#hexagrams

Beyond Taoism - Part 4
A Vector-based Probabilistic
Number System
Introduction

(continued from here)

Leibniz erred in concluding the hexagrams of the I Ching were based on a number system related to his own  binary number system.  He had a brilliant mind but was just as fallible as the rest of us.  He interpreted the I Ching in terms of his own thought forms,  and he saw the hexagrams as a foreshadowing of his own binary arithmetic.^[1]

So in considering the hexagram Receptive,  Leibniz understood the number 0; in the hexagram Return, the number 1; in the hexagram Army, the number 2; in the hexagram Approach, the number 3;  in the hexagram Modesty,  the number 4;  in the hexagram  Darkening of the Light, the number 5;  and so on, up to the hexagram Creative, in which he saw the number 63.^[2]  His error is perhaps excusable in light of the fact that the Taoists, though much closer to the origin of the I Ching in time, themselves misinterpreted the number system it was based on.^[3]

From our Western perspectiveI Ching hexagrams are composed of trigrams, tetragrams, bigrams, and ultimately yinandyang Lines. From the native perspective of the I Ching this order of arrangement is putting the cart before the horse.  Dimensions  and their interactions  are,  in the view of I Ching philosophy and mandalic geometry,  antecedent logically and materially to any cognitive parts we may abstract from them. Taoism in certain contexts has abstracted the parts and caused them to appear as if primary. It has the right to do so if creating its own philosophy,  but not as interpretation of the logic of the I Ching. It is a fallacy if so intended.^[4]

The Taoists borrowed from the I Ching two-dimensional numbers, treated them as one-dimensional and based their quasi-modular number system on  the dimension-deficient result.  This is the way they arrived at their seasonal cycle consisting of bigrams:   old yin (Winter),  young yang (Spring), old yang (Summer), young yin (Autumn), old yin (Winter),  and so forth. This represents a very much impoverished and impaired version of the original configuration in the primal strata of the I Ching.^[5]

The number system of the I Ching is not a linear one-dimensional number system like  the positional decimal number system  of the West; nor is it like the positional binary number system invented by Leibniz. It is not even like the quasi-modular number system of Taoism.  The key to the number system of the hexagrams is located not in the 64 unchanging explicit hexagrams,  but rather in the changing implicit hexagrams found only in the divination practice associated with the I Ching. These number 4032.^[6]  The manner in which these operate,  however,  is actually  fairly simple and is uniform throughout the system.  So once understood,  they can be safely relegated to the implicit background, coming into play only during procedures involving divination or in attempts to understand the system fully, logically and materially.  When dealing with more ordinary circumstances just the 64 more stable hexagrams need be attended to in a direct and explicit manner.

The Taoist sequence of bigrams is in fact a corruption of the far richer asequential multidimensional arrangement of bigrams that occurs in I Ching hexagrams and divination. There we see that change can occur from any one of the four stable bigrams to any other.  If this is so then no single sequence can do justice to the total number possible. The ordering of bigrams presented by Taoism is just one of many that make up the real worlds of nature and humankind.  Taoism imparts special significance to this sequence; the primal I Ching does not. It views all possible pathways of change as equally likely.^[7]

Next time around we will look further into the implications of this equipotentiality and see how it plays out in regard to the number system of the I Ching.

Section F_H(n)^[8]

(continuedhere)

Notes

[1] By equating yang with 1 and yin with 0 it is possibletosequence the 64 I Ching hexagrams according to binary numbers 0 through 63.  The mere fact that this is possible does not, however, mean that this was intended at the time the hexagrams were originally formulated. Unfortunately, this arrangement of hexagrams seems to have been the only one of which Leibniz had knowledge. This sequence was, in fact, the creation of the Chinese philosopher Shao Yong (1011–1077). It did not exist in human mentation prior to the 11th century CE.

