A new quantum logic: An interesting question raised

 

Did we or did we not uncover a broken symmetry in the previous post?

What we see there grows out of the asymmetry of the fundamental polarity (yin/yang) which arises at the 1-dimensional stage that breaks undifferentiated wholeness in two. That asymmetry presents again at the 2-dimensional stage of the bigrams but I think does not yet give there the appearance of a broken symmetry. So what makes this very same asymmetry appear to be a broken symmetry at the 3-dimensional stage of the trigrams? Can there be something about broken symmetries in general that we are misinterpreting? Are they sometimes due as much to bias of the conceptualizing mind as to the actual workings of nature?

I don’t have the answers to the questions raised here. But I reserve the right to have schlocky ideas. I have my fair share of them. Frequently the good ones grow out of the schlocky ones. All things are grist for the mill. Have you not noticed?

Vishnu

I need to think on these matters more. 

Image: Vishnu holding the lotus, also sitting on it and wearing a lotus-bud crown. See page for author [Public domain], via Wikimedia Commons

© 2014 Martin Hauser

A new quantum logic:

Symmetry and asymmetry - II

 

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(continued from here)

Let’s begin our advance by assailing the group of rotations of the cube. I’m going to start off with what amounts to a spoiler here so you will have plenty of time to chew on it, digest it, and expectorate it if you so please.

It is claimed that the rotations of the cube are not commutative. True enough. But my response to that is, “So what?” You want commutability, forget rotation. Go with multiple reflections across dimensions. It will get you there without all the fuss and bother of rotation, and the operations are fully commutative.(1) The way to go is with inversions across multiple dimensions. Physics must choose from available mathematics the course that provides the best result. Damn tradition if it prevents access to the truth. Quid pro quo. Something for something. The return is worth it.(2)

Let’s consider some actual operations of reflection now and see just where that leads us. For this we will have to refer back to what we have previously learned about the multiplication of trigrams.(3)  All described multiplication operations are commutative, making them easy to perform. Refer to the diagram above for trigram positioning and correspondences between the Taoist trigrams and their Cartesian ordered triad analogs.

Say we want to move WIND to FIRE. Instead of rotating by 180 degrees in either direction we’d do better by inverting through two dimensions, the horizontal and vertical. As a two-step process this could be performed in either direction, reflecting first horizontally then vertically or the reverse, first vertically then horizontally. The result in both cases is the same and looks quite similar to a rotation through 180 degrees. But there is an easier way to accomplish the same end by a concurrent double inversion through the two dimensions. This is clearly a more direct route though in quantum energetic terms it would require more energy input to bring it about. The trigram that can accomplish the feat is MOUNTAIN. It is the only trigram that can do so because it is the only one of the eight in possession of the two necessary inversion operator lines (yin) at positions 1 and 2 and the identity operator line (yang) at position 3. Take note that it is this exact combination of parameters that defines the trigram in 3-dimensional space.

To reverse the process and go from FIRE to WIND requires only reversal of the order of reflection or multiplication. Multiplying FIRE by MOUNTAIN gives WIND as the result. Fully commutative as promised. As a matter of fact, multiplying any two of these three trigrams will give the third as the result. Multiplying all three of them together will give the remaining vertex HEAVEN, the fourth vertex of this particular face of the cube, as result. Pushing this still further, multiplying any three vertex trigrams of this face of the cube (or further still, of any of the three faces of the cube in which the sole triplicate identity trigram HEAVEN occurs) will yield the fourth as the result and multiplying all four vertex trigrams together gives HEAVEN as result. For the other three faces of the cube - those which contain EARTH - multiplication of any three vertex trigrams also yields the fourth as result, but multiplying all four together gives HEAVEN not EARTH as might be anticipated. Have we stumbled upon a broken symmetry here and if so what caused it? The plot thickens.

