The Chinese are coming! The Chinese are coming!

 

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Lest some object that allowing the Taoist number line to reach our shores might endanger national security or at the very least the security of our hallowed number system, I must be quick to point out that the infiltration has already occurred. It happened well over half a century ago. As was the case with introduction of the Indo-Arabic number system to medieval Europe, the first to climb on the bandwagon were the merchants and moneylenders. Evidence once again that nothing ever really changes.

Our already deeply entrenched credit card system is fully conversant with, if not necessarily cognizant of its use of, the Taoist number line. The concept of using a card for purchases was exploited, if only in fiction, as early as 1887 by Edward Bellamy, the American author and socialist, in his utopian novel Looking Backward. [Wikipedia] In just over seventy years the fiction of the late 19th century was on the verge of becoming the fact of the mid 20th century. In 1958 the first successful recognizably modern credit card was introduced and the world has never looked back.

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Let’s now consider briefly how the credit card system is similar to the Taoist number line in its number usage. First, its central point of reckoning, which corresponds to the origin of the Taoist number line, the taijitu, is not zero but rather what is termed the credit line. This has a value which can be variable, often changing over time. But never in any of its various incarnations is it equal to zero. Unless of course the creditor fails to make the required payments and loses credit entirely.

To the right of the central point credit available exceeds credit used, corresponding to the yang domain of the Taoist number line. To the left of the central point credit used exceeds credit available, corresponding to the yin domain of the Taoist number line. In the example shown above each unit has a value of 100 dollars and the credit line is two thousand dollars. At every point along the line the sum of credit used and credit available (yin plus yang) equals the credit line, in this case 20 x 100 or $2000.

To the extreme right no credit has been used, all is still available for use (credit = 20; debit = 0). To the extreme left all credit has been used, none remains available (credit = 0; debit = 20). At this point further use of the credit card requires either making a payment or else obtaining an increase in the credit line. At the central point, credits and debits are perfectly in balance (credit = 10; debit = 10) and there is as much credit remaining for use as has already been used.(1)

 

(1) This example demonstrates among other things that use of the Taoist number line is possible and wholly practicable in ordinary commerce, i.e., the activity embracing all forms of the purchase and sale or exchange of goods and services. The question then arises as to why it would not be equally valid for defining and describing the events of the subatomic realm which are, when all is said and done, simply a commerce of another kind, one involving the exchange or interchange of particles and forces rather than debits and credits.

© 2014 Martin Hauser

The strange and convoluted history

of our number line

 

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Unlike the Taoist number line (1,2) which lacks a zero and which from its inception was entire and holistic, if not explicit, the number line that emerged in Western mathematics was the result of unplanned growth in which zero came to play a multiple and confusing role.(1)

First came the positive numbers (the counting numbers).(2) Negative numbers followed by centuries and were not introduced in Europe until the 15th century.(3) A practical quasi-mathematical zero had been used in antiquity (4), but this was not yet either the number zero or the zero of positional notation. When the Hindu-Arabic numeral notation system first entered Europe through Spain in the 10th century it did so without zero. The zero of position came later, and the number zero was a still later arrival to Europe.(5) There was no plan to any of this. It all sort of just materialized as what appears a patchwork of happenstance and pragmatic convenience, more a comedy of circumstance than a result of any rational forethought or perceived necessity.

Although today a 10-year old schoolchild is expected to master the basics of the number line, its birth was an uneasy one. From the start it had its critics and resisters. The notion of negative numbers representing debts must have been difficult enough to comprehend and accept initially, but with zero admitted to the number pantheon there was the additional challenge of dealing with something that was less than nothing. No easy task for a medieval craftsman, or scholar even, in the 13th-16th centuries.(6) Over the intervening centuries elapsed since then we all have become fully indoctrinated with the accepted dogma regarding the number line. So much so, we have forgotten about its adulterated origins. Along with the mathematicians we genuflect and call the number line a thing of beauty for its elegant simplicity and structural unity. It is nothing of the sort.

