**Trigram Mnemonics**

**The trigrams are the key** to understanding mandalic geometry. The trigrams are the handles by which to grasp the concepts embedded in the I Ching and mandalic geometry. Learn how to manipulate them and you will be much more fluent in working with both these disciplines. The mnemonic devices summarized above are intended to assist in accomplishing this mastery.

**Taoism regards the eight trigrams** as a sort of family composed of a mother and father, three sons and three daughters. Our interest in the eight trigrams viewed as a family stems from the fact these relationships are useful in distinguishing, recognizing, and remembering the various trigrams and the hexagrams as well. It is this ease of recognition and mental juggling that makes the trigrams superior to the Cartesian triads for certain purposes. Central to this thesis is the manner in which the trigrams make complex logical relationships immediately and readily apparent to the mind.**(1)**

**The eight trigrams do in fact constitute** a set the members of which stand in mathematical relationship to one another and therefore are in some sense a family. Through their various relationships and interactivity they present a number of highly important symmetries and asymmetries. The sixty-four hexagrams build upon the trigrams, extending them and their complexity to six dimensions. At one level all of the hexagrams are formed from two stacked trigrams, one above the other. Significantly then, the key to recognizing and remembering the hexagrams is by means of the set of eight trigrams. Although the hexagrams are related to one another through changes in component lines and bigrams as well, the easiest access to the hexagrams and their myriad relationships is through recognition of the trigrams and their interchangeability.

** (1) It is these logical relationships** which are entrenched in mandalic geometry that make it a viable candidate for understanding what takes place at the quantum level of reality.

© 2014 Martin Hauser

**A new quantum logic: Trigram multiplication - III**

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**So what is the one simple rule** needed for 3-dimensional multiplication of the trigrams of Taoism and mandalic geometry? Put simply, it is this:

- A negative multiplier (- / yin) always causes inversion through a point whereas a positive multiplier (+ / yang) does not.
**(1)**

**Most of us learned this in school** as four separate though related rules:

**Reduced to bare bone essentials** the easiest way to accomplish the multiplication of trigrams is this:

*That is everything needed to perform multiplication of trigrams accurately and easily.***(2)**

**Now we are in a better position to understand**** why,** for example, the trigram as multiplier causes trigrams in the lower octants of the cube to move up and trigrams in the upper octants to move down. It is because the sole negative line in this trigram is the middle line which determines the second dimension, the up/down y-axis dimension. The negative line here produces inversion through the origin reference point of the y-axis when the trigram is used as a multiplier.

And the trigram as multiplier causes *any* trigram to move front to back or back to front and at the same time left to right or right to left because it has negative upper and lower lines which determine front/back (z-axis) location and left/right (x-axis) location respectively. Therefore inversion through the central origin points of both these axes is produced simultaneously when this trigram is used as a multiplier.

**If so disposed, you might just for the fun of it** (and to assure yourself this really works as described if you still have doubts) work out yourself why the other trigrams used as multipliers in the previous post accomplish what they were stated to do. Having done that once there will never again be a need to refer back to the complicated conflated method of multiplication described there. Just knowing how to multiply by minus one (-1) is all you will ever need.**(3)**

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**Image: **Cubic crystal system, modified to demonstrate positioning of trigrams and Cartesian triads. Generalic, Eni. “Cubic crystal system.” *Croatian-English Chemistry Dictionary & Glossary*. 31 July 2014. KTF-Split.19 Aug. 2014. <http://glossary.periodni.com>.

**(1) The caveat here is that this rule deals with** multiplication of the sign portion of vectors exclusively. Mandalic geometry is, in essence, nearly a scalar-free geometry, based almost entirely on the single unit measurement one (1). Specifically, regarding the context at hand, all eight trigrams, like the Cartesian ordered triads of which they are analogues, are composed solely of -1 (yin) and/or +1 (yang) throughout three different dimensions. But in other contexts the rule described would still apply to the sign portion of vectors of *any* magnitude in *any* geometry or thought system. As magnitudes are scaled up or down the effect on sign and direction is null. This aspect of reality is independent of magnitude.

