#cartesian coordinate system

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 the Enlightenment Rationalists:
From imaginary to probable numbers - II

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When a geometric interpretation of imaginary numbers was at last proposed,  long after they were invented,  it was as though accomplished by central committee. The upshot was easily enough understood but also simplistic. In broad brushstroke here is what seems to have gone down.

The 3 dimensions of Descartes’ coordinate system-a number already deficient from the perspective of mandalic geometry-were reduced to just one.  Of the real number axes then  only the x-axis remained.  This meant from the get-go  that  any  geometric figure that ensued  could encompass only linearity in terms of real numbers and dimensions.  It was applicable only to a line segment,  so the complex plane that resulted  could describe just one real dimension and one imaginary dimension.  It consecrated the number line in a single dimension, to exclusion of its proper habitation in two others besides. Strike one for imaginary numbers.^[1]

With that as background let’s look now at the rotations described by this geometric interpretation of imaginary numbers in the context of the complex plane.

i in the complex or cartesian plane. Real numbers lie on the horizontal axis, and imaginary numbers lie on the vertical axis By Loadmaster (David R. Tribble) (Own work) [CC BY-SA 3.0orGFDL],via Wikimedia Commons

The number 1 is the multiplicative identity element for real numbers and the number -1  is the  reflection inversion element  for real numbers.  Put another way, the number one times any number equals that number;  the number -1 times any number is  a negative of that number  or  the inverse number through a reference point, usually taken as zero. Multiplying by 1 then leaves 1, -1, i and -i all unchanged. Multiplying by -1  changes  -1 to 1, 1 to -1, i to -i, and -i to i.  In terms of rotations in the complex plane, these changes  all involve a rotation through 180 degrees.  Multiplication of the number 1 by i changes it to i; i by i changes it to -1; -1 by i to -i; and -i by i to 1.  These changes all involve rotations through 90 degrees.  And finally, multiplication of 1 by -i changes it to -i; -i by -i changes it to -1; -1 by -i to i; and i by -i to 1: all changes involving rotations through -90 degrees.

The figure below shows another way to interpret these rotations that amounts to the same tbing: i¹ = i; i² =-1; i³ = -i; i⁴ = 1.  Click to enlarge.

Four numbers on the real line multiplied by integer powers of the imaginary unit, which corresponds to rotations by multiples of the right angle. By Keφr [CC0],via Wikimedia Commons

I think a committee of some sort must have come up with this resplendent plan. For certain it was an Academy of Mathematics and Sciences that endorsed and enthroned it. All bow to central authority.

I had planned to include a comparison of imaginary numbers and probable numbers in this post as well but because that is a long discussion itself, it will have to wait till the next post.  I might add it should prove well worth the wait.

(continuedhere)

Image: A drawing of the first four dimensions. On the left is zero dimensions (a point) and on the right is four dimensions  (A tesseract).  There is an axis and labels on the right and which level of dimensions it is on the bottom. The arrows alongside the shapes indicate the direction of extrusion. By NerdBoy1392 (Own work) [CC BY-SA 3.0orGFDL],via Wikimedia Commons

Notes

[1] Mathematician William Rowan Hamilton  subsequently addressed this deficiency in 1843 with his  quaternions,  a  number system  that  extends the complex numbers to three-dimensional space.  Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space  or,  in other words, as the quotient of two vectors.  This complicated matters even more by introducing a non-commutative multiplication operation to the system, though to be fair the quaternion coordinate system has found some useful applications mainly for calculations involving 3-dimensional rotations,  as in 3-dimensional computer graphics,computer vision, and crystallographic texture analysis. Still it becomes problematic when theoretical physics attempts use of quaternions in calculations pertaining to  atomic and subatomic spaces  where rotations do not actually take place.  The conclusion to be drawn here is that quaternions can be usefully,  if somewhat clumsily,  applied to 3-dimensional macro-spaces but are inapproriate for accurate description of higher dimensional spaces. What is here unfortunate and misleading  is that quaternions apparently do describe outcomes of events in the quantum realm to some partial degree,  if not the mechanisms of the events themselves.  Physicists would not long tolerate them were that not so.

[ADDENDUM (24 APRIL, 2016)
Since writing this I’ve learned that quaternions are not currently used in quantum physics nor were they ever, to any great degree, in the past.]

In other words, sometimes  the right answer  can be reached by a wrong method. In the case under discussion here, we should note that it is possible for a rotation to mimic inversion (reflection through a point). A 90° rotation in two dimensions can mimic a single inversion in a single plane through an edge of a square, and a 180° rotation in two dimensions can mimic a single inversion through a diagonal of a square  or  two successive inversions  through  two perpendicular edges of a square.  A 180° rotation in three dimensions  can mimic three inversions through three mutually perpendicular edges of a square;  a combination of  one inversion through a diagonal of a square  and another through an edge perpendicular to the plane of the first inversion;  or a single inversion through a diagonal of the cube. Subatomic paricles exist as discrete or quantized entities and would follow such methods of transformation rather than rotations through a continuous space.  Of course, transformations involving a diagonal would require more transformative energy than one involving a single edge.