The number line of mathematics is a Western cultural construct.There is nothing sacrosanct about it. It has its usefulness.(7) But in some ways it is more confining than a snake’s outgrown skin. In certain contexts it is misleading and counterproductive. To promote further advancement of modern physics, for example, a number line based on egality, balance and equilibrium rather than ascendancy and subordination would likely prove more advantageous.

Chemistry and biology, both in their own ways, have confronted and allied themselves with equilibrium as a powerful force and near omnipresent occurrence. Physics has as well to a degree. It needs to go much further though. It needs to go to the heart of the matter. It needs now to take on the Western cultural construct of the number line and detail its limited veracity. It is not enough to declare the vacuum of space is not empty. The time is ripe to disclose the zero of mathematics is not vacuous as we have been led to believe.(8) And that the negative sign appended to a class of numbers does not make them inferior or subordinate in magnitude, just different in direction. The signs appended to numbers specify the direction of vectors only and lack information pertaining to magnitude. The numbers themselves specify magnitude only and lack any information pertaining to direction.(9) Western thought’s love of conflation has led us far astray.

 

(1) By a process of conflation Western thought has used the symbol zero (0) both as a number and a numerical digit used to represent that number in its system of numeration (a writing system or mathematical notation for representing numbers in a consistent manner.) As a digit, 0 is used as a placeholder in place value systems such as our decimal or base ten numeral system. The earliest certain use of zero as a decimal positional digit dates to the 5th century in India. [Wikipedia] Confusing the issue still more, the word zero, in common usage, can mean the number, the symbol, or the word for the number. The use of 0 as a number should be distinguished from its use as a placeholder numeral in place-value systems. Mandalic geometry has no argument with the placeholder zero but only with the number zero.

(2) A mathematical notation to represent counting is believed to have first developed at least 50,000 years ago. [Wikipedia

(3) Chinese authors had been familiar with the idea of negative numbers by the Han Dynasty (2nd century CE), as seen in the The Nine Chapters on the Mathematical Art, much earlier than the first documented introduction of negative numbers in the 15th century in Europe. As recently as the 18th century it was common practice to ignore negative results of equations on the assumption that they were meaningless, just as René Descartes did with negative solutions in a Cartesian coordinate system. [Wikipedia] Gottfried Wilhelm Leibniz was the first mathematician to systematically employ negative numbers as part of a coherent mathematical system, the infinitesimal calculus. Calculus made negative numbers necessary and their dismissal as “absurd numbers” quickly faded. [Wikipedia] The earliest known use of the plus sign (+) occurs ca. 1360; of the minus sign (-), 1489. [Wikipedia] For short summaries of the history of negative numbers see here and here.

(4) In ancient Egypt, guidelines were used in construction of the pyramids and earlier still. These were labeled at regular intervals with integers and a zero level or zero reference line was part of this system. [Source] This is not to be construed as use of zero as a number but rather as an engineering and geometric operating principle, one related, it would seem, to constructional or architectural integrity.

(5) The concept of zero as a number rather than just a symbol or an empty space for separation is attributed to India, where, by the 9th century AD, practical calculations were carried out using zero, which was treated like any other number, even in case of division. [Wikipedia] Zero as a number which quantifies a count or an amount of null size in most cultures was identified before the idea of negative quantities that go lower than zero was accepted. [Wikipedia] The number zero arrived in the West circa 1200, brought by Italian mathematician Fibonacci who found it, along with the rest of the Arabic numerals during his travels to North Africa. [Source] He doesn’t yet recognize zero to be a number on equal footing with the other nine numerals of the Hindu-Arabic notational system though. In 1202 Fibonacci (aka Leonardo of Pisa) uses the phrase “sign 0”, indicating it is like a sign to do operations like addition or multiplication. [Wikipedia] For a short history of the number zero see here. For a description of the various attributes of the number zero and rules pertaining to the use of the number zero in mathematics see here.