**(2) It is necessary as always** to address each line level of the trigrams separately (lower line with lower line; middle line with middle line; upper line with upper line.) This is simply a matter of segregating the different dimensions. It is possible that mandalic geometry may address the subject of interdimensional multiplication in the future. That, however, is a topic of discussion for another day.

**(3) As an added bonus,** this method of multiplying the sign portion of vectors, is valid for any number of dimensions under consideration. That means, for instance, two or more hexagrams can be multiplied together easily by applying this one simple rule. And the calculation can be carried out in one’s own head without additional external paraphernalia. Try doing the same with Cartesian-type notation. Good luck with that!

© 2014 Martin Hauser

**A new quantum logic: Trigram multiplication - II**

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**When we inquire into multiplication of the trigrams** things get even more interesting as interactivity heats up throughout three dimensions. Just for fun and as a kind of experiment let’s start off this time with a mathematically conflated explanation of how this works. Once we have totally confused ourselves we’ll go back to step one to find out how it really works. Afterward the initial explanation may make some sense but only then. When the conflated version is the first and only one offered disaster, or at very least a deep down distaste for mathematics, must necessarily ensue.**(1)**

**Well that’s all as clear as mud!** There are still two more trigrams to go (**THUNDER** and **WIND**) but what’s the point. It’s not like we’re actually going to use any of this. You get the idea. This approach is worse than useless. It is positively off-putting. The irony here is that every statement made above is mathematically true. No information has been sacrificed in this presentation, just distorted beyond reasonable recognition. There are a lot of rules to remember and follow. But in all the confusion *the actual reason any of this works* has been disregarded and lost.**(2)**

**In the next module** we’ll look at the non-conflated version of how and more importantly *why* this all goes down. Remarkably there is only one simple rule to follow and we have all already learned it - - - long ago.

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**Image: **Cubic crystal system, modified to demonstrate positioning of trigrams and Cartesian triads. Generalic, Eni. “Cubic crystal system.” *Croatian-English Chemistry Dictionary & Glossary*. 31 July 2014. KTF-Split.19 Aug. 2014. <http://glossary.periodni.com>.

**(1) Be patient with the rest of this post.** It’s meant to confuse you. If it doesn’t I’ve done something wrong. Interestingly, this is a case where conflation ends up making the material appear rather *more* complicated than it actually is instead of less complex than it should be. Sometimes it works that way. Call it *inflation* if you like. The point is that regardless of whether information gets lost, distorted beyond recognition, or inflated to the degree that it is indigestible the end result is much the same - - - the truth of the matter ends up becoming incommunicable through the medium employed.

**(2) Though it is an effective moat** of sorts which like so many other similar constructions seems designed to keep the commoners out of the enchanting walled castle of mathematics that beckons enticingly just beyond the raised drawbridge.

© 2014 Martin Hauser

**A new quantum logic: Trigram Multiplication - I**

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**Here we begin to examine and compare** the Cartesian ordered triads and the Taoist trigrams**(1)** of the Book of Changes. We’re in totally new territory here. We’ve entered the third dimension. Whoopee! No need to be frightened. Just think of the trigrams and triads of the eight quadrants as the analogues in the cube of the ordered pairs and bigrams of the two dimensional square. The only difference here is that there is one more dimension to deal with. We’re already well on our way to our goal of six.

**Descartes uniquely identifies** every point in 3-dimensional space with his ordered triads. Mandalic geometry sets its eye on a somewhat smaller and self-contained goal. It limits its consideration to the range from minus one (-1) to plus one (+1) in each of the three dimensions. And because it is based upon a quantized or discretized geometry its entire universe of discourse is the unit cube as it morphs through its eight different identities in the eight octants of three dimensional Cartesian geometry.**(2)**

**Where Descartes identifies** the eight unique vertices of the *eightfold unit cube***(3)** with ordered triads (1,1,1; 1,1,-1; etc.) mandalic geometry identifies them with the eight trigrams of the Taoist I Ching. This is initially a matter of different notation but it grows into many other differences of great importance. Some of these have to do with the manner in which the human brain is better equipped to manipulate and interchange the Taoist symbols, a fact elaborated elsewhere in this blog. Along somewhat similar lines, the eight trigrams serve as excellent mnemonic devices, no mean accomplishment**(4)** and one which the Cartesian triads clearly fail to do.