This arrangement was set down by the Song dynasty philosopher Shao Yong (1011–1077 CE), six
centuries before Wilhelm Leibniz described binary notation. Leibniz published ‘De progressione
dyadica’ in 1679. In 1701 the Jesuit Joachim Bouvet wrote to him enclosing a copy of Shao Yong’s 'Xiantian cixu’ (Before Heaven sequence). [Source]

Note also that the author of Calling crane in the shade, the source quoted above, calls attention to confusion that exists about whether the “true binary sequence of hexagrams” should begin with the lowest line as the least significant bit (LSB) or the highest line. He points out that the Fuxi sequence as transmitted by Shao Yong in both circular and square diagrams takes the highest line as the LSB, although in fact it would make more sense in consideration of how the hexagram form is interpreted to take the lowest line as the LSB. My thinking is that either Shao Yong misinterpreted the usage of hexagram form or, more likely, the conventional interpretation of the Shao Yong diagrams is incorrect. Here I have chosen to use the lowest line of the hexagram as the LSB,  and I think it possible  Leibniz may have done the same.

If one considers the circular Shao Yong diagram,  the easier of the two to follow,  one can reconstruct the binary sequence,  with the lowest line as LSB,  by beginning with the hexagram EARTH at the center lower right half of the circle, reading all hexagrams from outside line (bottom) to inside line (top),  progressing counterclockwise to  MOUNTAIN over WIND at top center, then jumping to hexagram  MOUNTAIN over EARTH  bottom center of left half of the circle,  and progressing clockwise to hexagram  HEAVEN  at top center.  Of the two,  this is the interpretation that makes the more sense to me and the one I have followed here, despite the fact that it is not the received traditional interpretation of the Shao Yong sequence. Historical transmissions have not infrequently erred. Admittedly it is difficult to decipher all Lines of some of the hexagrams  in the copy Leibniz received due to passage of time and its effects on paper and ink.  Time is not kind to ink and paper, nor for that matter to flesh and products of intellect.

In the final analysis, which of the two described interpretations is the better is moot because neither conforms to the logic of the I Ching which is not binary to begin with. Moreover,  there is a third interpretation of the Shao Yong sequence that is superior to either described here.  It is not binary-based.  And why should it be? After all the Fuxi trigram sequence  which Shao Yong took as model for his hexagram sequence  is itself not binary-based. Perhaps we’ll consider that interpretation somewhere down the road. For now, the main take-away is that Leibniz, in his biased interpretation of the I Ching hexagrams made one huge mistake.  Ironically,  had he not some 22 years prior already invented  binary arithmetic, this error likely would have led him to invent it.  It was “in the cards” as they say. At least in certain probable worlds.

[2]ReceptiveandCreative are alternative names for the hexagrams EarthandHeaven, respectively. The sequence detailed can be continued ad infinitum using yin-yang notation, though of course this takes us beyond the realm of hexagrams into what would be, for mandalic geometry and logistics of the I Ching, domains of dimensions numbering more than six.  Keep in mind here though that Leibniz was not thinking in terms of dimension but an  alternative method  of expressing the prevalent base 10 positional number system notation of the West.  He held in his grasp the key to unlocking an even greater treasure but apparently never once saw that was so.  This seems strange considering his broadly diversified interests and pursuits in the fields of  mathematics,  physics,  symbolic logic,  information science,  combinatorics,  and in the nature of space.  Moreover,  his concern with these was not just as separate subjects of investigation.  He envisaged uniting all of them in a  universal language  capable of expressing mathematical,  scientific, and metaphysical concepts.

[3] Earlier in this blog I have too often confused Taoism with pre-Taoism. The earliest strata of the I Ching belong to an age that preceded Taoism by centuries, if not millennia.  Though Taoism was largely based on the philosophy and logic of the I Ching,  it didn’t always interpret source materials correctly,  or possibly at times it intentionally used source materials in new ways largely foreign to the originals. The number system of the I Ching is a case in point.