(6) We need to remember too that Europe during this time period was still reeling from the cultural shock of the newly introduced Hindu-Arabic numeral system which reached Spain in the 10th century (without zero). [Wikipedia] Until the late 15th century, Hindu-Arabic numerals seem to have predominated among mathematicians and moneylenders, while merchants preferred to use the Roman numerals. In the 16th century, they became commonly used in Europe, following on the heels of the printing press in the 15th century and undoubtedly related to introduction of that disruptive invention. [Wikipedia] In 1550 the first edition of Adam Riese’s Rechnung auff der linihen und federn… [Calculation by counter and pen…] is published. This work describes numerical calculations with Indian-Arabic digits in the vernacular German language rather than scholarly Latin. The intended readers are the apprentices of businessmen and craftsmen. The old Roman notational system has lost its stronghold, never to be regained.

(7) In the last few centuries, the European presentation of Arabic numbers has spread around the world and gradually become the most commonly used numeral system. Even in many countries which have their own distinct numeral systems, the European Arabic numerals are widely used in commerce and mathematics. [Wikipedia]

(8) Certainly there are concrete situational referents to which the number zero validly applies (I have no food to feed my child; I have no arrows in my quilt…). It is the validity of the abstract mathematical notion of zero which is in question here, particularly the version which treats it as a universal verity (There can be a natural subdivision of the universe that is completely empty, devoid of matter-energy and without space-time). 

(9) For an excellent discussion about scalars and vectors see here. (There are four sections to the discussion. This link leads to the first section. Click on through from there to the other three.)

Back to Basics: polarity and the number lines - II

 

(continued from here)

Taoism natively is more inclined to think in terms of 2-, 3-, and higher-dimensions than in 1-dimensional linear terms. Taoism has a number line analogue but an implicit one which is treated as an abstraction, more a distant consequence of real processes in the universe than a fundamental building block. Reference to higher dimensions is not fully relinquished even in this 1-dimensional abstraction. It is little used in isolation but features more prominently in Taoist diagrams of the analogue of the 2-dimensional Cartesian plane. In a sense this makes the Taoist number line much more robust than the number line of Western mathematics. Whereas from the narrowly focused perspective of Western mathematics the “number line” of Taoism might be viewed as “hyperdimensional” from the perspective of Taoism itself it is “dimension poor” and therefore degenerate.(1)

Regarding two distinct kinds of change, sequent and cyclic, Western thought is, in general, more concerned with sequent and Eastern thought with cyclic change.(2) Whereas the Western number line stretches out to infinity in both directions as in an orgiastic celebration of sequent change, the Taoist number line, exhibiting more restraint confines itself to some more realistic terminus of magnitude. It does so first because the taijitu (infinity analogue) of Taoism is non-polarized and exists at the center where Western thought places its “zero”. But also because it envisions change mainly in terms of cycles and invariably selects more realistic points of maximum and minimum extension than infinity.(3)

From the point of view of Taoism infinity though unbounded is also undifferentiated, existing in a non-polarized state of pure potential and potency whereas all differentiated states having polarity are limited in degree of potential and subject ultimately to constraint of extension.(4)

(to be continued)

 

(1) Taoism is a worldview based largely on relationships. From its very beginnings it likely considered a single dimension insufficient to express the full complexity of relationships possible. The I Ching, based largely upon the Taoist worldview, is a treatise which makes use of 64 hexagrams to correlate six dimensions of relationship. It may be the world’s earliest text on combinatorics and dimensionality. The true significance of this seminal work of humankind has sadly been too frequently overlooked.

(2) This is entirely a matter of degree and of preferred focus but has nevertheless profound consequences reflected in the resulting respective worldview of the different cultures. From an oversimplified bird’s eye view, Western thought regards significance best revealed by way of historical development through time experienced sequentially; Eastern thought, by way of recursive phenomena of nature expressed through cyclic time.

(3) This means also that there can be no single representative number line as there is in Western mathematics. Not at least if distances along the line are marked off in customary units of consecutive digits. For each specific Taoist number line unique complementary terminal maximum and minimum values must be selected. In the case illustrated above the value was chosen to be 20 so as to conform in terms of number of intervals to the Western number line segment shown (ten negative and ten positive intervals.) Had the value been chosen as 10 instead, the Taoist line would extend only from yin = 10; yang = 0 to yin = 0; yang = 10 and the number of intervals encompassed would have been a total of ten rather than the required twenty.

