**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**

(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**

(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**

**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**

(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**

(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

**Quantum Naughts and Crosses 11**

(continued from here)

**In this module we will engage in some play of the imagination **with the xz-planes. In the post to follow an xz-plane with all of its resident hexagrams will be presented. **(1)** Note first that the tetragrams shown above are composed from lines 1, 3, 4 and 6 of their hexagrams. Lines 2 and 5 which determine the vertical y-axis are constant throughout each of the two xz-planes (yang for y = +1; yin for y = -1) and are suppressed here to demonstrate the tetragrams more clearly. The ***** symbol is used here as a placeholder for a Cartesian coordinate, in this particular case the y-coordinate value.

**There are 16 possible tetragrams (2^4 = 16).** All appear in each of the six faces of the mandalic cube but are formed from from different lines of the hexagrams and occur at different locations in their respective planes except for complementary faces in which the formation and placement are identical in both planes. That is why tetragrams of complementary faces can be shown with a single diagram using placeholders for the two lines that do differ in corresponding hexagrams of the two planes.

**If in the vertex hexagrams we were to suppress** lines 4 and 6 also, replacing the placeholders at line 3 by yang lines, we would then see the resident trigrams of the xz-plane having y = +1. If instead we replace that same placeholder with yin lines we would see the resident trigrams of the xz-plane having y = -1. These are the trigrams that interact to compose the hexagrams of the xz-planes, four different trigram types in each of the xz-planes. The trigrams complementary to the four in one of the xz-planes are of course found in the other.

**If we exchange yang lines for the placeholders** in lines 2 and 5 above we arrive at the hexagrams of the xz-plane having y = +1. If instead we exchange yin lines for these same two placeholders we get the hexagrams of the xz-plane having y = -1. This is just like Cartesian coordinates but with an important twist, namely the Cartesian forms are derived from the more inclusive **(2)** mandalic forms by compositing. **(3)** From the higher perspective of mandalic geometry Cartesian dimensions are composite dimensions actualized from mandalic dimensions of potentiality.

**There is clearly a fixed unpremeditated but necessary order (4)** to placement of the hexagrams and all their components (i.e., lines, bigrams, trigrams, tetragrams, pentagrams.) Still, the holistic interrelated dynamic contextual placement of hexagrams allows for a considerable number of different spatiotemporal changes or processions. The large number of degrees of freedom present at all steps (quanta) along the way is quite naturally related to the perceived arrow of time. More about this will be said in future posts. **(5)**

**At this point it would be good to recall** that a hexagram represents a specific manner of intersection of six different dimensions of potentiality. It is like an evanescent will-o’-the-wisp, having only a fleeting existence but recurring persistently in spatiotemporal terms. A hexagram is in essence a snapshot of frozen potential spacetime. Its existence is not a fundamental but a derivative one, changing as its component parts coalesce, dissociate and commingle once again.

(continued here)

**(1) The xz-planes in the context of the mandalic cube **are the two planes directly below when the cube is viewed from above. In order to preserve Cartesian coordinate convention the plane with y = +1 (yang) is to be viewed from outside the cube whereas the plane with y = -1 (yin) is best viewed from a conceptual vantage point within the cube. Of course the conceptually adept can view this plane from above and outside the cube as well but for the rest of us the intervening xz-plane with y = +1 may interfere with adequate visualization, even in dealing with a non-solid transparent cube as we are, at least initially until one grows accustomed to doing so.

**(2) More inclusive in terms of both** information content and number of spatiotemporal possibilities or degrees of freedom permitted.

**(3) The term “compositing” as used here** refers to the mechanism by which a composite Cartesian (or Cartesian-like) dimension (dimension of actuality) is derived from two or more higher mandalic dimensions (here referred to as “dimensions of potentiality”.) Take note also of the related more general usages of the term as in graphics and video terminology.

**(4) The term “order” here does not refer** to any particular linear sequence or arrangement of succession of hexagrams as none such exists in the multidimensional probabilistic context of mandalic geometry. Rather it refers to a condition of methodical arrangement among component parts such that proper functioning is achieved by both parts and whole in a synchronous manner.