In the interest of full disclosure, I am not an expert in the history or philosophy of Taoism.  Taoist philosophies are diverse and extensive. No one has a complete set or grasp of all the thoughts, practices and techniques of Taoism. The two core Taoist texts, the  Tao Te ChingandChuang-tzu,   provide the philosophical basis of Taoism which derives from the eight trigrams (bagua) of Fu Xi, c. 2700 BCE, the various combinations of which created the 64 hexagrams documented in the I Ching.  The Daozang,  also referred to as  the Taoist canon,  consists of around 1,400 texts that were collected c. 400, long after the two classic texts mentioned. What I describe as Taoist thought then is abstracted in some manner from a huge compilation, parts of which may well differ from what is presented here. Similar effects of time and history can be discerned in Buddhism, Christianity, Islam and secular schools of thought like Platonism,Aristotelianism,Humanism, etc.

[4] Recent advances in the sciences have begun to raise new ideas regarding the structure of reality. Many of these have parallels in Eastern thought.  There has been a shift away from the reductionist view in which things are explained by breaking them down then looking at their component parts, towards a more holistic view. Quantum physics notably has changed the way reality is viewed. There are no certainties at a quantum level, and the experimenter is necessarily part of the experiment. In this new view of nature everything is linked and man is himself one of the linkages.

[5] It is not so much that this is incorrect as that it isextremelylimiting with respect to the capacities of the I Ching hexagrams.  A special case has here been turned into a generalization that purports to cover all bases. This may serve well enough within the confines of Taoism but it comes nowhere near elaborating the number system native to the I Ching. We would be generous in describing it as a watered down version of a far more complex whole.  Through the centuries both Confucianism and Taoism  restructured the I Ching to make it conducive to their own purposes.  They edited it and revised it repeatedly,  generating commentary after commentary,  which were admixed with the original,  so that the I Ching as we have it today,  the I Ching of tradition,  is a hodgepodge of many convictions and many opinions. This makes the quest for the original features of the I Ching somewhat akin to an archaeological dig.  I find it not all that surprising  that the oracular methodology of consulting the I Ching  holds possibly greater promise in this endeavor than the written text.  The  early oral traditions  were preserved better,  I think,  by the uneducated masses who used the I Ching as their tool for divination than by philosophers and scholars who,  in their writings,  played too often a game of one-upmanship with the original.

[6] A Line can be either yin or yang, changing or unchanging. Then there are four possible Line types and six Lines to a hexagram.  This gives a total of 4096 changing and unchanging hexagrams (4⁶ = 4096). Since there are 64 unchanging hexagrams (2⁶ = 64) there must be 4032 changing hexagrams (4096-64 = 4032).

[7] This calls to mind the path integral formulation of quantum mechanics which was developed in its complete form by Richard Feynman in 1948. See, for example, this description of the path integral formulation in context of the double-slit experiment, the quintessential experiment of quantum mechanics.

[8] 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. :)

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Beyond Taoism - Part 1
A Lost Logic of Chinese Antiquity

The 64 Hexagrams of the I Ching
(for interactive version click here)

(continued from here)

In speaking of “Taoist thought” I have often throughout this work used the term as a convenient shorthand for “primeval Chinese thought.” Strictly speaking, this usage is historically incorrect. Laozi is traditionally regarded as the founder of Taoism and is associated with “primordial” or “original” Taoism. Whether he actually existed is disputed.  According to tradition the classic text attributed to him, the Tao Te Ching, was written around the 6th century BCE. The oldest extant text, however, dates to the late 4th century BCE. The earliest strata of the I Ching predate both these historical periods by many centuries, if not millennia.  Taoism derived its cosmological notions from the philosophy of yin and yang,  and from that of the  Five Phases  or  Five Elements. Both these schools of thought were overwhelmingly  influenced and shaped  by exposure to the oldest known text of ancient Chinese classics, the I Ching.^[1]

The actual symbolic logic of Taoism,  although derived from the I Ching is extremely simplistic compared with that of the original upon which it is based. Whereas the philosophy of yinandyang as presented in the Tao Te Ching comprises little more than a two-dimensional cycle of two-valued elements,  in the I Ching these two represent vectors in a six-dimensional combinatorial manifold of 64 hexagrams (1,2). Clearly, it is a difference like that  between night and day.  It is,  in fact,  a literal comparing of 2² with 2⁶, the latter holding many more possibilities. The actual difference^[2]  in the  logic and geometry  emerging from the two is greater even than it appears at first. It eventuates not from just a simple geometric progression but from a mandalic intertwining and association of logical elements that give rise to different amplitudes of dimension as well as to a greater number of dimensions.  This mandalic interweaving leads also to a richer catalogue of relationship types.^[3]