One way to surmount this difficulty in labeling described would be to number the intervals along the Taoist number line in terms of percentages rather than specific sequent intervals. Were this procedure followed every Taoist number line would extend from yin = 100%; yang = 0% to yin = 0%; yang = 100% with the central point of origin (corresponding to “zero” in the Western number line) labeled as yin = 50%; yang = 50%.

The two “zeros” that occur at the extreme ends of the Taoist line (yang = 0 to the left; yin = 0 to the right) should not be viewed as numbers but rather in a sense similar to that in which “zeros” are used as unit ten placeholders in our decimal number system. 

(4) In any case, labeling of the central origin point with either specific sequent intervals having identical absolute values or equal percentages (yin 50%/yang 50%) signifies the potential of the non-polarized and unbounded taijitu (infinity) to change by means of polarization into its polarized, bounded aspect. This process can be viewed also in terms of pair production (as understood by both Taoism and particle physics.)

© 2014 Martin Hauser

Back to Basics: polarity and the number lines - I

 

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(continued from here)

The number line is represented as a straight line on which every point is assumed to correspond to a single real number and every real number to a single point in a 1:1 mapping. It can be graphed along either a horizontal axis or a vertical axis. In the former case, negative is shown to the left of zero and positive to the right by convention. In the latter case, negative is shown below zero and positive above.

The number lines of Western mathematics and Taoism are similar in that both make constant use of the concept of polarity. That is where the similarities end. Even the manner in which this concept is used in the two worldviews differs.

For mathematics the basic polarity entrenched in its number line(1,2) is that between positive and negative. These two polarities are thought of as opposite and mutually exclusive. They are mediated by the concept of “zero”, a sort of no-man’s-land, the boundary between the two which belongs rightfully to neither. It is thought of as being in a sense an empty buffer zone between the polarities of positive and negative and functions not so much to balance or unify the two as to keep them apart from one another, or failing that to nullify both. Hence zero generally is treated as having no sign. Additionally, zero in the number line of mathematics and in the one-dimensional line of Cartesian geometry has neither magnitude nor preferred direction.(1)

It should come as no surprise that division by zero is not possible in Western mathematics when “zero” has itself been conceived as a kind of singularity.(2) This fact of itself should have indicated something amiss with the way “zero” has been conceptualized.(3) The “zero” of Western thought works well in the field of finance and most everyday practical fields of endeavor in general. Where the concept falls short is in the attempt to apply it indiscriminately in modern physics and certain other fields of science.

A close corollary here is the misapprehension of the actual manner in which mathematical signs (and hence all polarities) operate in the real world. In place of division by a non-existent “zero” Taoism advances the concepts of “polarization”, “depolarization”, and “repolarization”, all of which its “zero” alternative is fully capable of accomplishing. In a very real sense this “zero” alternative represents pure potential, both in the mathematical and physical senses. In the physical world it corresponds to the taijitu which may legitimately be considered pure potential energy as opposed to differentiated matter. Taoism in its way realized long before Einstein that the two are interchangeable.

With the preceding as background we are ready to consider next what the Taoist number line equivalent might look like.

(continued here)

Image: The Number Line. art: Zach Sterba/mC. writer: Kevin Gallagher/mC. [Source]

 

(1) This combination of parameters, comprising as it were a particular worldview, contrasts markedly with the worldview of Taoism. Unlike the demilitarized buffer zone that “zero” represents in Western mathematics, the taijitu of Taoism which occurs in its place both mediates between the polarities of negative and positive and gives rise to those same polarities repeatedly by the process of polarization. In place of the additive inverse negation operation of “zero” we have the operations of “depolarization” and “repolarization”. Having more in common with the worldview of Taoism than that embraced by Western mathematics, mandalic geometry treats the central buffer zone as a point of equilibrium, balance and potentiality which lacks neither content nor direction. It has in fact two directions, one centripetal, one centrifugal, which over the course of time can be alternatively chosen as a preferred direction repeatedly in cyclic fashion.

(2) This naturally brings to mind the informal rule of Computer Science holding that integrity of output is dependent upon the integrity of input. [GIGO (1960-1965): acronym for garbage in, garbage out.]