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**Image:** Cubic crystal system, modified to demonstrate positioning of trigrams and Cartesian triads. Generalic, Eni. “Cubic crystal system.” *Croatian-English Chemistry Dictionary & Glossary*. 31 July 2014. KTF-Split.19 Aug. 2014. <http://glossary.periodni.com>.

**(1) ****Also known as bagua.**

The bagua are eight trigrams used in Taoist cosmology to represent the fundamental principles of reality, seen as a range of eight interrelated concepts. Wikipedia

The Wikipedia article, however, does not broach the hidden or implied mathematical interrelationships of the eight trigrams. It is those very geometric/logical interrelationships missing in the article that mandalic geometry is about. As a point of interest it should be noted that even the I Ching itself fails to disclose these logical interrelationships in a fully explicit and satisfying manner. They are there to be sure, but in implicit form only. Also note that though historically the trigrams have existed in two different arrangements, both described in the Wikipedia article cited, mandalic geometry is based exclusively on the one arrangement that is commensurable with Cartesian coordinates, namely the “Earlier Heaven” arrangement.

**(2) The only scalar quantities** mandalic geometry is concerned with are one (1) and a few scalar numbers found in nature like 2, 4, 8, pi, square root 2, square root 3, etc. It is the sign portion of vectors that is of more interest to mandalic geometry and which it is principally concerned with. The stated range of scalar 2 for mandalic geometry is much too modest. It’s bigger on the inside than the outside. Though the range of mandalic geometry is only scalar 2 as described in a three dimensional system, the system of the I Ching on which it is based is actually a six dimensional one so it is destined to squeeze a lot more in those eight unit cubes than one might expect. But not just yet. Though if you are eager to know more about the subject right now you *could* check out earlier entries in this blog where a lot has already been said regarding these matters.

**(3) This is not as far as I know an official geometric term.** I am using it as a kind of shorthand for “the larger cube composed of eight small unit cubes occupying all eight octants of three dimensional Cartesian geometry and mutually tangent at the single origin point of the coordinate system.”

**(4) Particularly as they do so also** in their alter egos as components of the 64 hexagrams. Recognizing and remembering how to distinguish eight closely related forms is difficult enough, and sixty-four all the more so. A mathematical wizard might be able to accomplish the 8-form feat using the Cartesian triads. I doubt that the same could be done with the Cartesian equivalent hexads that would be required for the hexagrams, other than possibly by a savant.

© 2014 Martin Hauser

**A new quantum logic - Bigram multiplication**

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**We are ready now to look more closely at** 2-dimensional multiplication. In terms of Taoist notation and philosophy this involves multiplication of the bigrams. The easiest and most direct way to do this is in simple chart form. With the previous few posts as preparation the risk of mathematical conflation**(1)** here has been largely nullified.

**The important thing** to keep in mind when doing these simple bigram multiplications is that first and second dimensions are kept separate.**(2)** The lower line of the multiplier bigram multiplied by the lower line of the multiplicand bigram gives the lower line of the product bigram. The upper line of the multiplier bigram times the upper line of the multiplicand bigram gives the upper line of the product bigram. Nothing else is required here. Bigram multiplication reduces to two easy linear yin/yang multiplications.

**In the multiplication charts below** the bigram in the pink rectangle at the left is the multiplier and the bigrams in the blue rectangles across the top the multiplicands. The product of each individual multiplication is found in the white rectangle where the two intersect. Incidentally, these charts work for division as well. I’ll let you figure out how that is done.**(3)**

**The top chart of these four** shows bigram multiplication with the identity bigram of Quadrant I as the multiplier.

**The second shows** multiplication with the bigram of Quadrant II (which produces inversion of the horizontal dimension only) as the multiplier.

