#bigram
Beyond the Enlightenment Rationalists:
From imaginary to probable numbers - VI
(continued from here)
“O Oysters, come and walk with us!” The Walrus did beseech. “A pleasant walk, a pleasant talk, Along the briny beach: We cannot do with more than four, To give a hand to each.”
* * *
“The time has come,” the Walrus said, “To talk of many things: Of shoes–and ships–and sealing-wax– Of cabbages–and kings– And why the sea is boiling hot– And whether pigs have wings.”
-Lewis Carroll, The Walrus and the Carpenter
In this segment, probable numbers will be shown to grow out of a natural context inherently rather than through geometric second thought as transpired in the history of Western thought with imaginary numbers and complex plane. To continue with development of probable numbers it will be necessary to leave behind, for the time being, all preoccupation with imaginary numbers and complex plane. It will also be necessary to depart from our comfort zone of Cartesian spatial coordinate axioms and orientation.
Probable coordinates do not negate validity of Cartesian coordinates but they do relegate them to the status of a special case. In the probable coordinate system the three-dimensional coordinate system of Descartes maps only one eighth of the totality. This means then, that the Cartesian two-dimensional coordinate plane furnishes just one quarter of the total number of corresponding probable coordinate mappings projected to a two-dimensional space.[1] It suggests also that Cartesian localization in 2-space or 3-space is just a small part of the whole story regarding actual spatial and temporal locality and their accompanying physical capacities, say for instance of momentum or mass, but actually encompassing a host of other competencies as well.
Although this might seem strange it is a good thing. Why is it a good thing? First, because nature, as a self-sustaining reality, cannot favor any one coordinate scheme but must encompass all possible - if it is to realize any. Second, because both the Schrödinger equationandFeynman path integral approaches to quantum mechanics say it is so.[2] Third, because Hilbert space demands it. This may leave us disoriented and bewildered, but nature revels in this plan of probable planes. Who are we to argue?
So how do we accomplish this feat? Well, basically by reflections in all dimensions and directions. We extend the Cartesian vectors every way possible. That would give us a 3 x 3 grid or lattice of coordinate systems (the original Cartesian system and eight new grid elements surrounding it), but there are only four different types, so we require only four of the nine to demonstrate. It is best not to show all nine in any case because to do so would place our Cartesian system at direct center of this geometric probable universe and that would be misleading. Why? Because when we tile the two-dimensional universe to infinity in all directions, there is no central coordinate system. Any one of the four could be considered at the center, so none actually is. Overall orientation is nondiscriminative.[3]
LOOKING GLASS CARTESIAN COORDINATE QUARTET
The image seen immediately above shows four Looking House Cartesian coordinate systems, correlated within a mandalic plane. This mandalic plane is one of six faces of a mandalic cube, each of which is constructed to a different plan but composed of similar building blocks, the four bigrams in various positions and orientations. A 2-dimensional geometric universe can be tiled with this image, recursively repeating it in all directions throughout the two dimensions.[4] It should not be very difficult for the reader to determine which of the four mandalic moieties references our particular conventional Cartesian geometric universe.[5]
It remains only to be added here and now that potential dimensions, probable planes, and probable numbers arise immediately and directly from the remarks above. In some ways it’s a little like valence in chemical reactions. We’ll likely take a look at that combinatory dynamic in context of mandalic geometry at some time down the road. Next though we want to see how the addition of composite dimension impacts and modifies the basic geometry of the probable plane discussed here.[6]
(to be continued)
Top image: The four quadrants of the Cartesian plane. These are numbered in the counterclockwise direction by convention. Architectonically, two number lines are placed together, one going left-right and the other going up-down to provide context for the two-dimensional plane. This image has been modified from one found here.
Notes
[1] To clarify further: There are eight possible Cartesian-like orientation variants in mandalic space arranged around a single point at which they are all tangent to one another. If we consider just the planar aspects of mandalic space, there are four possible Cartesian-like orientation variants which are organized about a central shared point in a manner similar to how quadrants are symmetrically arranged about the Cartesian origin point (0,0) in ordinary 2D space. But here the center point determining symmetries is always one of the points showing greatest rather than least differentiation. That is to say it is formed by Cartesian vertices, ordered pairs having all 1s, no zeros. That may have confused more than clarified, but it seemed important to say. We will be expanding on these thoughts in posts to come. Don’t despair. For just now the important takeaway is that the mandalic coordinate system combines two very important elements that optimize it for quantum application: it manages to be both probabilistic and convention-free (in terms of spatial orientation, which surely must relate to quantum states and numbers in some as yet undetermined manner.) At the same time, imaginary numbers and complex plane are neither.