Find all that difficult to believe? Try it for yourself and prove whether or not it is true.(4)

(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) The strange thing about this is that when all is said and done the end result looks exactly like the result of a rotation or rotations would. It is only the way one gets there that is different. So do we follow the difficult route or the one that is easier? The caveat here is that this applies only to discretized Planck scale physics. We are not talking here about operations in the continuous space or at the scale of magnitude that geometry generally deals with. Mandalic geometry is not concerned particularly with those. These thoughts are being offered in the hope of stimulating a new vision and direction in a physics that has been stalled for awhile due in part I believe to misappropriation of mathematical ideas.

(2) I deem it highly unlikely in any case that quantum jumps are based upon rotations of any kind. What would the particles be rotating through? Empty space? There is none. The rotation takes place only in our minds. It is conceptual. If it gets in the way of truth the solution is to discard the culpable concept, plain and simple. Reflection needs no intervening space. Reflection trumps rotation. Leave the concern for rotation with the pure mathematicians where it belongs, along with their imaginary and complex numbers and quaternions. And while we’re on the subject, wouldn’t it make sense that a discrete mathematics and discrete geometry would better apply to quantum physics than a continuous mathematics and geometry? The very reason the operation of reflection requires no intervening space is because it is based from ground up on interoperability of these discrete mathematical and geometric entities which do not presuppose existence of continuous space. Regarding the meaning of reflection refer here (1,2,3).

(3) Recall that the trigrams encode information about magnitude and direction in three dimensions. Though magnitude appears severely confined from the perspective of Cartesian geometry it suffices quite adequately for the purposes of mandalic geometry. The latter requires only the magnitude of scalar 1 to specify Planck scale position. It is the direction part of the trigram vectors, the yin/yang gradients, we are mainly concerned about.

(4) Confusing as all this might sound it is actually quite simple and straightforward. The significant point to understand here is that these reflections and multiplications can be thought of as mapping particle exchanges and interactions involving changes in quantum numbers or states. Mandalic geometry is attempting here something Cartesian geometry never thought to do. Complex numbers and quaternions attempted the same but clumsily and with only partial success. And there’s that thing about the commutative property which mandalic geometry excels at and which quaternion geometry totally lacks. I understand that I may in part be reinventing the wheel here but all the same I think something new of value has been added. If you can detect an error in any of this please let me know. I’m still learning here myself. To be frank, at the time of the writing of the previous post I wasn’t expecting to come across a broken symmetry quite this soon - - - if we have, that is.

 © 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. :)

A new quantum logic: Symmetry and asymmetry - I

 

Isfahan Lotfollah mosque ceiling symmetric

(continued from here)

Physics considers symmetries quite important as they often carry in concealment laws of nature. Asymmetries are important too. These are at times broken symmetries which physics also makes much of and which can instruct us in important ways as well. We will look at what appears to be one of these broken symmetries and attempt its demystification. We will be describing what must surely be one of the most basic asymmetries of dimensional numbers, one which involves the most fundamental polarity of nature. Could this be one of the earliest broken symmetries? And could dimensional numbers be the fount of symmetry, asymmetry and broken symmetry alike?(1)

The way mandalic geometry views symmetry differs slightly from the way physics does. This must be explained before we attempt the broken symmetry demystification mentioned above. To accomplish this we will need to discuss a number of other topics, including that of the rotation group of the cube, composite dimension, isotropic and anisotropic space, variant perspective, and invariance under Planck scale interchangeability. We have a long road ahead of us before we reach the fabled Fortress of Broken Symmetry and attempt to penetrate its formidable encasement.

(continued here)

Image: The ceiling of the Sheikh-Lotf-Allah mosque in Isfahan, Iran. By Phillip Maiwald (Nikopol) (Own work) [GFDL or CC-BY-SA-3.0], via Wikimedia Commons

 

 (1) Within mathematics symmetry in geometry is very closely linked to group theory.  Mandalic geometry at its current stage of development is concerned principally with the symmetries and asymmetries inherent in the cube, sphere, and their extrapolations to six dimensional space, or rather to the composited 3D/6D space of the mandalic cube and mandalic sphere. Wikipedia has an excellent introductory article on groups in mathematics which can be found here. An equally excellent article dealing with both symmetry and symmetry breaking appears in The Stanford Encyclopedia of Philosophy. For a discussion of the relation between groups and graphs I think you can do no better than this original tumblr post.

