Inversion Transforms of the Cube - I

 

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As we begin our exploration of the inversion transforms of the cube it is well to keep in mind that from the Taoist point of view this is the same as the 3-dimensional inversive transforms of the trigrams. (1) The broken(yin) line and solid(yang) line serve respectively as the inversion (2) and identity elements of multiplication or transformation throughout all dimensions. In practical terms this means that two simple rules suffice to characterize all transformations through any number of dimensions:

  1. The broken yin line produces inversion through a point of reference(3) when acting upon either a solid yang line or a second yin line.
  2. The solid yang line produces no inversion, leaving the operand unchanged regardless of its sign.

 

This can be summarized as

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which are the same rules we all learned in grade school using minus(-) and plus(+) in place of yin and yang.

As noted previously, one of the great advantages of using the Taoist yin/yang notation over the Cartesian +/- notation is that manipulation of the trigrams is far easier than the analogous manipulation of the Cartesian ordered triads. And when we come to consideration of manipulations through 6 dimensions using the 64 hexagrams, which is the true meat of the matter, there is no real Cartesian analogue available to use.(4)

(continued here)

 

(1) Which is to say, the transforms of yin and yang through three dimensions. Taoist inversive transforms can encompass any number of dimensions. The hexagrams of the I Ching involve transformations which occur throughout six dimensions. At some future time we will consider those transformations. Here, however, we are restricting our focus to the three dimensions of the ordinary cube and of the Taoist trigrams.

(2) We are speaking here literally of inversion of direction or sign and are concerned exclusively with the direction or sign aspect of vectors. Magnitude is of no concern within this focused context.

(3) For both Cartesian geometry and mandalic geometry this point of reference is either the origin point of the coordinate system or the origin point of a dimensional subdivision thereof (i.e., center of cube, center of cube face or center of cube edge for a 3-dimensional coordinate system.)

(4) One could be easily enough constructed but why go out of one’s way to make life difficult?

© 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 Teaser

 

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A possible solution to the still-hanging-in-air question as to whether we uncovered a broken symmetry in an earlier post has occurred to me. Here is a provisional answer and a teaser of sorts:

  • A broken symmetry cannot exist where there was no symmetry to break in the first place.

We will elaborate on that idea after our look at the group of inverse transforms of the cube which begins in the post immediately following.

As we progress through the inverse transforms of the cube keep your eye on the bouncing ball and see if you can distinguish any symmetries from asymmetries. Mandalic geometry views the latter as being just as important as the former, if not more so, particularly where the asymmetries are related to one another in specific recurring manners.

Classic 4beats passing 2juggler 6balls side

continued here)

Image: World record holding club passers Christoph and Manuel Mitasch passing 11 clubs in Linz, Austria. By Cmitasch (Own work) [CC-BY-SA-3.0 or GFDL], via Wikimedia Commons

Animation: Classic_4beats_passing_2juggler_6balls_side.gif By Own work [GFDL or CC-BY-SA-3.0-2.5-2.0-1.0], via Wikimedia Commons

© 2014 Martin Hauser

The way I see it

 

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Quantum entanglement is a fact. Physics has shown it so despite Einstein’s objection to it as being “spooky action at a distance.”(1) But physicists can offer no explanation of how or why it works as it does. Perhaps they are looking in the wrong directions for explanation. Maybe they are making the whole thing out to be much more complicated than it actually is.

Physicists need look no further than the closest playground for a useful model of how this “spooky action at a distance” might work in principle. The model they seek would likely be found in most children’s playgrounds - - - the common see-saw. An equally valid model would be the balance or scale though these are not seen nearly as frequently in our highly technological society as they previously were. All of these objects are based upon the simple principle of the lever, a principle which has been known and used in many different applications of the arts and sciences since ancient times.

What occurs with these mechanisms involves reciprocity of motion in response to slight differences in applied force to the two limbs of the lever balanced on a fulcrum. The two ends of the lever by nature both exist and move in reciprocal relationship to one another. When one limb heads down the second will always head up. It is not actually a matter of transference of information because the whole is all of a piece.(2) Down does not need to tell up which way to head because there is no choice. Down demands up; up demands down. They are necessary reciprocals of one another. Up and down are one and the same in a higher scheme of things than we are used to or comfortable dealing with. Our minds seem to want to see them as separate whereas in reality they are not. Were they actually so then information would necessarily have to pass between the two to keep them coordinated and that information would indeed need to obey the limit in speed of transmission dictated by the finite velocity of light. That is, if any transmission of information were actually involved.

