drawing which usually looks awful when resized. This would be fine except profiles are still set to 500 so if a 540 width image will look crisp and sharp on the dash, but blurry and shitty on my profile page. It’s maddening. Is there anyway I can fix this? If I could just go to the edit html page and copy/paste something to change the max image width on my profile that would be just swell.

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

(continued from here)

The four Cartesian quadrants provide the two-dimensional analogue of the number line and its graphic representation in Cartesian coordinate space.  This is the true native habitat of the square and, by implication, of square root.  Because  Enlightenment mathematicians  found fit to define square root in a different context inadvertently  -that of the number line- we will find it necessary to devise a different name for what ought rightly to have been called square root,  but wasn’t.  I propose that we retain the existent definition of tradition and refer to the new relationship between opposite numbers in the square,  that is to say,  opposite vertices through two dimensions or antipodal numbers, as contra-square root.^[1]

Modified from image found here.

Given this fresh context - one of greater dimension than the number line - it soon becomes clear with little effort that a unit number^[2]ofany dimension multiplied by itself gives as result the identity element of that express dimension. For the native two-dimensional context of the square the identity element is OLD YANG,  the bigram composed of two stacked yang (+) Lines,  which corresponds to yang (+1),  the identity element in the  one-dimensional context  of the number line. In a three-dimensional context,  the identity element is the trigram HEAVEN which is composed of three stacked yang (+) Lines.  The crucial idea here is that the identity element differs for each dimensional context,  and whatever that context might be,  it produces no change when in the operation of multiplication it acts as operator on any operand within the stated dimension.^[3]

As a corollary it can be stated that any number in any dimension n composed of  any combination  of  yang Lines (+1) and yin Lines (-1) if multiplied by itself (i.e., squared) produces the identity element for that dimension.  In concrete terms this means, for example, that any bigram multiplied by itself equals the bigram OLD YANG; any of eight trigrams multiplied by itself  equals the trigram HEAVEN;  and  any of the sixty-four hexagrams multiplied by itself  equals the hexagram HEAVEN; etc. (valid for any and all dimensions without exception). Consequently, the number of roots the identity element has in any dimension n is equal to the number 2ⁿ, these all being real roots in that particular dimension.

Similar contextual analysis would show that the inversion element of any dimension n  has  2ⁿ roots of the kind we have agreed to refer to as contra-square roots in deference to the Mathematics Establishment.^[4]

That leads us to the possibly startling conclusion that in every dimension n  there is an  inversion element  that has the same number of roots as the identity elementandall of them are real roots.  For two dimensions the two pairs that satisfy the requirement are bigram pairs

For one dimension there is only a single pair that satisfies. That is (surprise, surprise)  yin(-1)/yang (+1).  What it comes down to is
this:

If we are going to continue to insist on referring to square root
in terms of the one-dimensional number line, then

• +1 has two real roots of the traditional variety, +1 and -1
• -1 has two real roots of the newly defined contra variety,
  +1/-1 and -1/+1

So where do imaginary numbers and quaternions fit in all this? The short answer is they don’t.  Imaginary numbers entered the annals of human thought through error.  There was a pivotal moment^[5]  in the history of mathematics and science, an opportunity to see that there are in every dimension two different kinds of roots - - - what has been called square root and what we are calling contra-square roots.  Enlightenment mathematicians and philosophers  essentially allowed the opportunity to slip through their fingers unnoticed.^[6]

Descartes at least saw through the veil.  He called the whole matter of imaginary numbers ‘preposterous’.  It seems his venerable opinion was overruled though. Isaac Newton had his say in the matter too. He claimed that roots of imaginary numbers “had to occur in pairs.” And yet another great mathematician, philosopher opined.  Gottfried Wilhelm Leibniz,  in 1702 characterized √−1 as  “that amphibian between being and non-being which we call the imaginary root of negative unity.” Had he but preserved such augury conspicuously in mind he might have elaborated the concept of probable numbers in the 18th century.  If only he had truly understood the I Ching,  instead of dismissing it as a primitive articulation of his own binary number system.

(continuedhere)

Image: The four quadrants of the Cartesian plane. By convention the quadrants are numbered in a counterclockwise direction.  It is as though two number lines were placed together, one going left-right, and the other going up-down to provide context for the two-dimensional plane. Sourced from Math Is Fun.

Notes

[1] My preference might be for square root to be redefined from the bottom up, but I don’t see that happening in our lifetimes. Then too this way could be better.

[2] By the term unit number,  I intend any number of a given dimension that consists entirely of variant elements of the number one (1) in either its positive or negative manifestation.  Stated differently,  these are vectors having various different directions within the dimension,  but all of scalar value -1 (yin) or +1 (yang). All emblems of I Ching symbolic logic satisfy this requirement. These include the Line, bigram, trigram, tetragram, and hexagram.  In any dimension n there exist 2ⁿ such emblems.  In sum, for our purposes here, a unit number is any of the set of numbers, within any dimension n, which when self-multiplied (squared) produces the multiplicative identity of that dimension which is itself, of course, a member of the set.

ADDENDUM (01 MAY 2016): I’ve since learned that mathematics has a much simpler way of describing this. It calls all these unit vectors. Simple, yes?

[3] I think it fair to presume that this might well have physical correlates in terms of quantum mechanical states or numbers. Here’s a thought: why would it be necessary that all subatomic particles exist in the same dimension at all times given that they have a playing field of multiple dimensions, - some of them near certainly beyond the three with which we are familiar? And why would it not be possible for two different particles to be stable and unchanging in their different dimensions,  yet become reactive and interact with one another when both enter the same dimension or same amplitude of dimension?