(3) This has far-reaching consequences, more importantly in the fields of modern physics and epistemology than that of mathematics. It is my contention that physics since the introduction of quantum mechanics and the theory of relativity (or “invariance” as Einstein preferred to refer to it) has been thwarted in its development by (among other things) too strict adherence to a limited concept of “zero” unsuited to its purposes. For Taoism and mandalic geometry “zero” is not a number but a polarizing function, itself without magnitude as commonly understood or permanent sign. Modern physics already partially leans toward this point of view but has not yet entirely shed the old outgrown “skin” that the number “zero” represents.

© 2014 Martin Hauser

Back to Basics: space, time and dimension

 

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(continued from here)

Throughout most of the history of Western thought space and time were regarded as independent aspects of reality. (1) Dimension was considered an attribute of both. Space was viewed as consisting of three independent linear dimensions pictured as being mutually perpendicular. Each of these spatial dimensions consisted of two opposite directions and movement in either direction was possible. Different spatial dimensions were independent of one another and space could be traversed in one, two or three dimensions at once. Time was viewed as having a single dimension progressing in a forward direction only. Most often sequent time was emphasized preferentially to cyclic time. (2) Material objects were viewed as distinct entities occupying space and time but independent of them. (3)

In the worldview of Taoism space, time and dimension were never viewed as existing apart from one another but as all intimately related. Furthermore, dimensions are viewed as interrelated, not independent of one another. In general neither space nor time is conceived in terms of single linear dimensions but as interrelated composites of two or more dimensions. Direction in Taoism has to do not exclusively with opposing pairs but also with interdependent polarities. Time like space is considered to be bidirectional. Cyclic change plays a role of at least equal importance to sequent change. Time in the cyclic sense develops in directions of both expansion and contraction. Both evolution and involution, activation and deactivation are all ever-present possibilities. All possible combinations of relationship are explored and the probable eventual future outcomes of each occurrence are always taken into consideration for purposes of understanding events and planning actions. (4)

Image: A simple cycle. Author: Jerome Tremblay, writeLaTex. (This is used here to illustrate in the most elementary manner possible the basis of cyclic change and cyclic time. The more complex nature of these will be elaborated more fully in future posts.)

(continued here)

 

(1) It was not until the early 20th century when Einstein introduced his Special theory of relativity, that space and time were fully integrated in a single concept, spacetime.

(2) Historically the "cyclic" view of time was of great importance in ancient thought and religions in the West as well as in the East. Attention was certainly paid to periodic recurring cycles related to the lunar month and, with the rise of agriculture, to the solar year. With the subsequent ascendance of the historically based religions however and in more modern times as technological achievements have taken center stage this acute awareness of periodicity and cyclic time has largely declined in the West.

(3) Leibniz believed that space and time, far from having independent existence, were determined by these material objects which he supposed were not contained in space and time but rather created them through their positioning relative to one another. (1,2,3) Leibniz, however, was familiar with the I Ching and it is unlikely that his thought processes would not have been influenced to some degree by the relational and relativistic Taoist worldview he found therein.

(4) In fact, the Taoist I Ching can be considered first and foremost an exhaustive compendium of the combinatorial probabilities of spacetime relationships in six dimensions. Its alternate title The Book of Changes attests to this. The fact that it has also been used over the centuries as a method of divination should not in the least detract from its more comprehensive value to human knowledge and epistemology.

© 2014 Martin Hauser

Back to Basics: the fundamental polarity

 

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For Taoism the fundamental polarity is that between yin and yang. This is usually presented along a vertical axis when considered in a single dimension. Yang is always above and associated with the South compass direction. Whenever two dimensions are treated simultaneously the yang polarity of the second dimension is presented along a horizontal axis to the left by convention and is associated with the East compass direction. (1)

For Western mathematics the fundamental polarity is that between negative and positive. When considered in context of one dimension this may be presented either along a horizontal axis (positive to the right) or vertical axis (positive up). When two dimensions are under consideration the horizontal axis is generally referred to as the x-axis and the vertical axis, the y-axis, both with directions labeled as noted above.