**The third shows** multiplication with the inversion bigram of Quadrant III (which produces inversion of both vertical and horizontal dimensions) as the multiplier.

**The fourth shows** multiplication with the bigram of Quadrant IV (which produces inversion of the vertical dimension only) as the multiplier.

**Now that we fully understand** where we’re coming from we can restate these results in a manner that involves some mathematical conflation.**(4):**

**Any bigram multiplied by itself** gives the identity bigram as product.

**Any bigram multiplied by its inversion (polar opposite) **gives the inversion bigram as product.

**The identity bigram multiplied by any other** gives back that same other as product.

**The inversion bigram multiplied by any other** gives back the inversion (polar opposite) of that other as product.

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**(1) See footnote (1) here for a definition of conflation.**

**(2) Recall here that in the bigram** the lower line specifies the first dimension or x-axis coordinate and the upper line the second dimension or y-axis coordinate.

**(3) Hint: 2-dimensional bigram division reduces to** two simple linear divisions and you already know how to deal with the vector forms of the number one (1). Okay, I’m not actually being condescending here, just slightly facetious.

**(4) Not a very good thing to do** as has been pointed out previously. But this will hopefully begin to make clear how and why mathematics goes bad. In this particular case no actual information has been lost. It has merely been twisted and distorted into a form which shifts focus from what is actually taking place to a specialized description which forces the reader to deal with issues of translation rather than matters of true importance. Welcome to the world of mathematical conflation. May you all be spared further exposure to it. (Not very likely though. All of us are occasional innocent perpetrators of conflation though I sometimes wonder whether at least some mathematical language isn’t intended mainly as a barrier to dissuade the *peasants* and other* commoners* from entry. That may just be my paranoid self talking though.)

© 2014 Martin Hauser

**A new quantum logic - Numbers and dimension**

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**We need to understand that numbers** possess dimension. Or let me rephrase that - *numbers exist in all dimensions* and take on the special character of the particular dimension they are inhabiting. Descartes uses both 2-dimensional numbers, his ordered pairs, and 3-dimensional numbers, the ordered triads, to uniquely identify all resident points of the plane, the 2-dimensional version of his coordinate system, and of the 3-dimensional version respectively.

**Mandalic geometry does something similar **using Taoist notation. But it is more interested in how different dimensions relate to one another than in merely locating points. For mandalic geometry it is dimensions that are real while points are just the evanescent phantasmagorical embodiments of their interactions. In other words, points are the temporary expression of the intersection of various dimensions.**(1)** Points are entities existing in spacetime in a manner similar to how fictional characters exist in books and films. As such they are subject to frequent whims and alterations of various sorts and degrees.**(2)**

**The bigrams of the I Ching can be viewed** in a manner similar to the ordered pairs of Descartes and treated similarly though with a few specific exceptions.**(3) **Fundamentally they refer to systems of two dimensions or to 2-dimensional parts of higher-dimensional systems. With all systems, but particularly with nonlinear systems (those having dimension greater than one), consideration must be given to units of measurement, point of reference and orientation. These three specifications are all required to completely and uniquely identify a situation or event.**(4)** Those wishing to know more about *dimensional context* can find a good article dealing with it in Wikipedia here.

**The first or lower line of a bigram** corresponds to the x-axis coordinate of Cartesian geometry; the upper line to the y-axis coordinate. Like the Cartesian ordered pair, the bigram is a 2-dimensional or planar number. Multiplication of bigrams, however, can be treated as multiplication of the individual linear parts. There is a multiplication of the first dimensional part to consider and a multiplication of the second dimensional part. The end result is another bigram which may or may not differ from the original two. This depends solely upon which of the four bigrams are used as multiplier and multiplicand. When both or one of these is the identity operator of bigram multiplication (i.e., +1 or yang in both dimensions of the bigram) the result will be the other initial bigram. As is so with linear numbers, multiplication of like vectors yields a positive vector while multiplication of unlike vectors yields a negative vector.**(5)**