[4] Since in any contra-pair (antipodal opposites) of any dimension, either member of the pair must be regarded  once as operator  and  once as operand. So for the two-dimensional square, for example, there are two antipodal pairs (diagonals) and either vertex of each can be either operator or operand.  So in this case, 2 x 2 = 4.  For trigrams there are four antipodal pairs, and 2 x 4 = 8. For hexagrams there are thirty-two antipodal pairs and 2 x 32 = 64. In general, for any dimension n there are 2 x 2ⁿ/2 = 2ⁿ antipodal pairs or contra-roots.

[5] Actually lasting several centuries, from about the 16th to the 19th century. Long enough,  assuredly,  for the error  to have been  discovered and corrected. Instead,  the 20th century dawned with error still in place,  and physicists eager to explain the newly discovered bewildering quantum phenomena compounded the error  by latching onto  √−1 and quaternions  to assuage their confusion and discomfiture.  This probably took place in the early days of quantum mechanics when the Bohr model of the atom still featured electrons as traveling in circular orbits around the nucleus or soon thereafter, visions of minuscule solar systems still fresh in the mind. At that time rotations detailed by imaginary numbers and quaternions may have still made some sense. Such are the vagaries of history.

[6] I think an important point to consider is that imaginary and complex numbers were, -to mathematicians and physicists alike,- new toys of a sort that  enabled them  to  accomplish certain things  they could not otherwise. They were basically tools of empowerment which allowed manipulation of numbers and points on a graph more easily or conveniently.  They provided
their controllers a longed for power over symbols, if not over the real world itself. In the modern world ever more of what we humans do and want to do involves manipulation of symbols. Herein,  I think,  lies the rationale for our continued fascination with and dependence on these tools of the trade. They don’t need to actually apply to the world of nature,  the noumenal world,  so long as they satisfy human desire for domination  over the world of symbols it has created for itself and in which it increasingly dwells, to a considerable degree apart from the natural world’s sometimes seemingly too harsh laws.

© 2016 Martin Hauser

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

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

-Page 310-

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.

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

Imaginary numbers arose in the history of mathematics as a result of misunderstanding the dimensional character of numbers.  There was a failure to acknowledge that numbers exist in a context of dimension. This has earlier been addressed at length.^[1]  Simply put, numbers exist always in a particular dimensional context.  Square numbers pertain to a context of two dimensions and therefore to a plane,  not a line.  Square roots then ought justly reference a two-dimensional geometrical context rather than the linear one mathematics has maintained ever since mathematicians of the Age of Enlightenment decreed it so.  Square roots contrary to the way mathematics would have it can neither exist in nor be found in any single line segment,  because they do not originate in the number line but in the two-dimensional square.

Algebra, not geometry, provided the breeding ground for imaginary numbers.  They were given a geometric interpretation as an afterthought only, long after the fact of their invention. Rationalist algebraists, feeling compelled to give meaning to equations of the form b² = -4 came up with the fantastic notion of imaginary numbers. Only indirectly did these grow out of nature, by way of minds of men obsessed with reason.^[2]

Descartes knew of the recently introduced square roots of negative numbers. He thought them preposterous and was first to refer to the new numbers by the mocking name imaginary, a label which stuck and which continues to inform posterity of the exact manner in which he viewed the oddities.  It is one of the ironies of history that when at last a geometrical interpretation of square root of negative numbers was offered it involved swallowing up Descartes’ own y-axis. Poetic justice? Or ultimate folly?

Had the essential dimensional nature of numbers been recognized there would have been no need to inquire what the square root of -1 was. It would have been clear that there was no square root of -1 nor any need for such as +1 also has no square root.  As linear numbers,  neither -1 nor +1 can legitimately be said to have a square root.  Both, though, have two-dimensional analogues and these do have square roots, not recognized as such unfortunately by the mathematics hegemony.^[3]

In the next post we will look at a comparison between imaginary numbers,  which were formulated in accordance with this misconstrual about how numbers relate to dimensions,  and probable numbers which grow organically out of a consideration of how numbers and dimensions actually relate to one another in nature.^[4]  The first of these approaches can be thought of as rational planning by a central authority; the second, as the holistic manner in which nature attends to everything, all at once, and without rational forethought.

(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] See the series of about nine posts that begins here.

[2] The Rationalists missed here a golden opportunity to relate number and dimension by defining square root much too narrowly. They seem to have been so mesmerized by their algebraic equations that they failed to pursue the search into deeper significance pertaining to essential linkages between dimension and number that intuition and imagination might have bestowed.

[3] As Shakespeare correctly pointed out, a rose by any name would smell as sweet. Plus one times plus one certainly equals plus one but that has nothing to do with actual square root really, just with algebraic linear multiplication.  Note has often been made in these pages of the difference between mathematical truth and scientific truth. Whereas mathematics demands only adherence to its axioms and consistency,  science requires empirical proof.  Mathematics defined square root in a certain manner centuries ago, and has since been devoutly consistent in its adherence to that definition.  In so doing it has preserved a cherished doctrine of mathematical truth, as though in formaldehyde.  It has also for many centuries contrived to be consistently scientifically incorrect.  The problem lies in the fact it has converted physicists and near everyone else to its own insular worldview.

[4] For an early discussion of the probable plane, potential dimensions, and probable numbers see here.