** (5) For the time being suffice it to say** that our linear notion of time is as faulty as are our linear notions of space and number.

© 2014 Martin Hauser

**Quantum Naughts and Crosses 10**

(continued from here)

**Previously we looked at the tetragrams of the xy plane** in context of their respective hexagrams with placeholders for the third Cartesian dimension (z) at lines 3 and 6 while imagining these lines to be yang lines and z therefore equal to +1. Here we demonstrate these same tetragrams in context of their respective hexagrams, shown in their entirety, and with z = -1 (yin lines at lines 3 and 6.) **(1)**

**Note that the vertex hexagrams differ from one another** horizontally and vertically by two lines (or Cartesian units of measurement), diagonally by four lines. The edge center hexagrams differ from those in the adjacent vertices in either direction by one line. The hexagrams of the center of the square (or face of the cube) can change to one of the hexagrams found in the small squares adjacent horizontally and vertically by change of a single line and to the hexagrams found adjacent diagonally in the vertex squares by change of two lines.

**The line changes corresponding to position changes described** are precisely those required of a coordinate system commensurate with the Cartesian coordinate system. What is important to note here is that the Taoist notation conveys more information than does the Cartesian. It is a more generally inclusive system in terms of spatiotemporal parameters and allowing of greater degrees of freedom in movement through spacetime and a correspondingly greater number of possibilities. This means it would likely be capable of describing quantum states and changes more precisely than the Cartesian notational system.

**We will soon see also** that this more expansive coordinate notational system of mandalic geometry does not require introduction of imaginary and complex numbers nor the method of quaternions to accomplish what those abstruse mathematical creations of earlier centuries do and without their folie de grandeur. It does so using ordinary multiplication with its simple commutative operations that any reasonably bright third grade student could manage once the notation was explained and mastered. **(2)**

(continued here)

**(1)** **Although we are currently considering** the six planes of mandalic geometry individually, more or less in isolation, it must be kept in mind that these planes exist in context as the six faces of the mandalic cube. They are all intricately related one to another as well also to and through the central origin point of the cube which does not itself appear in any of the six planes. Whereas in the post seen here we took the xy-plane with z = +1 as our reference plane and viewed it from without the cube as though the cube were set upon a bridge table with the reference plane facing us, we now are viewing the complementary face of the cube (xy-plane with z = -1) from a vantage point within the cube. The sole reason we are doing this is to maintain the convention of Cartesian coordinates which considers the positive direction of the x-axis to be always to the right of the origin of the axis. This is not a law of nature but purely a human convention devised for consistency and ease of communication.

**Were we as observers** to assume a position on the opposite side of the bridge table we could view this same plane from outside the cube but no longer with the Cartesian convention described. The positive direction of the x-axis would now be to the left of the origin. Also the directions of the z-axis found in the Cartesian convention would have changed to the opposite. In blindmen6.tumblr.com the mandalic cube was presented using this convention of ancient Chinese cartography which is the coordinate convention used in the I Ching as well. It is a little like entering Lewis Carroll’s Looking-Glass House. It is important to understand though that nothing has changed except our own perspective. Reality meanwhile goes on unchanged as ever. Nature has no need of a conventional coordinate system of any flavor or variety. It simply is what it is. This is not to say that reality does away with coordinate systems entirely but that it uses any and all in whatever manner necessary for continued existence. It is human convention that nature has no need for.

**(2) The difference in the two approaches** is, of course, both notational and contextual. In mandalic geometry each composite dimension carries with it two dimensions of potentiality or possibility. Ordinary Cartesian dimensions, lacking these, require the invention of imaginary and complex numbers or something similar. The wonder here is that the mathematics community has so long accepted the artificial ex post facto geometric explanation of the imaginary number *i* as being related to rotation and the physics community has swallowed whole this simple-minded explanation as somehow related to the strange goings-on in the subatomic quantum realm. We need to remember there is a decided difference between an orbit and an orbital. If there were not there would have been no need to invent the new term for use in quantum theory in its early days.

© 2014 Martin Hauser