Such patterns of relationship and transformation could no doubt be described in terms of quantum states and quantum numbers without too much difficulty by a knowledgeable theoretical physicist.  Surely doing so could be no more difficult than using quaternions,  which may give a correct answer while also misleading and limiting knowledge of the the true workings of the quantum realm by using an incorrect mechanism, one non-commutative to boot. Nature doesn’t approve of hat tricks like that.

Beyond Taoism - Part 3
A Multidimensional Number System

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Descartes modeled his coordinate system on the Western number line,  itself an extension of the decimal number system to include the new negative numbers, and upon the Euclidean notion of a three-dimensional geometry.  All these events took place in historical times.  In approaching the I Ching and its number system we are dealing mostly with events that took place before recorded history so it is impossible to say with certainty how anything involved came about.  We can’t so much as be sure whether the I Ching was based on an antecedent number system,  or predated and foreshadowed a subsequent number system of Chinese antiquity possibly contingent on it. We view all such things as through a glass, darkly.^[1]

It is clear, though, that the number system of the I Ching is one far more complex than that of Western mathematics.  The number system of the West is unidimensional (linear).  Descartes,  in his coordinate system, extends it for use in three dimensions. The number system of the I Ching, on the other hand,  is  in origin  multidimensional.  It is mandalic as well, which is to say it consists of multiple dimensions interwoven in a specific manner which can best be characterized as mandalic in form,  possessing a number of interlaced and interlinked concentric shells or orbitals about a unifying center.

At the important origin of Descartes’ coordinate system is found his triple zero ordered triad (0,0,0).  Descartes  views this point,^[2] asall his points, primarily in terms of location, not relationship.  The matter of relationship is left to analytic geometry,  the geometry Descartes codified based on his coordinate system.^[3] The coordinate system itself seems not to care how points are formed or related beyond the most elementary and trivial operations of addition and subtraction throughout what essentially remain predominantly isolated dimensions.^[4] In the end this becomes an effective and prodigious mind snare.^[5]

In contrast to the Cartesian approach,theI Ching offers a unified coordinate system and geometry in a single entity which emphasizes the relationship of “points” and other “parts” (e.g.,  lines,  faces) as much,  if not more,  than location,  beginning with wholeness and ending with the same.  In between,  all sorts of  complex and interesting interactions and changes take place.  In analyzing these,  it is best to begin at the origin of the coordinate system of the I Ching,  the unceasing wellspring  of  being that supplants the triple vacuity of Descartes and Western mathematics.

Section F_H(n)^[6]

(continuedhere)

Notes

[1] My thinking is that the I Ching was originally primarily a non-numerical relationship system that subsequently was repurposed to include,  as one of its more important contextual capacities, numerical relationships. That said, from a contemporary perspective,  rooted in  a comprehensive awareness that spans combinatorics,  Boolean algebra,  particle physics, and the elusive but alluring logic  of  quantum mechanics and the Standard Model,  it would seem that this relationship system is an exemplary candidate for an altogether natural number system, one that a self-organizing reality could readily manage.

[2] As do most geometers who follow after Descartes.

[3] Strictly speaking, this approach is not in error, though it does seem a somniferous misdirection.  Due to the specific focus and emphasis enfolded in Descartes’ system, certain essential aspects of mathematical and physical reality tend to be overlooked. These are important relational aspects,  highly significant to particle physicists among others. These remarks are in no way intended to denigrate  Cartesian  coordinates and geometry,  but to motivate physicists and all freethinkers  to investigate further in their explorations of reality.

[4] The Cartesian system neglects, for instance, to express anywhere that the fact  the algebra of the real numbers  can be employed to yield results about the linear continuum of geometry relies on  the Cantor–Dedekind axiom,  which in mathematical logic

has been used to describe the thesis that the real numbers are order-isomorphic to the linear continuum of geometry. In other words, the axiom states that there is a one to one correspondence between real numbers and points on a line.