Beyond Taoism - Part 5
A Vector-based Probabilistic
Number System
Part II

(continued from here)

Taoism and the primordial I Chingare in agreement that temporal changes have two different aspects: sequent and cyclic.  Western thought in general follows suit. The I Ching differs from the other two in asserting that  the direction of change - for both sequent and cyclic change - is fully reversible,  with the proviso  that sufficiently small units of measurement are involved.^[1]  The probability that reversal can be achieved  diminishes proportionately to the magnitude of change that has taken place.^[2]

Taoist appropriation of bigrams and trigrams of the I Ching to model such phenomena as change of seasons and phases of the moon  is plausible if not quite legitimate. The natural phenomena so modeled are macroscopic and vary continuouslyandinexorably throughout an ever-repeating cyclic spectrum. And there’s the rub.

As they occur and function in the I Ching bigrams and trigrams are dicontinuous discrete elements,  formed by other similarly discontinuous discretized entities,  and they follow evolutionary courses which are most often nonrepetitive. So the Taoist usage is misleading at best, annihilative at worst. Unfortunately, as the I Ching itself evolved through centuries of commentaries and reinterpretations,  it became  ever more contaminated and tainted by these Taoist corruptions of meaning, at the same time that it was being inundated by  Confucian sociological and ethical reworkings.  What we have today is an amalgam, the various parts of which do not sit well with one another.^[3]

Though it may in part be hyperbole to prove a point,  the stark difference between the two approaches,  that of Taoism and that of the I Ching, is epitomized by comparison of the Taoist diagram of the cycle of seasons with diagrams at the top and bottom of the page,  which are based on  the  number,  logic,  and coordinate systems of   The Book of Changes.^[4] The increased complexity of the latter diagrams should not prove a stumbling block, as they can be readily understood in time with focus and attention to detail.  The  important take-away  for now is that in the I Ching bigrams  exist within a larger dimensional context  than the Taoist diagram avows,  and this context makes all their interactions more variable,  conditional,  and complex. As well, the same can be said of trigrams and hexagrams.

One of the more important aspects of these differences has to do with the notion of equipotentiality.  As bigrams and trigrams function within  higher dimensional contexts  in the  I Ching,  this introduces a possibility of multiple alternative paths of movement and directions of change.  Put another way,  primordial I Ching logic encompasses many more  degrees of freedom  than does the logic of Taoism.^[5]  There is no one direction or path  invariably decreed or favored.  An all-important element of conditionality prevails.  And that might be the origin of what quantum mechanics has interpreted as indeterminism or chance.

Next up, a closer look at equipotentiality and its further implications.

Section F_H(n)^[6]

(continuedhere)

Notes

[1] There are exceptions. Taoist alchemy describes existence of certain changes that admit reversibility under special circumstances.  Other than the Second Law of Thermodynamics (which is macroscopic in origin,  not result of any internally irreversible microscopic properties of the bodies), the laws of physics neglect all distinction between forward-moving timeandbackward-moving time. Chemistry recognizes existence of certain states of equilibrium in which the rates of change in both directions are equal. Other exceptions likely occur as well.

[2] Since change is quantized in the I Ching, which is to say, it is divided into small discretized units,  which Line changes model,  the magnitude of change is determined by the number of Line changes that have occurred  between Point A and Point B in spacetime.  Reversal is far easier to achieve if only a single Line change has occurred than if three or four Lines have changed for example.

[3] Ironically, Taoism itself has pointed out the perils of popularity. Had the I Ching been less popular, less appealing to members of all strata of society, it would have traveled through time more intact.  Unless,  of course,  it ended up buried or burned. What is fortunate here is that much of the primordial logic of the I Ching can be reconstructed by focusing our attention on the diagrammatic figures and ignoring most of the attached commentary.

[4] These diagrams do not occur explicitly in the I Ching. The logic they are based on, though, is fully present implicitly in the diagramatic structural forms of hexagrams, trigrams, and bigrams and the manner of their usage in  I Ching divinatory practices.

[5] Or, for that matter, than does the logic of Cartesian coordinate space if we take into account the degrees of freedom of six dimensional hexagrams mapped by composite dimensional methodology to model mandalic space. (See Note [4] here for important related remarks.)

[6] This is the closest frontal section to the viewer through the 3-dimensional cube using Taoist notation.  See here for further explanation.  Keep in mind this graph barely hints at the complexity of relationships found in the 6-dimensional hypercube which has in total 4096 distinct changing and unchanging hexagrams in contrast to the 16 changing and unchanging trigrams we see here. Though this model may be simple by comparison,  it will nevertheless serve us well as a key to deciphering the number system on which I Ching logic is based as well as the structure and context of the geometric line that can be derived by application of reductionist thought to the associated mandalic coordinate system of the I Ching hexagrams. We will refer back to this figure for that purpose in the near future.

© 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 300-

Beyond Taoism - Part 3
A Multidimensional Number System

(continued from here)

Descartes modeled his coordinate system on the Western number line,  itself an extension of the decimal number system to include the new negative numbers, and upon the Euclidean notion of a three-dimensional geometry.  All these events took place in historical times.  In approaching the I Ching and its number system we are dealing mostly with events that took place before recorded history so it is impossible to say with certainty how anything involved came about.  We can’t so much as be sure whether the I Ching was based on an antecedent number system,  or predated and foreshadowed a subsequent number system of Chinese antiquity possibly contingent on it. We view all such things as through a glass, darkly.^[1]

It is clear, though, that the number system of the I Ching is one far more complex than that of Western mathematics.  The number system of the West is unidimensional (linear).  Descartes,  in his coordinate system, extends it for use in three dimensions. The number system of the I Ching, on the other hand,  is  in origin  multidimensional.  It is mandalic as well, which is to say it consists of multiple dimensions interwoven in a specific manner which can best be characterized as mandalic in form,  possessing a number of interlaced and interlinked concentric shells or orbitals about a unifying center.