The two thought systems can be made commensurate in terms of mathematics. For instructional purposes here the Western conventions of direction are followed. (2) Also used here exclusively is the right-hand rule convention of three-dimensional vector geometry. Since the letter “x” is used to refer to the horizontal dimension and the letter “y” to the vertical dimension, the third dimension or “z” dimension must then necessarily have its positive direction toward the viewer. (3)

In physics, polarity is an attribute with two possible values. An electric charge, for example, can have either a positive or negative polarity. A voltage or potential difference between two points of an electric circuit has a polarity determined by which of the two points has the higher electric potential. A magnet has a polarity with the two poles distinguished as “north” and “south” pole. More generally, the polarity of an electric or magnetic field can be viewed as the sign of the vectors describing the field. (4)

(continued here)

Image: Yin yang. Public domain.

 

(1) Early Chinese cartography traditionally placed South above, North below, East to the left and West to the right. Though all reversed from Western presentations these are clearly conventional choices rather than matters of necessity. Many other ancient conventional associations of “yin” and “yang” have been preserved in Taoism. Most of the ancient traditional associations of “positive” and “negative” have long since been lost to Western thought.

(2) It should be noted that blindmen6.tumblr.com, this blog’s predecessor, presented instead the conventions used in the I Ching. That choice of convention has been abandoned here in favor of the Western convention in order to avoid unnecessary confusion.

(3) “Necessarily” only because the die has already been cast by choice of the directions of the horizontal and vertical axes and choice of adherence to the generally accepted right-hand rule. These, though, are all matters of convention. That should be kept in mind, if only because foresight suggests at a certain stage of development mandalic geometry may find it necessary to give the boot to some conventions and possibly as well to the use of any convention at all. Indeed the ultimate goal is a convention-free geometry though we are very far from that at this point in time.

(4) Although the text of this blog often equates the “yang” polarity with “+1” and the “yin” polarity with “-1” that is to be taken as a shorthand of sorts used instead of referring to “the positive sign of the vector +1” and “the negative sign of the vector -1” each time. Although doing so is most decidedly a convenience it is not strictly correct as these Taoist concepts actually refer to the entire poles of positivity and negativity. It is possible to use this shorthand only because to this point and for the foreseeable future the discretized number system of mandalic geometry requires only +1, -1 and 0 in terms of Western mathematics. It can be extrapolated to higher scalar values but will not be in the near time frame.

© 2014 Martin Hauser

Quantum Naughts and Crosses 13

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(continued from here)

This is the yz-plane with x = +1. It is the face of the mandalic cube that would be presented were the cube rotated 90 degrees clockwise.(1) Its resident tetragrams are formed from lines 2, 3, 5 and 6. Lines 1 and 4 are not included in the resident tetragrams because the x-dimension value is unchanging (+1) throughout this face of the mandalic cube. The z-axis (lines 3 and 6) is presented horizontally here, positive toward the viewer’s left. The y-axis (lines 2 and 5) is presented vertically, positive toward the top. The corresponding Cartesian triples are shown directly beneath the hexagram(s) they relate to.

The complementary(2) face of the mandalic cube, the yz-plane with x = -1, can be generated by changing lines 1 and 4 in every hexagram above from yang(+1) to yin(-1). Were we to do that and also view the resulting plane from a vantage point inside the cube we would then see a patterning of resident tetragrams identical to that in the plane above. The only difference apparent would be the substitution of yin lines for yang lines at positions 1 and 4.

We might have justifiably started out here by viewing the yz-plane with x = -1 from without the cube and followed with viewing the yz-plane with x = +1 from inside the cube had the die not already been cast. The problem with that attack given the present circumstances is that we have previously begun our consideration of the members of the the other two face pairs with the positive member from outside the cube. By preserving that consistency we end up with a jigsaw puzzle the parts of which can readily be fitted together to recreate the whole. Any inconsistency at this point can only result in failure.(3)

 

(1) This assumes that we begin with the reference face we have been using (xy-plane, z = +1) toward the observer seated at the bridge table.