This axiom is the cornerstone of analytic geometry. The Cartesian coordinate system developed by René Descartes explicitly assumes this axiom by blending the distinct concepts of real number system with the geometric line or plane into a conceptual metaphor. This is sometimes referred to as the real number line blend. [Wikipedia]

Neither mandalic geometry nor the I Ching,  upon which it is based,  accept this axiom as true in circumstances other than those restrictive settings, such as Cartesian geometry, where it is explicitly demanded as axiomatic to the system. In other words,  they do not recognize the described one to one correspondence between number and geometric space as something that reality is contingent on. The assumption contained in this axiom, however, has been with us so long that we tend to see it as a necessary part of nature.  Use of the stated correspondence may indeed be expedient in everyday macro-circumstances but continued use in other situations,  particularly to describe subatomic spatial relations,  is illogical and counterproductive, to paraphrase a certain Vulcan science officer.

[5] For an interesting take on the grounding metaphors at the basis of the real number line and neurological conflation see  The Importance of Deconstructing the Real Number Line.  Also on my reading list regarding this subject matter  is Where Mathematics Comes From:How the Embodied Mind Brings Mathematics into Being(1,2,3) by George Lakoff and Rafael Nuñez. Neither of the authors is a mathematician, but sometimes it is good to get an outside perspective on what is in the box.

[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.  Simple by comparison though this model may be it will nevertheless serve us well as a key to deciphering the line derived from the mandala of I Ching hexagrams, and we will be referring back to this figure for that purpose in the near future.

© 2015 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 298-

Beyond Descartes - Part 6

The Fiction Formerly Known As the Line

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Rereading the last post a moment ago I see I fell into the same old trap, namely describing a concept arising from an alternative worldview in terms of our Western worldview.  It is so astonishingly easy to do this. So it is important always to be on guard against this error of mind.

In saying that the Taoist number line is the basis of its coordinate system I was phrasing the subject in Western terminology,  which doesn’t just do an injustice to the truth of the matter,  it does violence to it,  in the process destroying the reality:  that within Taoism, the coordinate system is primary.  It precedes the line,  which follows from it.  What may be the most important difference between the Taoist apprehension of space and that of Descartes lies encoded within that single thought.

Descartes continues the fiction fomented in the Western mind by Euclid that  the point and the line  have independent reality. Taking that to be true,  Descartes constructs his coordinate system using  pointsandlines  as the elemental building blocks. But to be true to the content and spirit of Taoism, this fabrication must be surrendered.  For Taoism,  the coordinate system, which models space, or spacetime rather, is primary. Therefore to understand the fictional Taoist line we must begin there, in the holism and the complexity of its coordinate system where dimension, whatever it may be, reigns supreme.^[1]

And that means we can no longer disregard composite dimension, postponing discussion of it for a later time,  because it is the logical basis on which the I Ching is predicated. It is related to what we today know as combinatorics,Boolean algebra, and probability,  and is what gives rise to what I have called the plane of potentiality. It is the very pith of mandalic geometry, what makes it a representation of mandalic spacetime.^[2]

(continuedhere)

Notes

[1] In my mind, dimension is a category of physical energetic description before it is a category of geometrical description.  When particle physicists speak about “quantum numbers” I think they are actually, whether intended or not, referring to dimensions. If this is true, then our geometries should be constructed to reflect that primordial reality, not arbitrarily as we choose.

[2] In speaking of logic and the I Ching in the same breath I am using the term in its broadest sense as any formal system in which are defined axioms and rules of inference. In reference to the I Ching,  the logic involved is far removed from the rationalism bequeathed to Descartes by his times.  It is a pre-rationalist logic that prevailed in human history for a very long time before the eventual splitting off of the irrational from the rational.  This means also that the I Ching is among other things a viable instrument to access strata of human minds long dormant in historical times,  other than possibly,  at times,  in poetry and art and the work of those select scientists who make extensive use of intuition in the development of their theories.

Note to self:  Two contrasting systems of thought based on very different worldviews can never be adequately explained in terms of one another. At times though, for lack of anything better, we necessarily fall back on just such a strategy, however limited, and make the best of it we can.

© 2015 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 282-

Earlier to Later Heaven: Fugue VII Beyond Descartes - Part 2
A Different Zero

(continued from here)

Mandalic geometry has been formulated in such a manner as to be fully commensurate with Descartes’ coordinate system. Firstly, because it can be.  Beyond that,  because Descartes’ system is known throughout the world, and is endorsed by all conversant in disparate fields of science and mathematics. Moreover, the Cartesian coordinate system is a special case of the mandalic coordinate system,  bearing a relationship to it analogous to that which Newtonian mechanics does to quantum mechanics.

One of the fundamental differences lies in the way the two regard zero locations. Descartes, taking his cue from the Western number line, constructs a coordinate system which envisages a single common origin to all three dimensions, while maintaining between those dimensions a rigid uncompromising distinction. Mandalic geometry views dimension as primary rather than points, lines, or two or three dimensional figures. It does not regard dimensions as intrinsically separate in the manner in which they  exist and relate  to one another.  This allows for a far greater degree of flexibility of what we view as parts of the system, including the possibility of folding each into another,  through different dimensions as well as the same dimension.