Long viewed as mainly an ancient text of Chinese divination,theI Chingencompasses many more categories of thought - - - among them symbolic logic, geometry, and combinatorics.  As a treatise which deals with combinatorics alone, it soars without equal, the first known compendium of combinatorial elements and still one of the finest. The logic and geometry  that are embedded in the  hexagram system  of the I Ching are best understood in terms of dimensions and vectors akin to those in Cartesian systematics, and of logic gates analogous to the truth tables of Boolean algebra. And still the cognoscente will want to explore beyond the pertinency of these disciplines as also beyond Taoism to find the full meaning and intent of the I Ching.^[4]

Having existed for millenia,  and itself a treatise regarding change^[5] in its many aspects, it would be inconceivable that the I Ching as we have it today is as it was in its beginnings. Popular at all societal levels through its entire existence,  reinterpretations and reworkings  have been myriad. Confucianism in particular interlaced its own brand of philosophical and “ethical-sociopolitical teachings”  during and after  the fifth century BCE. Other schools of thought added their unique perspectives to what became essentially  a massive melting pot of schematization,  one not always self-consistent by any means.

When one attempts to uncover the original face of the I Ching the difficulties encountered soon appear insurmountable. If involved in such a venture,  it is imperative to bear in mind the bedrock strata of the work were in some ways more ingenuous, and in some more intricate, than the traditional version that has come down to us.  The earliest layers arose in context of a preliterate oral tradition with all the many unique aspects of being that entails. In some ways the golden age of the I Ching ended with coming of the written word and literacy. The multidimensional logic that was readily accommodated by an oral tradition foundered and eventually was all but lost in the unrelenting techno-sociological onslaught of script with its associated inevitable linearity. Anyone who hopes to excavate the buried multidimensional logic of the primordial I Ching can expect to do a good deal of laborious digging.

(continuedhere)

Image:Source. Originally from Richard Wilhelm’s and Cary F. Baynes translation “I Ching: Or, Book of Changes” [3rd. ed., Bollingen Series XIX, (Princeton NJ: Princeton University Press, 1967, 1st ed. 1950)]

Notes

[1] Two diagrams known as bagua (or pa kua) that figure prominently in the I Ching and its Commentaries predate their appearances in the I Ching. The Lo Shu Square is sometimes associated with the  Later Heaven arrangement  of the bagua or trigrams, and the  Yellow River Map  is sometimes associated with the Earlier Heaven arrangement of trigrams. Both are linked to astronomical events of the sixth millennium BCE. Although part of Chinese mythology, they played an important role in development of Chinese philosophy.  The Lo Shu Square is intimately connected with the legacy of the most ancient Chinese mathematical and divinatory traditions.  The Lo Shu is the  unique normal magic square (1,2) of order three (every normal magic square of order three is derived from Lo Shu by rotation or reflection). [Wikipedia]

[2] Taking into account both changing lines and unchanging lines of hexagrams there are four possible variants for each line:  unchanging yin,  unchanging yang, changing yin,  and changing yang.  This results in a total of  4⁶  or 4096  possible different line combinations for each six-line figure.  This allows for an enormous number of logical / geometric configurations,  all of which map to various points of the mandalic cube or, in terms of  Cartesian coordinates,  to discretized points of the three-dimensional cube bounded by  the eight Cartesian triads which have coordinates of  +1  and/or  -1  in all possible combinations (corresponding to the eight trigrams.)