(2) Mandalic geometry views opposite faces of the mandalic cube as being complementary rather than antagonistic or adversarial. This seems almost unnecessary to point out when the six planes that constitute the Cartesian cube are viewed as a single complex whole. There is a synergy of action simultaneously involving all component parts of the whole and there is an even greater degree of complex interactivity involving the component parts of the higher dimension mandalic cube.

The parts may indeed at times be in conflict or opposition with one another but at other times work together to create an effect. For a possible analogy think here of the constructive and destructive interference in which two or more wave fronts may participate. The I Ching, although it does not explicitly view the hexagrams and their component trigrams and tetragrams in the context of a geometric cube, nonetheless attributes these alternative and alternating reciprocal capacities to yin and yang and to all the line figures formed from them.

(3) This is much more than a simple matter of human convention. In this case we really are dealing with actual laws of nature, however cryptic and concealed they might be. This is not the right time to elaborate fully on what is involved here. Suffice it for now to point out that the approach we have chosen allows the three Cartesian and six additional mandalic dimensions to conform together with one another to certain combinatorial principles that nature demands they do.

For example, the three faces of the cube in which the hexagram consisting of six yang lines is found must fit together at a single point which forms one of the eight vertices of the cube by superimposing the three occurrences of this hexagram in the three different Cartesian planes at that single point. A similar requirement exists for all the other vertices of the cube as well. When all these various requirements are met all six faces can fit together snugly to form the cube. Were even just one of the requirements not satisfied the cube as a structural and functional whole would be unable to form.

We are talking here not simply about geometric shapes but about energetic physical phenomena as well. Ultimately this is not just a matter of composing a cube but of confronting the reality that dimensions fit together and force fields interact only in specific predetermined ways which we have no power to change. Moreover, this is just one indication that mandalic geometry describes more than literal locations existing in a topological space. It also corresponds in some sense to a state space, an abstract space in which different “positions” represent states of some physical system.

© 2014 Martin Hauser

Quantum Naughts and Crosses 12

image

(continued from here)

Here we have the xz-plane with y = +1 with all its resident hexagrams. This is the face the mandalic cube presents when we view it from directly above, with all of the planar Cartesian coordinate conventions maintained. The z-axis is positive toward the viewer and the x-axis positive toward the viewer’s right. (1)

The xz-plane with y = -1, which is to say the opposite or complementary face of the mandalic cube, could be viewed by simply lifting the roof face above off of the cube and then looking down at what has now become the floor of the cube. Once again, this ploy preserves the Cartesian coordinate conventions. Also, this is the only way to view this opposite face of the cube in such a manner that its tetragrams are all congruent to those in the hexagram patterning pictured above. (2)

We could always view this lower face from a vantage point outside the cube, as we did the upper face, but not without disregarding one of the Cartesian coordinate conventions, that of either the x-axis or the z-axis. We wouldn’t be breaking any laws of nature were we to do that, but some of us would, initially at least, be somewhat confused. My suggestion in that case would be, “Brush up on your Lewis Carroll (3) and in particular on his Looking Glass House.”

(continued here)

 

(1) It is good always to keep in mind that nature has neither respect for nor allegiance toward human convention. These artificially fabricated coordinate conventions play no fundamental role in any of the geometric descriptions found in this blog, but the pretense that they somehow do really matter must still be maintained to make consistent communication between human minds possible. I’m betting that more highly advanced alien civilizations and reality itself would view this whole formulation of things in a conventional manner as somewhat quaint.

(2) As used here the term “congruent” refers to the situation in which all the lines of the figures of concern (here all the tetragrams positioned opposite in the vertical y-dimension) are identical. It is the fact that the hexagrams in the opposite planes differ only in the value of y, which is to say in lines 2 and 5, that gives rise to this congruence. The hexagrams of the lower plane can be generated by substituting yin lines for the yang lines at positions 2 and 5 of the hexagrams shown in the diagram above.

(3) In addition to his literary works Lewis Carroll penned a good number of mathematical works under his real name Charles Lutwidge Dodgson as well, mainly in the fields of mathematical logic, geometry, linear and matrix algebra, and recreational mathematics.

© 2014 Martin Hauser