[2] Even if physics doesn’t yet (circa 2016) realize this to be true.
[3] It is an easy enough matter to extrapolate this mentally to encompass the Cartesian three-dimensional coordinate system but somewhat difficult to demonstrate in two dimensions. So we’ll persevere with a two-dimensional exposition for the time being. It only needs to be clarified here that the three-dimensional realization involves a 3 x 3 x 3 grid but requires just eight cubes to demonstrate because there are only eight different coordinate system types.
[4] I am speaking here in terms of ordinary dimensions but it should be understood that the reality is that the mandalic plane is a composite 4D/2D geometric structure, and the mandalic cube is a composite 6D/3D structure. The image seen here does not fully clarify that because it does not yet take into account composite dimension nor place the bigrams in holistic context within tetragrams and hexagrams. All that is still to come. Greater context will make clear how composite dimension works and why it makes eminent good sense for a self-organizing universe to invoke it. Hint: it has to do with quantum interference phenomena and is what makes all process possible.
ADDENDUM (12 APRIL, 2016)
The mandalic plane I am referring to here corresponds to the Cartesian 2-dimensional plane and is based on four extraordinary dimensions that are composited to the ordinary two dimensions, hence hybrid 4D/2D. It should be understood though that any number of extra dimensions could potentially be composited to two or three ordinary dimensions. The probable plane described in this post is not such a mandalic plane as no compositing of dimensions has yet been performed. What is illustrated here is an ordinary 2-dimensional plane that has undergone reflections in x- and y-dimensions of first and second order to form a noncomposited probable plane. The distinction is an important one.
[5] This is perhaps a good place to mention that the six planar faces of the mandalic cube fit together seamlessly in 3-space, all mediated by the common shared central point, in Cartesian terms the origin at ordered triad (0.0.0) where eight hexagrams coexist in mandalic space. Moreover the six planes fit together mutually by means of a nuclear particle-and-force equivalent of the mortise and tenon joint but in six dimensions rather than two or three, and both positive and negative directions for each.
[6] It should also be avowed that tessellation of a geometric universe with a nondiscriminative, convention-free coordinate system need not exclude use of Cartesian coordinates entirely in all contextual usages. Where useful they can still be applied in combination with mandalic coordinates since the two can be made commensurate, irrespective of specific Cartesian coordinate orientation locally operative. Whatever the Cartesian orientation might be it can always be overlaid with our conventional version of the same. More concretely, hexagram Lines can be annotated with an ordinal numerical subscript specifying Cartesian location in terms of our local convention should it prove necessary or desirable to do so for whatever reason.
On the other hand, before prematurely throwing out the baby with the bath water, we might do well to ask ourselves whether these strange juxtapositions of coordinates might not in fact encode the long sought-after hidden variables that could transform quantum mechanics into a complete theory. In mandalic coordinates of the reflexive nature described, these so-called hidden variables could be hiding in plain sight. Were that to prove the case, David Bohm andLouis de Broglie would be immediately and hugely vindicated in advancing their pilot-wave theory of quantum mechanics. We could finally consign the Copenhagen Interpretation to the scrapheap where it belongs, along with both imaginary numbers and the complex plane.
ADDENDUM (24 APRIL, 2016)
Since writing this I’ve learned that de Broglie disavowed Bohm’s pilot wave theory upon learning of it in 1952. Bohm had derived his interpretation of QM from de Broglie’s original interpretation but de Broglie himself subsequently converted to Niels Bohr’s prevailing Copenhagen interpretation.
© 2016 Martin Hauser
Please note: The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)
Scroll to bottom for links to Previous / Next pages (if existent). This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added. To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering. To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)
-Page 311-
Beyond Taoism - Part 5
A Vector-based Probabilistic
Number System
Part II
(continued from here)
Taoism and the primordial I Chingare in agreement that temporal changes have two different aspects: sequent and cyclic. Western thought in general follows suit. The I Ching differs from the other two in asserting that the direction of change - for both sequent and cyclic change - is fully reversible, with the proviso that sufficiently small units of measurement are involved.[1] The probability that reversal can be achieved diminishes proportionately to the magnitude of change that has taken place.[2]
Taoist appropriation of bigrams and trigrams of the I Ching to model such phenomena as change of seasons and phases of the moon is plausible if not quite legitimate. The natural phenomena so modeled are macroscopic and vary continuouslyandinexorably throughout an ever-repeating cyclic spectrum. And there’s the rub.
