#complex systems

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.

Bootstrapping Neo-Boolean - I

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So yes. This is very much a work in progress. And we have strayed now as it happens  into  unfamiliar territory.  Terra incognita.  Therewill be dragons.^[1]  Dragons  are  errors.  Errors  are  dangerous,  and we must slay them.  But  all  in  good  time.  First,  we should scout out the terrain. That would be prudent.

Descartes in constructing his system of coordinates built upon the bedrockofelementary algebra and the number line. We’vepreviously called attention to the important  but mostly overlooked issue of the 1:1 congruence between number and geometric/spatial position he incorporated implicitly in the logic of his coordinates and questioned the validity of such correspondence, at least with respect to subatomic scales.

Working two centuries later but very much under the influence of Descartes’ thought,  George Boole introduced his own unique brand of algebra.  A second major influence on the development of his symbolic logic was the binary number system of Leibniz, himself influenced to a large degree by Descartes. We need to carefully follow and connect the dots here. Great advances in human cognition rarely,  if ever,  occur in isolation and seclusion. There is a fine line to tread though. If progress requires the shoulders of giants to stand on,  it is still difficult at times not to be overly influenced by those who came before.

Boole’s new logic, constructed in the wake of what by his time were firmly entrenched systematizations of thought by two of the most highly regarded philosopher mathematicians, was devised in such a manner as to conform to both of these conventions of system design.  Significant to our purposes here are the facts that first, Boolean logic echoes Cartesian convention of attributing to each and every location in geometric space a single unique number,  and second, it adheres to Leibniz’s convention of using a modulo-2 number system based on binary elements 1 and 0.^[2]

The symbolic logic systems of mandalic geometry and the I Ching do not abide by either of these conventions.  Instead they are based on what is best described as  composite dimensions with four unique truth values (or vector directions) each, ranging from -1 through two distinctive zeros (0_a; 0_b) to +1, and assignment of numbers to spatial locations through all dimensions by means of probability distributions in place of a simple and simplistic 1:1 distribution.  To accommodate these alternative conceptual concepts, we will need to expand and modify traditional Boolean logic as we have already done as regards Cartesian coordinate theory.

For starters here we should doubtless add, the mandalic form is the probability distribution through all dimensions, and the probability distributions are the mandalas.  And movement through either or both can only be accomplished by  discretized stepwise maneuvers  between different amplitudes of dimension separated by obscure quantum leaps of endless being and becoming and being and unbecoming, toward and away from  the centers and subcenters of holistic systems,  the parts of which are always aiming towards some kind of equilibrium never quite within reach. Which then makes error also a necessary aspect of reality and not simply the fearful monster we imagined.  It is error that makes achievement possible.^[3][4]

(continuedhere)

Image:Here Be Dragons Map. Detail of he Carta marina (Latin “map of the sea” or “sea map”), drawn by  Olaus Magnus  in 1527-39.  This is the earliest map of the Nordic countries that gives details and place names, by Olaus Magnus [Public domain], via Wikimedia Commons. The map was in production for 12 years.  The first copies were printed in 1539 in Venice. [Wikipedia]

Notes

[1] Mapmakers during the Age of Exploration sometimes placed the phrase “here be dragons” at the edges of their known world,  presumably to warn of the dangers lying in wait for sailing vessels  and  travelers by land who strayed too far from well-traveled routes.  Here is a list of all known historical maps on which these words appear.

[2] Or in Boole’s case, we might say,  attributing to each proposition in concept space a single truth value:  TRUE or FALSE (var YES or NO;  or, in electronics applications,  ON or OFF.)  What we have here, I believe, is in many instances a false dilemma  or the old Aristotelian dichotomy of  either/or.  Quantum physics demands and deserves better.  OK, true enough,  Boole gets around to extending possibilities  by means of multi-term propositions,  which his system can readily handle.  The question here, though,  is whether  nature  can or does  handle such similarly.  I think not.  I think it approaches the question  at a more fundamental level of reasoning and reality: at the most basic level of spacetime itself.

[3] This echoes the view of cybernetics,  a transdisciplinary approach for exploring regulatory systems, their structures, constraints, and possibilities.

Cybernetics is relevant to the study of systems, such as mechanical, physical, biological, cognitive, and social systems. Cybernetics is applicable when a system being analyzed incorporates a closed signaling loop; that is, where action by the system generates some change in its environment and that change is reflected in that system in some manner (feedback) that triggers a system change, originally referred to as a “circular causal” relationship. [Wikipedia]

[4] This entire blog and its predecessor are in some sense the chronology of a journey from the familiar shoreline into largely uncharted waters.  Hesitant at first, increasingly more daring as time has gone on and I’ve come to see  errors  to be stepping stones along the way. And there have beenmanyerrors along the way. Some I am not yet cognizant of.  But of those I am aware,  I have left most intact in spite of since being superseded by ideas superior, more correct or better formulated.  I’ve done this  because I think it  important  to  map the course  of a conceptual journey,  how the ideas evolved from A to B to C to D.  It also allows readers to participate,  to a degree,  in the thrill of an exciting adventure of mind, should they so choose. Happy travels.

© 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 303-

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-

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 - 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-