**This multiplication of planar numbers** just represents a dimensional evolution of the multiplication of linear numbers. In fact, as described above, it involves two separate linear or one-dimensional multiplications. Mandalic geometry, concerned as it is with Planck-scale units, deals almost exclusively with the scalar number 1 and its *natural derivatives***(6) **unlike Cartesian geometry which deals with scalars ranging to infinity. It is the sign portion (+/-; yang/yin) of vector numbers that is of most interest to mandalic geometry. Both Cartesian and mandalic geometries recognize two fundamental signs which represent the positive and negative polarities of nature. Descartes uses signs only for the purpose of locating points in space. Mandalic geometry uses signs also as a jumping off place for an investigation into the nature of dimension and how dimensions relate to one another both spatially and temporally.

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**Image:** Aid to defining fractal dimensions. By Brendan Ryan.Nazlfrag at en.wikipedia [Public domain], from Wikimedia Commons

**(1) Euclid proclaimed points to be dimensionless.** It is the view of mandalic geometry that they participate in *all* dimensions. This may be a matter only of differing perspective. But small differences in perspective snowball eventually into varying worldviews which may differ considerably. Physics shares many of the perspectives held by Western mathematics as historically this was where it sourced them from. Though all of these viewpoints may be true in the mathematical sense they are likely not uniformly valid in the manner required by physics to satisfy the stricter definition of scientific truth. Physics would do well, I think, to do some of its sourcing in the vast storehouses of biology and chemistry. See here for an excellent short discussion of the differences between three categories of *truth* - subjective, deductive, and inductive (or scientific truth).

**(2) This may be true of material particles as well.** Like points, different particles possess a dynamic and energy specific to themselves and as with points are interchangeable one to another. The interchangeability of points is one of the thought forms expressed in both analytic geometry and the Taoist I Ching. The one graphs all of the varied changes decreed by the equations of mathematics; the other is a compendium or microcosm of all possible changes and combinatoric variants in a universe of six dimensions.

**(3) In the context of higher dimension systems** found in the I Ching and mandalic geometry the bigrams are not restricted to a single plane as are the Cartesian ordered pairs but are capable of entering and dwelling in any of the planar structures which may occur within the context of those disciplines. This will be explained fully in posts to follow. For now it is only necessary to note that whereas Cartesian ordered pairs are confined to and in a sense define a particular plane, Taoist bigrams are transposable among the many different planes of any number of dimensions and are themselves defined by the dimensional context in which they exist. At some level this is related to the subject of fractals but in the interest of clarity and parsimony we’ll save that discussion for another time.

**(4) For Descartes a point is** an object with no properties other than location; a space is a collection of locations; spaces can be characterized by their degrees of freedom. For mandalic geometry none of these three is separate from the others. That being so, none exists in itself or by itself alone. The most general of the three is degrees of freedom which amounts to the same thing as *dimension*. Then dimension determines both location of points and the spaces in which those points are embedded. None of this is static. All is fluid, constantly in process of change. A point that changes cannot be determined by location in space alone. Not even by location in spacetime except as a moving, changing point of reference. *That* is what the graphs of analytic geometry and the changes of The Book of Changes actually describe - *changing* location and relationships through spacetime.

** (5) This is to be understood as** a general principle which will be found to hold true throughout all dimensional contexts. In a sense it is a simple mathematical tautology. If +1 times +1 equals +1 and -1 times -1 equals +1 in a single linear dimension then the same holds true in a set of any number of linear dimensions. Also if +1 times -1 equals -1 and -1 times +1 equals -1 in a single linear dimension then the same holds true in a set of any number of linear dimensions. The qualifier here, if any exists, is that mandalic geometry views every dimension as a set of linear dimensions in relationships specific to that particular dimension.

**(6) As used here the term natural derivatives** refers neither to natural numbers specifically nor to the derivatives of differential calculus. It refers rather to those numbers, such as pi, square root 2, and square root 3, which occur in nature. Some of the natural numbers do qualify here as well - - - the integers 2, 3, 4, 8 for example as well as certain others.

© 2014 Martin Hauser