As they occur and function in the I Ching bigrams and trigrams are dicontinuous discrete elements, formed by other similarly discontinuous discretized entities, and they follow evolutionary courses which are most often nonrepetitive. So the Taoist usage is misleading at best, annihilative at worst. Unfortunately, as the I Ching itself evolved through centuries of commentaries and reinterpretations, it became ever more contaminated and tainted by these Taoist corruptions of meaning, at the same time that it was being inundated by Confucian sociological and ethical reworkings. What we have today is an amalgam, the various parts of which do not sit well with one another.[3]
Though it may in part be hyperbole to prove a point, the stark difference between the two approaches, that of Taoism and that of the I Ching, is epitomized by comparison of the Taoist diagram of the cycle of seasons with diagrams at the top and bottom of the page, which are based on the number, logic, and coordinate systems of The Book of Changes.[4] The increased complexity of the latter diagrams should not prove a stumbling block, as they can be readily understood in time with focus and attention to detail. The important take-away for now is that in the I Ching bigrams exist within a larger dimensional context than the Taoist diagram avows, and this context makes all their interactions more variable, conditional, and complex. As well, the same can be said of trigrams and hexagrams.
One of the more important aspects of these differences has to do with the notion of equipotentiality. As bigrams and trigrams function within higher dimensional contexts in the I Ching, this introduces a possibility of multiple alternative paths of movement and directions of change. Put another way, primordial I Ching logic encompasses many more degrees of freedom than does the logic of Taoism.[5] There is no one direction or path invariably decreed or favored. An all-important element of conditionality prevails. And that might be the origin of what quantum mechanics has interpreted as indeterminism or chance.
Next up, a closer look at equipotentiality and its further implications.
Section FH(n)[6]
(continuedhere)
Notes
[1] There are exceptions. Taoist alchemy describes existence of certain changes that admit reversibility under special circumstances. Other than the Second Law of Thermodynamics (which is macroscopic in origin, not result of any internally irreversible microscopic properties of the bodies), the laws of physics neglect all distinction between forward-moving timeandbackward-moving time. Chemistry recognizes existence of certain states of equilibrium in which the rates of change in both directions are equal. Other exceptions likely occur as well.
[2] Since change is quantized in the I Ching, which is to say, it is divided into small discretized units, which Line changes model, the magnitude of change is determined by the number of Line changes that have occurred between Point A and Point B in spacetime. Reversal is far easier to achieve if only a single Line change has occurred than if three or four Lines have changed for example.
[3] Ironically, Taoism itself has pointed out the perils of popularity. Had the I Ching been less popular, less appealing to members of all strata of society, it would have traveled through time more intact. Unless, of course, it ended up buried or burned. What is fortunate here is that much of the primordial logic of the I Ching can be reconstructed by focusing our attention on the diagrammatic figures and ignoring most of the attached commentary.
[4] These diagrams do not occur explicitly in the I Ching. The logic they are based on, though, is fully present implicitly in the diagramatic structural forms of hexagrams, trigrams, and bigrams and the manner of their usage in I Ching divinatory practices.
[5] Or, for that matter, than does the logic of Cartesian coordinate space if we take into account the degrees of freedom of six dimensional hexagrams mapped by composite dimensional methodology to model mandalic space. (See Note [4] here for important related remarks.)
[6] This is the closest frontal section to the viewer through the 3-dimensional cube using Taoist notation. See here for further explanation. Keep in mind this graph barely hints at the complexity of relationships found in the 6-dimensional hypercube which has in total 4096 distinct changing and unchanging hexagrams in contrast to the 16 changing and unchanging trigrams we see here. Though this model may be simple by comparison, it will nevertheless serve us well as a key to deciphering the number system on which I Ching logic is based as well as the structure and context of the geometric line that can be derived by application of reductionist thought to the associated mandalic coordinate system of the I Ching hexagrams. We will refer back to this figure for that purpose in the near future.
© 2016 Martin Hauser
Please note: The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)
Scroll to bottom for links to Previous / Next pages (if existent). This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added. To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering. To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)
-Page 300-