To this point changing lines have not been discussed to avoid overcomplicating already complicated matters too soon.  Changing lines play an indispensable role in all changes of yin lines to yang lines and vice versa,  and therefore, in changes of one hexagram to another.  They are also essential elements in formation of the geometric line segment generated by the I Ching hexagrams which I have earlier referred to as the  "Taoist line“  and which we have yet to broach fully. Mandalic line segments uniformly comprise sixteen interrelated elements,  hexagrams with changing and/or unchanging lines.  Though various mandalic line segments have different compositions in terms of six-dimensional hexagrams,  these hexagrams can always be reduced in logical and geometrical terms to  sixteen bigram forms containing changing and/or unchanging lines. These bigram sets are all identical. No other variants are possible, since 4² equals 16. In this sense there is a single species of mandalic line segment but one which takes on different characteristics in different dimensional contexts.  Every hexagram has a commentary appended to each of its six lines,  which annotation is intended to be reflected upon only if the line is a changing one at time of consulting the oracle. Justly put, this system is brilliant beyond belief.

[3] Understand here that ‘relationship types’ may variously refer to human relationships in a society, to particle relationships in context of the atom, or to any other species of relationship one might imagine.

[4] For an exhaustive listing of linkstoI Ching related materials on the Web see here.

[5] Indeed, an alternative name of the I Ching in English is Book of Changes. The ensconced multidimensional logic encoded in the original work purports to be a microcosm describing all possible pathways of change, and their incessant changing relationships in the greater macrocosm of the universe.

© 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. :)

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Russia’s barbaric invasion of Ukraine is a nightmare turned into reality. I feel compelled to write a post about it as my personal reflection, but also as my small contribution to the joint learning process. The ancient time-tested wisdom of the I Ching could perhaps help us to understand and resolve the complex issues and conflicts that lead to violence and war. The I Ching is an ancient Chinese text and divination system which counsels appropriate action in the moment for a given set of circumstances. For 5000 years, people have turned to the I Ching to help them uncover the meaning of their experience and to bring their actions into harmony with the interests of society and the cosmos as a whole.

In the I Ching, there are several hexagrams that offer insight into war. One is Hexagram 6: Conflict, and another is Hexagram 8: Unity. Hexagram 6 describes a tense situation with a high level of contention and strife. Conflict develops when one feels himself to be in the right and runs into opposition. Escalating conflict is a no-win situation, therefore the hexagram counsels compromise and resolution. To carry on the conflict to the bitter end has exceedingly harmful effects even when one is in the right. Conflicts in which one party is not sincere inevitably lead to subterfuge and distortions. Conflicting parties can profit from the advice of a truly wise mediator. Clarification will bring about understanding and resolution. There is little chance of success without a unity of forces.

Conflict, in essence, is the absence of unity. We live in a conflicted world and very often we experience conflict ourselves. In fact, conflict is so pervasive in our polarized world that we take it for granted and deem it to be an inevitable part of life. This perspective has significant consequences; among them is the fact that by taking conflict for granted, our efforts to resolve it often fail and conflict turns into violence.

Hexagram 8 essentially describes unity as the binding force within society. It represents the idea of union between the different members and classes of a state and how it can be secured. Unity is a conscious and purposeful convergence of two or more diverse entities in a state of harmony, integration, and cooperation to create a new and evolving entity or entities. The hexagram portends that a leader with a strong and guiding personality will be the center of union. It emphasizes that joining people and things through recognizing their essential qualities is the adequate way to handle it. It counsels that those who do not seek to promote and enjoy union until it is too late will be left out in the cold. Conflict within weakens the power to conquer danger without.

Unity is the fundamental law of existence. Life takes place in the context of unity, and when the law of unity is violated, conflict and violence is the outcome. Everything that exists is the outcome of the law of unity. At the physical level, the law of unity ensures order and stability in the way subatomic particles, atoms, molecules, stars, and galaxies cohere and operate in a harmonious and integrated manner. At the biological level, the very process of formation and continuation of life is dependent on the proper operation of the law of unity. The same is true at the social level. Families are happy, healthy, and stable when unity exists between all its members. Communities prosper and are safe in the context of unity, and nations advance in every area when peace is present. At all levels of human life, unity, rather than conflict, is the fundamental operative and creative force.

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. :)

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