© 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. :)

All fixed set patterns are incapable of adaptability or pliability. The truth is outside of all fixed patterns. Bruce Lee (1940 -1973)

(via rhizinspirations)

Interlude 4

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The mathematical structure upon which mandalic geometry is based is the mandalic cube/hypercube. This maps a combined 3D/6D spacetime in which the 3D and the 6D aspects are both superimposed and composited, resulting in a mandala. In three dimensions this mandalic spacetime can be represented in configuration of either a cube or a sphere but owing to the discretized quantum nature of Plank scale spacetime these two are in fact congruent. In other words, point for point the mandalic cube/hypercube can be superimposed upon the mandalic sphere/hypersphere. In reality the two are one. It is only our rational minds that insist on distinguishing them.(1)

More precisely it presents as a set of four nesting cubes or spheres which in their interactions create the hypercubes or hyperspheres. It is as though the mandala is composed of nesting Chinese boxes, placed one within another, with the stipulation that all the discretized points of each “box” are able to interact with any and all those in all the other “boxes” and interchange location with any of them in the spacetime defined by mandalic geometry.(2) For particle physics this may entail in some sense the exchange of quantum numbers.

One of the most essential characteristics of mandalic geometry is its seamless holistic nature, all parts of which can nevertheless be extracted conceptually and rearranged in countless different manners. This results in a discretized probabilistic spacetime which seems not to respect any requirement that insists on restriction to a given dimension. Not only are the parts of the mandala interchangeable but this interchangeability can take place in any number of different dimensions both concurrently and consecutively (both terms which I believe must ultimately be given new definitions in the context of this geometry.)(3)

More specifically, instantaneous jumps can occur, for example, from two dimensions to three dimensions or from two to four or three to six and vice versa. This is not simply a conceptual trick of the mind. Nature itself participates in all these maneuverings described and many more. This is one of the characteristics of nature that makes quantum events appear so strange and unsettling, demanding that we view them in a probabilistic manner.(4)

Image: Stacking boxes - GRIMM’S kleiner Kistensatz blau-grün (Small Set of Boxes, blue)

 

(1) In truth though, to view mandalic spacetime in terms of either a cube or sphere is almost like the way stars in the night sky are viewed in terms of constellations which are really no more than conceptual areas grouped around patterns which represent the shapes that give the name to the constellations. Though not quite because the points mandalic geometry describes do bear multiple real relationships in mandalic spacetime. They are just not fixed in the sense that the points of a cube or sphere are and there is a lot of nothingness between them other than these relationships, more that is than seems apparent between stars in a constellation.

(2) Up till now only the cube/hypercube form has been described in this blog but previously the sphere/hypersphere form was described in detail in blindmen6.tumblr.com. The first in that series of posts can be located here.

(3) The mandalic configuration imposes upon spacetime a gradient between actual and potential. Similar to the way in which the yin/yang gradient works in both Taoist philosophy and mandalic geometry, this actual/potential gradient possesses bidirectional and multidimensional capabilities which trigger a layering of a probability distribution on reality and have significant effects on how time is expressed and experienced. This being so, it would seem the traditional definitions of concurrence and succession are inadequate in the context of mandalic geometry. The entire notion of sequence may need to be reeevaluated, more stringently even than Einstein did in his special theory of relativity. This subject will be addressed in greater detail in future posts.

(4) For a good discussion regarding the tesseract or four-dimensional hypercube see here. Keep in mind though that the mandalic cube we are describing in this blog is a 6-dimensional analog of the cube and therefore all the more complex in its construction.

© 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 - III

 

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[Click here to enlarge image.]

(continued from here)

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

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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)

(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) 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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(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.

(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) 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

  

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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