**Family Relationships - II**

(continued from here)

**Two helpful ways to view the trigram family relationships are:**

(1) as single planes or cube faces, of which there are six, all having four trigrams or vertices, each of which is shared by three planes equally; and

(2) as sets of three planes which intersect at a single vertex, of which there can be eight variant sets, the most important of which are the two sets in which the three component planes mutually intersect either at the **EARTH** vertex or at the **HEAVEN** vertex, these two being the trigrams which encode the primary polarity of nature for 3-dimensional constructs.

**No plane in these latter two sets** is shared by both sets. However the two sets do share six of twelve lines or edges of the cube and all trigrams or vertices other than **EARTH** and **HEAVEN** themselves, these two being diametrically opposite trigram points through the center of the cube. The logic of construction seems determined to keep such diametrically opposed trigrams, complete inversions of each other, as far apart as possible.**(1)**

**The relations described above** will prove of enormous significance for point interactions in mandalic geometry and particle interactions in quantum physics. These subjects will be addressed in future posts.

(to be continued)

**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) It is difficult here to avoid comparison** with particle-antiparticle annihilation from entering the mind. Are we being directed by the manner of interactivity of these multi-dimensional numbers toward a deep down feature of reality existing even at Planck scale? For the numbers at least maximal separation of completely opposite entities is clearly obligatory.

© 2014 Martin Hauser

**Family Relationships - I**

**We look here at some of the regularities** that can be observed in the relationships between members of the family of trigrams. These involve both similarities and dissimilarities, symmetries and asymmetries. Refer to the chart summarizing the family of trigrams found here in following the rest of this post.

**The trigrams EARTH and HEAVEN**, mother and father, are exceptional in that they are the only trigrams having all three of their lines alike: mother three yin lines; father three yang lines. Mathematically speaking, there can be only two such trigrams as there are only two species of lines, yin and yang or negative and positive. The deeper significance here is that both these trigrams result from intersections of similar vectors (either yin or yang) in three dimensions but are dimensional tri-vectorial opposites**(1)**.

**The other six trigrams have either** one yang and two yin lines or one yin and two yang lines. The three sons each have a single yang line: the first or lowest line in the first son; the second line in the second son; and the third line in the third son. Similarly, the three daughters each have a single yin line: the first line in the first daughter; the second line in the second daughter; and the third line in the third daughter. The key to distinguishing among these six is to note which of the three lines differs from the other two lines and whether the line that differs is yin or yang.

**Understanding how the eight trigrams** are differently structured makes recognition of the individual trigrams easy and immediate. Once recognition of the trigrams is accomplished, identification of the sixty-four hexagrams will follow naturally with little or no effort. The hexagrams are composed of two trigrams stacked one above the other and fall into logical higher dimension families based upon their component trigrams. This makes the hexagrams nearly as easy to distinguish from one another as are the trigrams.**(2)**

**Although each of the sixty-four hexagrams** is given a distinguishing name in the I Ching it is not necessary to learn these because mandalic geometry makes essentially no use of them, referring to each hexagram rather by the trigrams composing it.

(continued here)

**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) Or complements as Taoism prefers** to think of them. They do sometimes work against one another and appear to be *misaligned*, but in the larger scheme of things they are always in the alignments that bring about ongoing reality which requires both constructive and destructive phases. In Taoism there is no *up* without a corresponding *down*.

**(2) If you think any of this superfluous** imagine how differently Western mathematics and physics might have developed had Descartes formed his ordered triads into a family and given its members distinguishing names. But as that never happened in our reality I suppose it wasn’t meant to be. The wonder is that Leibniz who followed Descartes by half a century and who both knew of the I Ching and had a deep interest in combinatorics never thought to follow a course such as this. In his defense though it is true that his mind was largely preoccupied with creating calculus and the binary number system. He also became one of the most prolific inventors in the field of mechanical calculators.

© 2014 Martin Hauser

**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**

(continued from here)

**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)**

(continued here)

**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**

(continued from here)

**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.

(continued here)

**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**

(continued from here)

**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.

(continued here)

**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. 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**

(continued from here)

**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.

(continued here)

**(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