**Family Relationships - III**

[Click here to enlarge image.]

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**In the operation of multiplication** the trigrams are fully commutative. The order of multiplication does not alter the result so does not matter.

**There are only three significant differences** between the Indo-Arabic multiplication and the Taoist multiplication seen above. The most obvious difference is the difference in notation. Less obvious perhaps is the fact that the first is a linear multiplication involving a single dimension while the second is cubic, involving three dimensions. The third difference is that the first multiplication shown involves only scalars (magnitudes without direction), which here differ from one another, while the second involves vectors (magnitude and direction). In the case of the vector trigrams the magnitudes involved are all identical, namely magnitude one(1) and the direction is either yin(negative) or yang(positive).

**Taoist geometric notation patterns** (e.g., bigrams, trigrams), like their Cartesian counterparts (ordered pairs, ordered triads), always encode both magnitude and direction and so are multi-dimensional vectors. The critical difference is that whereas members of Cartesian ordered pairs and triads can assume any scalar magnitude, the Taoist geometric entities always have a magnitude of exactly one(1) in all dimensions. This is true of the tetragrams and hexagrams also.**(1)**

**Multiplication using Taoist notation** is commutative regardless of the number of dimensions involved in the multiplication. This means, for example, that it holds true for bigrams, tetragrams, and hexagrams as well as for trigrams. Also, Taoist notation multiplication remains fully commutative regardless of the number of members in the series of multiplication operators. This last is true of Indo-Arabic notation as well but the significant difference is that the former encodes and preserves both magnitude and direction whereas the latter encodes and preserves only magnitude.**(2)**

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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) If this strikes you as somewhat limiting,** remember that mandalic geometry describes events at the Planck scale and the most important magnitude is the unit magnitude. It is entirely possible to scale up from there in a manner similar to Cartesian coordinates but at this early stage of development of mandalic geometry, and likely for a long time to come, the unit magnitude will prove sufficient to our purposes.

**(2) Cartesian notation multiplications, **of course, encode and preserve both magnitude and direction as well but are cumbersome in usage and difficult for the mind to manipulate without employing external accessory calculating paraphernalia. A corollary to this last idea is the fact that historically certain important relationships among the various operators have been overlooked conceptually due to the difficulty that exists in discerning them in the first place.

© 2014 Martin Hauser

**Please note -** This is a beta post meaning that the content and/or format may not yet be in finalized form. A reblog as a **TEXT** post now will contain this caveat to warn readers to refer to the current version which appears in the source blog. This note will not however appear in a **LINK** post which itself accomplishes the same. Thank you for not removing this statement and for your understanding. :)

**Family Relationships - II**

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**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.

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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) 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.

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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) 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

**Tabular Summary of Trigram Multiplication**

**The trigrams and multiplications in the charts above** correspond to three-dimensional locations of these trigrams and interactivity in their movements in Cartesian 3-dimensional space. For an explanation of how and why these multiplications work as they do and to better understand the geometric implications of these three-dimensional multiplications refer to the diagram and post here.

**Note that all these trigram multiplication operations** are commutative. This would, of course, be anticipated as trigram multiplication just involves single dimension multiplications of real numbers throughout three different dimensions, though using the Taoist notation which admittedly is unfamiliar to most of us. The same could be accomplished with Cartesian notation but considerably more difficulty as regards manipulation and remembering. The trigrams as well as being multiplication operators here serve as their own mnemonics. More on that in the post to follow.

**As one investigates these charts one will begin to discover** certain regularities and symmetries. For example, any trigram multiplied by itself gives as the result; any trigram multiplied by its complement gives as the result. There are many more such awaiting discovery. More will be said about these important relationships in future posts as they begin to build a logical system ripe for use in quantum theory.

**Before they will be ready for that though** we need to extrapolate the relationships to six dimensions, which is to say to the hexagrams. We also need to expand the subject of composite dimension which has been only briefly touched upon before. Cartesian coordinates turn out to be the composite forms of higher dimension coordinates. In the absence of that understanding they are merely degenerate falsehoods when they speak about space and time. And into this mix we must as well throw the notions of potentiality and actuality and the workings of probabilities. Every good quantum dish needs some of those to spice it up.

© 2014 Martin Hauser

In answer to obywan’s question:The physicist Richard Feynman called the formula it is derived from “one of the most remarkable, almost astounding, formulas in all of mathematics”.

I’m sure you’ve seen all the symbols in the identity before, but isn’t it weird how they are not…

(via theguybehindthescreenn)