At the important origin of Descartes’ coordinate system is found his triple zero ordered triad (0,0,0).  Descartes  views this point,^[2] asall his points, primarily in terms of location, not relationship.  The matter of relationship is left to analytic geometry,  the geometry Descartes codified based on his coordinate system.^[3] The coordinate system itself seems not to care how points are formed or related beyond the most elementary and trivial operations of addition and subtraction throughout what essentially remain predominantly isolated dimensions.^[4] In the end this becomes an effective and prodigious mind snare.^[5]

In contrast to the Cartesian approach,theI Ching offers a unified coordinate system and geometry in a single entity which emphasizes the relationship of “points” and other “parts” (e.g.,  lines,  faces) as much,  if not more,  than location,  beginning with wholeness and ending with the same.  In between,  all sorts of  complex and interesting interactions and changes take place.  In analyzing these,  it is best to begin at the origin of the coordinate system of the I Ching,  the unceasing wellspring  of  being that supplants the triple vacuity of Descartes and Western mathematics.

Section F_H(n)^[6]

(continuedhere)

Notes

[1] My thinking is that the I Ching was originally primarily a non-numerical relationship system that subsequently was repurposed to include,  as one of its more important contextual capacities, numerical relationships. That said, from a contemporary perspective,  rooted in  a comprehensive awareness that spans combinatorics,  Boolean algebra,  particle physics, and the elusive but alluring logic  of  quantum mechanics and the Standard Model,  it would seem that this relationship system is an exemplary candidate for an altogether natural number system, one that a self-organizing reality could readily manage.

[2] As do most geometers who follow after Descartes.

[3] Strictly speaking, this approach is not in error, though it does seem a somniferous misdirection.  Due to the specific focus and emphasis enfolded in Descartes’ system, certain essential aspects of mathematical and physical reality tend to be overlooked. These are important relational aspects,  highly significant to particle physicists among others. These remarks are in no way intended to denigrate  Cartesian  coordinates and geometry,  but to motivate physicists and all freethinkers  to investigate further in their explorations of reality.

[4] The Cartesian system neglects, for instance, to express anywhere that the fact  the algebra of the real numbers  can be employed to yield results about the linear continuum of geometry relies on  the Cantor–Dedekind axiom,  which in mathematical logic

has been used to describe the thesis that the real numbers are order-isomorphic to the linear continuum of geometry. In other words, the axiom states that there is a one to one correspondence between real numbers and points on a line.

This axiom is the cornerstone of analytic geometry. The Cartesian coordinate system developed by René Descartes explicitly assumes this axiom by blending the distinct concepts of real number system with the geometric line or plane into a conceptual metaphor. This is sometimes referred to as the real number line blend. [Wikipedia]

Neither mandalic geometry nor the I Ching,  upon which it is based,  accept this axiom as true in circumstances other than those restrictive settings, such as Cartesian geometry, where it is explicitly demanded as axiomatic to the system. In other words,  they do not recognize the described one to one correspondence between number and geometric space as something that reality is contingent on. The assumption contained in this axiom, however, has been with us so long that we tend to see it as a necessary part of nature.  Use of the stated correspondence may indeed be expedient in everyday macro-circumstances but continued use in other situations,  particularly to describe subatomic spatial relations,  is illogical and counterproductive, to paraphrase a certain Vulcan science officer.

[5] For an interesting take on the grounding metaphors at the basis of the real number line and neurological conflation see  The Importance of Deconstructing the Real Number Line.  Also on my reading list regarding this subject matter  is Where Mathematics Comes From:How the Embodied Mind Brings Mathematics into Being(1,2,3) by George Lakoff and Rafael Nuñez. Neither of the authors is a mathematician, but sometimes it is good to get an outside perspective on what is in the box.

[6] This is the closest frontal section to the viewer through the 3-dimensional cube using Taoist notation.  See here for further explanation.  Keep in mind this graph barely hints at the complexity of relationships found in the 6-dimensional hypercube which has in total 4096 distinct changing and unchanging hexagrams in contrast to  the 16 changing and unchanging trigrams we see here.  Simple by comparison though this model may be it will nevertheless serve us well as a key to deciphering the line derived from the mandala of I Ching hexagrams, and we will be referring back to this figure for that purpose in the near future.

© 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 298-

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-

Quantum Naughts and Crosses Revisited - II

(continued from here)

This post builds on orientational material offered in the previous post.  An explanation of the procedural method of graphic demonstration used in this post and those following can be found there,  and it would be helpful to review that earlier post, if not already done,  before proceeding further.

Due in part to the challenging subject matter,  in part to arduous graphic demonstration,  we’ll approach this investigation in three stages of progressive difficulty. In the first stage we’ll just dangle our feet in the water by looking at how the  "slicing methodology"  works with ordinary three-dimensional  Cartesian coordinates.  In the second stage,  we’ll go waist-deep, and consider the same Cartesian coordinates in their Taoist notation transliteration equivalents.  And in the final stage,  we’ll go for full immersion,  with graphic representation of true mandalic geometry, that is,  plotting all 64 hexagrams  in a hybrid 6D/3D coordinate system using the methodology of composite dimension which, of course, has no analogue in purely Cartesian terms.