As is the case with entangled particles, the two ends of the see-saw or lever could be at opposite ends of the universe and if proper and adequate forces were applied distance would be no deterrent. Changes would occur instantaneously without any transmission of information. This of course would appear paradoxical and be dismissed as impossible or unexplainable by disbelieving or skeptical eyes viewing the process from their restrictive base in the universe.(3) It seems to me that what physics is overlooking here and what it needs to find is the missing link, the hidden connection that unites what appear to be two distinct entities into one. Find that and quantum entanglement will cease to be a mystery.

Image: Turquoise Eyes by thepeachpeddler CC BY 2.0

 

(1) The phrase Einstein actually used was “spukhafte Fernwirkung” (in a letter to Max Born 3 March 1947). Like Schrödinger, Einstein was dissatisfied with the concept of entanglement,because it seemed to violate the speed limit on the transmission of information implicit in the theory of relativity. The irony here is that all the bold predictions Einstein made in his own work have proved true thus far, yet he was unable to accept this equally bold assertion of quantum mechanics which demands redefinition of our concept of space in a manner no more radical than Einstein’s own Special and General Theories of Relativity had done some years prior.

(2) All of a piece is an idiom and it means that there is an invisible link.

(3) The scale at which we humans exist restricts our view of both the realm of the very large and the realm of the very small.

© 2014 Martin Hauser

sciencealert:

Maths is the language of nature.Image via AsapSCIENCE

The more I think about it, the more I conclude that it is an illusion of sorts that mathematics is the language of nature. The language of nature, I think, is more likely a combination of fields, forces, energy, mass, transformations and interrelationships. Mathematics is the result of man recreating nature in terms of his own mental imagery.

sciencealert:

Maths is the language of nature.

Image via AsapSCIENCE

The more I think about it, the more I conclude that it is an illusion of sorts that mathematics is the language of nature. The language of nature, I think, is more likely a combination of fields, forces, energy, mass, transformations and interrelationships. Mathematics is the result of man recreating nature in terms of his own mental imagery.

(via cantbuymepoo)

haayyad:

What is Math? Including a comparison to Science

Interlude 5

 

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In a sense the rationalists of the Enlightenment were like children playing with new toys not fully understood.(1) They were enchanted with the new geometric method of representing dimensions and positive and negative numbers offered by Cartesian geometry but they didn’t yet quite comprehend the intricacies involved or the awesome power unleashed by the methodology. Descartes himself likely did not.

Though Descartes’ method of geometry offered what seems from our modern perspective a ready-made access to dimensional numbers and to the number plane and number 3-space, the bait appears not to have been taken. Thought processes remained rooted for an inordinately long time in the well-worn tracks of linearity, and the number line, in spite of the obvious superiority of the more inclusive approach of higher dimensionality, reigned supreme in the minds of Enlightenment thinkers.(2)

We will soon see that the number line as we know it is just one variant in a spectrum of simultaneously existent species of number lines which all dovetail together to form a higher-dimensional geometry. Quite possibly spacetime itself is constructed so at Planck scale. And in some paradoxical sense, at that scale uniformity and non-uniformity of space are one and the same once one manages to get beyond the conventions of labeling and unitary perspective. Symmetry and asymmetry also combine at some level and relate to one another singularly in some higher dimensional sense. That takes us well beyond the usual perspective by which we view things. Possibly the most important rule to keep in mind throughout all of this is the one that states, “Nature is never simplistic but always, wherever and whenever it can, takes the most simple direct route available.”(3)

There are more loose ends still to be tied up and they will be eventually. But next we will tackle the large challenge involving the inversion transforms of the cube. With that accomplished we will be better equipped to evaluate the question left hanging in mid-air regarding whether or not we unearthed a broken symmetry during our recent explorations. 

Image: Tunnel vision, a focus too narrow. License: CC0 Public Domain/FAQ

 

(1) The same, of course, could be said today about both theoretical and experimental physicists and any number of other categories of contemporary thinkers. Likely it is always so in times of rapid change and innovation.

(2) Though perhaps the perspective expressed is anachronistic as rationalists of the 17th and 18th centuries were still largely uncomfortable with holistic syncretic approaches to investigation and understanding of natural processes. Their view was that comprehension of nature required breaking it into its component parts. Possessed of such an analytic view of things they must have felt more comfortable with lines and points than with cubes and squares. I see no convincing evidence that they ever conceived of the number line as existing in the larger contexts of 2- and 3-dimensional space. Someone more familiar with the history of mathematics than myself may view this matter differently. I am simply trying to understand from a cultural/psychological perspective how it came about that this opportunity was missed, at least until much later when rationalism no longer held sway.

(3) I should probably add parenthetically that a simple direct route is not always among the options available.

© 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. :)