At each stage - Cartesian, Taoist transliteration, and mandalic - we’ll look at the respective cube in  frontal,transverse, and sagittal slices, always in that order and always progressing from identity face containing Cartesian (1,1,1),  trigram HEAVEN,  or hexagram HEAVEN  to inversion face, containing Cartesian (-1,-1,-1),  trigram EARTH, or hexagram Earth, as the case may be.

To accomplish our purpose we will require an effective, consistent way to refer to the individual “slices” and each of the 27 Cartesian points. There are three “slices” for each type of sectioning of the “cube”, so a total of nine. I propose that we uniquely identify each “slice” by labeling it with the first letter of the section type  (frontal, transverse, or sagittal)  and the subscript letters “H” for planes containing trigram or hexagram HEAVEN but not Earth, “E” for planes containing trigram or hexagram EARTH but not HEAVEN, and “HE” for planes containing both trigram forms.^[1]

The labels of the sections, then, will be:

• F_H     frontal section containing HEAVEN but not EARTH
• F_HE   frontal section containing both HEAVEN and EARTH
• F_E     frontal section containing EARTH but not HEAVEN
• T_H    transverse section containing HEAVEN but not EARTH
• T_HE   transverse section containing both HEAVEN and EARTH
• T_E     transverse section containing EARTH but not HEAVEN
• S_H     sagittal section containing HEAVEN but not EARTH
• S_HE   sagittal section containing both HEAVEN and EARTH
• S_E      sagittal section containing EARTH but not HEAVEN

For the 27 individual discretized Cartesian points, I propose the following labeling convention:

Each point is to be first identified as to type.  There are four point types: vertex(V), edge center(E), face center(F), and cube center(O).  The cube center corresponds to the Cartesian triad (0,0,0), the origin point of the Cartesian coordinate system. In the Cartesian/Euclidean cube there are 8 vertices, 12 edge centers, 6 face centers, and a single cube center.  The higher dimensional mandalic cube has many more of each of these.

Vertices

Having identified the point type, each point is then further identified by a subscript consisting of the first letter of the name of  trigram or hexagram that is resident at the point.  The single exception to this will be  WATER. To differentiate between  WATER  and  WIND,  I propose using the letter “A” (first letter of “aqua”, Latin for “water”) to specify WATER.  This plan allows us, then, to discriminate among the various vertex points, and also to distinguish them from the other point types.  Accordingly,  we arrive at these labels for the 8 vertex points:

• V_H  HEAVEN
• V_E   EARTH
• V_T  THUNDER
• V_W WIND
• V_A  WATER
• V_F   FIRE
• V_M  MOUNTAIN
• V_L   LAKE

Edge centers

Edge centers will be labeled “E” along with a subscript consisting of the first letter of its two vertices, “A” being used instead of “W” for WATER. Though this may initially seem excessively complicated,  the reasons for setting things up this way will soon be made clear, and it will all become second nature. The 12 edge centers will be labeled as below:

• E_HW
• E_HF
• E_HL
• E_ET
• E_EA
• E_EM
• E_TF
• E_TL
• E_AW
• E_AL
• E_MW
• E_MF

Face centers

There are six face centers.  Three occur in  identity faces  of the cube that contain the trigram or hexagram HEAVEN; three, in inversion faces that contain the trigram or hexagram EARTH. Labeling will be with the letter “F” and a subscript consisting of either “E” for EARTH along with one of its companion diagonal vertices, “W” for WIND, “F”, FIRE, “L”, LAKE or “H” for HEAVEN,  along with one of its companion diagonal vertices, “T” for THUNDER, “A”, WATER, “M”, MOUNTAIN.  So these six face center labels are:

• F_EW
• F_EF
• F_EL
• F_HT
• F_HA
• F_HM

Cube center

The cube center, which is singular in Cartesian terms but a multiple composite in terms of mandalic geometry, will be labeled as:

• O

identifying it as the origin of the coordinate system, that is to say, of both the Cartesian coordinate system and the mandalic coordinate system.

With that, let the games begin!

(continuedhere)

Notes

[1] There are no sections among those described that include both the hexagram HEAVEN and the hexagram EARTH.

© 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 287-

Quantum Naughts and Crosses Revisited - I

(continued from here)

Because nature is ever playful, grokking mandalic geometry is much like a game.  We view it as a largely serious one, though, one that involves combinatorics, Boolean logic, and magic squares and cubes. Groundwork for what appears in this post, and several to follow, was laid in May, 2014 in a series titled “Quantum Naughts and Crosses” which began here.

The game is played on a board or field made of three-dimensional coordinates of the Cartesian variety upon which are superimposed the six additional extraordinary dimensions unique to mandalic coordinates. For convenience and ease of representation,  the board will be displayed here in two dimensional sections abstracted from the Cartesian cube and from the superimposed mandalic hypercube in a manner analogous to the way computed tomography renders sections of the human body.

The sections commonly used^[1] in computed tomographyandmagnetic resonance imaging (MRI) are

• Frontal
• Transverse
• Sagittal

For our purposes here, these can be thought of as

• Planes perpendicular to the z-axis viewed from front to back of cube
• Planes perpendicular to the y-axis viewed from top to bottom of cube
• Planes perpendicular to the x-axis viewed from side right to left of cube

These “cuts” will produce square sections through xy-, xz-, and yz-planes, respectively,  of the Cartesian cube and,  in the case of the mandalic cube, analogous sections of higher dimension.

These choices of sections are made largely for convenience and ease of communicability. They are mainly of a conventional nature.^[2]  On the other hand,  there is special significance in the fact that all three section types progress from identity faces of the cube, containing the trigram or hexagram HEAVEN, to inversion faces, containing trigram or hexagram EARTH.  Some manner of consistency of this sort is necessary.  The one chosen here will make things easier as we progress.

Ourgameboard has 27 discretized Cartesian points,  centered in 3 amplitude levels about the Cartesian origin (0,0,0).^[3] Each point in the figure on the right above is represented by a single small cube,  but in the two-dimensional sections we’ll be using for elaboration,  they will appear as small squares.  So the gameboard is “composed of” 27 cubes arranged in a 3x3x3 pattern. But in descriptions of sections, we will view 9 squares in a 3x3 pattern. This configuration will appear as

But keep in mind each small square in this figure is actually a small cube representing one of the 27 discretized Cartesian points we’ve described.

Until next time, then.

(continuedhere)

Notes

[1] The origin of the word  "tomography"  is from the Greek word “tomos” meaning “slice” or “section” and “graphe” meaning “drawing.” A CT imaging system produces cross-sectional images or “slices” of anatomy,  like the slices in a loaf of bread.  The “slices” made are transverse  (cross-sections from head to toes or, more often, a portion thereof), but reconstructions of the other types of sections described above are sometimes made,  and MRI generates all three types natively.

[2] Admittedly, I’ve chosen the convention here myself and to date it is shared by no one else.  Perhaps at some future time it will be a shared convention.  One can only hope.

[3] These three discrete amplitude levels of potentiality in the mandalic 9-cube correspond geometrically to face centers, edge centers and vertices of the 3-cube of Cartesian coordinates.  They are encoded by the six new potential dimensions interacting with the three ordinary Cartesian dimensions in context of the hybrid 6D/3D mandalic cube. They are a feature of the manner of interaction of all nine temporospatial dimensions acting together in holistic fashion. This should begin to give an idea why there is no Taoist line that can generate a 9-cube in a fashion analogous to the way the Western number line is used to generate the Cartesian / Euclidean 3-cube. The 9-dimensional entity is primeval and a variety of different types of  "line"  can be derived  from it.  Similarly,  the  mandala  of the  I Ching  hexagrams cannot be derived from the logic encoded in any linear structure.  An overarching perspective is required to derive first the mandala of hexagrams and then  from it,  a variety of  Taoist line types.  Nature may be playful,  but it is not nearly as simplistic  as our Western science, mathematics, and philosophy would have 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 286-

Beyond Descartes - Part 9:
The Potential Plane
and Probable States of Change

Composite Dimension and
Amplitudes of Potentiality
Episode 3

(continued from here)

We have seen that an imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit i,  which is defined by its property ixi =−1.  The square of an imaginary number bi is −b².  For example,  6i is an imaginary number,  and its square is −36.^[1] Other than 0,  imaginary numbers yield negative real numbers when they are squared.^[2]

Turning now to potential numbers, we can similarly define a unit of potentiality p by the property p x -p = -1. [Long pause here waiting for the other shoe to drop.] Just a minute, you say, that’s just like 1 x -1 = -1.  Yes, it is. And that is just the point. All real numbers. Nothing to imagine. And Descartes finally vindicated after all these years - imaginary numbers just imaginary after all.  But how does this work? Or does it even work?  What exactly is the point? Is this a joke? It’s no joke, I assure you.  It’s an easier and better way to achieve the same ends - - - and more. Muchmore.

The secret is in the sauce, I say slyly. Really? Well, yes - in a way. Though imaginaries use a sauce with nearly identical ingredients.  The recipe is p + (-p) = 0. And, of course, i + (-i) = 0 as well.  The trick is in how - - - and where - - - the sauce is applied.  In the potential plane the sauce is applied more liberally in more locations for greater lubrication.

Levity aside. (This is after all a TST^[3].) The complex plane uses a single axis.  This axis represents a new dimension, wholly distinct from the x, y and z dimensions.  Strangely,  we’re never informed where this axis/dimension might be located,  just that it is somewhere other than where x, y and z are located. Stranger still, the complex plane allocates the y-axis of the Cartesian plane for its own use in location of its points. Although never specifically mentioned, to my knowledge, I surmise the imaginary dimension exists in what mathematics and physics both call phase space.^[4]

The mandalic or potential plane uses no such underhanded plan. It openly posits the existence of six new dimensions, allocated equally with two accompanying each of the Cartesian dimensions,  all overtly evident. (All nine spatial dimensions in plain sight together, that is.)  Nothing left to the imagination. As the new dimensions are made commensurate with the old in a hybrid geometric display,  no imaginary dimension is needed. Coordinates of  all potential dimensions  are  readily communicable  with the real number system through all of the ordinary Cartesian dimensions concurrently along with the Cartesian coordinates.  Moreover,  mandalic geometry conjectures that the ordinary Cartesian dimensions may in fact originate in  interactions among number species  of potential dimensions filtered through impacts on inherited biological sensory mechanisms.^[5] This raises yet another interesting possibility.^[6]

In the long convoluted history of mathematics, the imaginary numbers were introduced as a correlative to the number line with its real numbers. That meant, among other things, that they were linear, consisting of a single dimension.  The  complex plane  related the two
in a kind of hybrid geometry that consisted of one real dimension and one imaginary dimension.  Mathematician  William Rowan Hamilton in 1843 proffered the  quaternions,  a number system that extends the complex numbers to three dimensions, whereupon things went, to my mind, from bad, to very much worse.

Quaternions came with certain dysfunctional characteristics, among them,  the fact that multiplication of two quaternions is noncommutative. This is problematic.  The imaginary and complex numbers,  at least,  had both been commutative.  Nevertheless, physics endorsed the quaternions as it earlier had imaginary and complex numbers.

Why? Because the quaternions do in fact give partly correct results, and when investigating a dimly illuminated region of reality, such as the subatomic world still is today, even partial results are heartily welcomed if that is all that can be had.  The sad consequence of this, is that physics has been led astray in its quest for truth for over a century now,  because partial truths can be much more misleading than complete errors. Total error is often uncovered much sooner than partial truth, which can pass undiscovered, depending upon circumstances, for a very long time.

Mandalic geometry will be shown to be free of the difficulty posed by noncommutative multiplication. It is fully commutative throughout its nine dimensions (three ordinary, six extraordinary). It was not composed that way from a number line,  with elements that could be commutatively multiplied with one another. It came that way fully formed from the start, in its primeval embodiment  as a multidimensional structure,  expressing behavior intrinsic to holistic nature.

Next time around, we’ll begin to look under the hood of the mandalic approach to geometry and see if we can grokit.

(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] 6²xi² = 36 x (-1) = -36.

[2] Zero (0) is considered both real and imaginary, and both the real part and the imaginary part are defined as real numbers. (If that makes little sense to you, don’t blame me. I’m just the messenger here, reporting what the mathematicians have stated to be the case.) This seems to me to be purely an arbitrary definition, and it confuses me as much as it probably does you.  Could it be they did this to avoid the situation where 0² x (-1) = -0?  I think I would find that definition less disturbing, welcome even.

[3] Newly coined Internet acronym for Truly Serious Topic. (Not to be confused with TSR Totally Stupid Rules.)

Speaking about “greater lubrication”(wewere a moment ago, remember?), I use the phrase not simply as  a figure of speech,  or a simile,  but rather,  as a metaphor.  "Spicing" of mandalic geometry with all those zeros of potentiality makes for a very “fluidic dish” which, I believe, reflects the changeable nature of reality far better than the stricter, strait-laced coordinates of Descartes or the complex plane are able to do. And it’s not just a matter of fluidity involved here. The mandalic form so begotten is, in fact, a probability distribution through the three Cartesian dimensions concurrently,  which feature alone  makes mandalic geometry an ideal candidate for application to quantum physics.

[4] A phase space of a dynamical system is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space. In a phase space every degree of freedom or parameter of the system is represented as an axis of a multidimensional space; a one-dimensional system is called a phase line, while a two-dimensional system is called a phase plane.  For every possible state of the system (that is to say, any allowed combination of values of the system’s parameters) a point is included in the multidimensional space. [Wikipedia]

[5] I am speaking here of the hybrid 6D/3D formulation of mandalic geometry which combines the features of  dimensional numbers,  potential numbers,  and composite dimension,  this being a fully open access geometric system that has nothing hidden, nothing held back. What you see is what you get. (WYSIWYG)

[6] It is tempting to wonder whether there might be a close connection between the composite dimensions/potential coordinates  proposed by mandalic geometry and the pilot wave theoryorde Broglie–Bohm theory of quantum mechanics. At least there seems to be a correlation  between  David Bohm’s implicate/explicate order and the manifest/unmanifest (potential) coordinates of mandalic geometry.

© 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 285-

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 6

The Fiction Formerly Known As the Line

(continued from here)

Rereading the last post a moment ago I see I fell into the same old trap, namely describing a concept arising from an alternative worldview in terms of our Western worldview.  It is so astonishingly easy to do this. So it is important always to be on guard against this error of mind.

In saying that the Taoist number line is the basis of its coordinate system I was phrasing the subject in Western terminology,  which doesn’t just do an injustice to the truth of the matter,  it does violence to it,  in the process destroying the reality:  that within Taoism, the coordinate system is primary.  It precedes the line,  which follows from it.  What may be the most important difference between the Taoist apprehension of space and that of Descartes lies encoded within that single thought.

Descartes continues the fiction fomented in the Western mind by Euclid that  the point and the line  have independent reality. Taking that to be true,  Descartes constructs his coordinate system using  pointsandlines  as the elemental building blocks. But to be true to the content and spirit of Taoism, this fabrication must be surrendered.  For Taoism,  the coordinate system, which models space, or spacetime rather, is primary. Therefore to understand the fictional Taoist line we must begin there, in the holism and the complexity of its coordinate system where dimension, whatever it may be, reigns supreme.^[1]

And that means we can no longer disregard composite dimension, postponing discussion of it for a later time,  because it is the logical basis on which the I Ching is predicated. It is related to what we today know as combinatorics,Boolean algebra, and probability,  and is what gives rise to what I have called the plane of potentiality. It is the very pith of mandalic geometry, what makes it a representation of mandalic spacetime.^[2]

(continuedhere)

Notes

[1] In my mind, dimension is a category of physical energetic description before it is a category of geometrical description.  When particle physicists speak about “quantum numbers” I think they are actually, whether intended or not, referring to dimensions. If this is true, then our geometries should be constructed to reflect that primordial reality, not arbitrarily as we choose.

[2] In speaking of logic and the I Ching in the same breath I am using the term in its broadest sense as any formal system in which are defined axioms and rules of inference. In reference to the I Ching,  the logic involved is far removed from the rationalism bequeathed to Descartes by his times.  It is a pre-rationalist logic that prevailed in human history for a very long time before the eventual splitting off of the irrational from the rational.  This means also that the I Ching is among other things a viable instrument to access strata of human minds long dormant in historical times,  other than possibly,  at times,  in poetry and art and the work of those select scientists who make extensive use of intuition in the development of their theories.

Note to self:  Two contrasting systems of thought based on very different worldviews can never be adequately explained in terms of one another. At times though, for lack of anything better, we necessarily fall back on just such a strategy, however limited, and make the best of it we can.

© 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 282-

i spent 50k down in mexico city

got a spot in condesa, fresa tell me im pretty

you already know, say that she can come with me

madboy land on her body that aint prada or fendi

she aint with no violence but she cop the machete

she really blew my mind said relleno con sesos

riding down the eje, uber color huesos

tuetano got me zooted or maybe thats the weedsmoke

wonder where my brain go say i never listen

dimensions where im ascending to a higher plane

lost the weight of ego can barely hear my name

how we coming back to the banal

she say shut up bitch change the channel 

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.

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

Imaginary numbers arose in the history of mathematics as a result of misunderstanding the dimensional character of numbers.  There was a failure to acknowledge that numbers exist in a context of dimension. This has earlier been addressed at length.^[1]  Simply put, numbers exist always in a particular dimensional context.  Square numbers pertain to a context of two dimensions and therefore to a plane,  not a line.  Square roots then ought justly reference a two-dimensional geometrical context rather than the linear one mathematics has maintained ever since mathematicians of the Age of Enlightenment decreed it so.  Square roots contrary to the way mathematics would have it can neither exist in nor be found in any single line segment,  because they do not originate in the number line but in the two-dimensional square.

Algebra, not geometry, provided the breeding ground for imaginary numbers.  They were given a geometric interpretation as an afterthought only, long after the fact of their invention. Rationalist algebraists, feeling compelled to give meaning to equations of the form b² = -4 came up with the fantastic notion of imaginary numbers. Only indirectly did these grow out of nature, by way of minds of men obsessed with reason.^[2]

Descartes knew of the recently introduced square roots of negative numbers. He thought them preposterous and was first to refer to the new numbers by the mocking name imaginary, a label which stuck and which continues to inform posterity of the exact manner in which he viewed the oddities.  It is one of the ironies of history that when at last a geometrical interpretation of square root of negative numbers was offered it involved swallowing up Descartes’ own y-axis. Poetic justice? Or ultimate folly?

Had the essential dimensional nature of numbers been recognized there would have been no need to inquire what the square root of -1 was. It would have been clear that there was no square root of -1 nor any need for such as +1 also has no square root.  As linear numbers,  neither -1 nor +1 can legitimately be said to have a square root.  Both, though, have two-dimensional analogues and these do have square roots, not recognized as such unfortunately by the mathematics hegemony.^[3]

In the next post we will look at a comparison between imaginary numbers,  which were formulated in accordance with this misconstrual about how numbers relate to dimensions,  and probable numbers which grow organically out of a consideration of how numbers and dimensions actually relate to one another in nature.^[4]  The first of these approaches can be thought of as rational planning by a central authority; the second, as the holistic manner in which nature attends to everything, all at once, and without rational forethought.

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Image: A drawing of the first four dimensions. On the left is zero dimensions (a point) and on the right is four dimensions  (A tesseract).  There is an axis and labels on the right and which level of dimensions it is on the bottom. The arrows alongside the shapes indicate the direction of extrusion. By NerdBoy1392 (Own work) [CC BY-SA 3.0orGFDL],via Wikimedia Commons

Notes

[1] See the series of about nine posts that begins here.

[2] The Rationalists missed here a golden opportunity to relate number and dimension by defining square root much too narrowly. They seem to have been so mesmerized by their algebraic equations that they failed to pursue the search into deeper significance pertaining to essential linkages between dimension and number that intuition and imagination might have bestowed.

[3] As Shakespeare correctly pointed out, a rose by any name would smell as sweet. Plus one times plus one certainly equals plus one but that has nothing to do with actual square root really, just with algebraic linear multiplication.  Note has often been made in these pages of the difference between mathematical truth and scientific truth. Whereas mathematics demands only adherence to its axioms and consistency,  science requires empirical proof.  Mathematics defined square root in a certain manner centuries ago, and has since been devoutly consistent in its adherence to that definition.  In so doing it has preserved a cherished doctrine of mathematical truth, as though in formaldehyde.  It has also for many centuries contrived to be consistently scientifically incorrect.  The problem lies in the fact it has converted physicists and near everyone else to its own insular worldview.

[4] For an early discussion of the probable plane, potential dimensions, and probable numbers see here.

Ask Ethan: Does Our Universe Have More Than 3 Spatial Dimensions?

“My question is has the 4th [spatial] dimension been proven to exist or is it just theoretical? If it does exist how was it proven to exist? If it’s theoretical why do we think it might exist?”

Having three spatial dimensions is nice, but just like when you go to a restaurant, wouldn’t it be nice if they gave you something, I don’t know, a little bit “extra” for your enjoyment? The fourth spatial dimension hasn’t been proven to exist, as we have no direct evidence for it. But it’s worth considering, and there are plenty of both physical and mathematical reasons for it.

Take a look inside and see what you think for yourself!