For Descartes, zero is the empty location, the no man’s land where positive and negative vectors of each dimension invert or fail to invert.  A negative vector acting on a positive vector or another negative vector will cause inversion.  A positive vector, acting on a negative vector or another positive vector, will not. For mandalic geometry, zeros are that, but more. They are dimension interchange lanes,  and also locations of dimensional amplitudetransition.^[1]

Descartes, influenced still by the number line, proceeds to build a geometric universe based largely on scale. It is an imposing edifice nearly purely divergent,  constructed from three largely independent linear axes of evolutionary zeal.  Taoist cosmology and mandalic coordinates equally eschew an impressive but mundane number line in pursuance of complex twisting and intertwining of parts evolved on the underlying principles of modularity, repetition, reflection, relationship and recursion.^[2]

These are two very different universes of logic.  Descartes’ approach leads to a description of space as being homogenous, isotropic, and fixed while that of mandalic geometry leads alternatively to a spacetime which is inhomogeneous, anisotropic and dynamically variable.^[3] For Descartes space is a background arena,  the theater in which all events transpire.^[4] For mandalic geometry,  space-time is foreground and background both. It is the sole ground which defines the nature of reality.

(continuedhere)

Notes

[1] The first,  dimensional interchanges,  occur in the Cartesian coordinate system but are generally neither recognized nor treated as such. Dimensional amplitude transition locations do not occur in Cartesian coordinates,  nor are they found in the simple 3D trigram Cartesian equivalent,  reproduced in the upper diagram above, as they are a manifestation only of compositing of two or more dimensions. They are attributes of all hybrid composite dimensional systems,  for our purposes here, either the 6D/3D hybrid mandalic system of hexagrams,  the 4D/2D hybrid mandalic system of tetragrams,  or the 2D/1D hybrid mandalic system of bigrams.

[2] An important consequence here is that Descartes’ number line-based axes each contain a single zero. When mandalic coordinates are scaled up beyond the basic modular unit, every even number maintains all characteristics of the initial zero, including, most significantly, its multipotentiality. This is a basic axiomatic result of the intermingling, sharing nature of mandalic structure.

[3] It is this variability and dynamism of mandalic coordinates that make the method potentially suitable to mappings of subatomic particles as these are similarly variable and dynamic,  sharing importantly also the ability of exchanges / interchanges among their diverse numbers.

[4] Witness for example how Descartes exploits his newly formed coordinate system to stage, what was then, a cutting-edge geometric exposition of algebra, now referred to as analytic geometry. Mandalic geometry employs coordinates which are pre-invested with the ability to directly impart information regarding spatial transmutations themselves, without requirement of any intermediary.

© 2015 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
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Earlier to Later Heaven: Fugue VI Beyond Descartes - Part 1

(continued from here)

In this post we take a short detour within our current central topic, that of relationship of Earlier Heaven and Later Heaven arrangements of the trigrams. The new material included here grew out of ruminations on the aforesaid primary topic though,  and is actually not so much a detour as a preparing the way for what I hope will be the eventual solution of our problem at hand.

Mandalic geometry, as we’ve seen, is fully commensurate with the coordinate system of Descartes, but its principal forebears lie elsewhere. It is derived largely from Taoist and pre-Taoist thought structures, most importantly the I Ching,  the earliest strata of which were formed before the separation of rational and irrational thought in the history of human cognition. As a result it is capable of far exceeding the possibilities of the Cartesian coordinate system, a product of the Enlightenment and Age of Rationalism. It offers geometry the possibility of a structural fluidity and a functional variability that Cartesian geometry lacks.^[1]

From the very beginning of this project I’ve been much puzzled by the lack in traditional Chinese thought  of a symbol corresponding to the zero of the Western number line and number theory.^[2] Traditional Asian thought does not uniformly lack a zero symbol.^[3] And yet the I Ching and Taoism manage well enough without one, electing to base their numerical relationships instead entirely on combinatorics involving permutations of yinandyang – what we in the West call  negativeandpositive – through multiple dimensions. It is an entirely different perspective arising out of a very different worldview.^[4]

What Taoism invented in the process was a unique,  thoroughly self-consistent brilliant system of logic/geometry/combinatorics which has been masquerading, all these many centuries,  as “just a method of divination.”^[5]  In essence, Chinese thought invented a discrete number system and geometry, one based on vectors rather than scalars, a vector geometry that can be extrapolated to any desired number of dimensions. The I Ching settles for just six,  the first whole number multiple of three. That is complicated enough.^[6][7]

(continuedhere)

Notes

[1] For one example of the advantages such variability and fluidity offer, in this particular case in creating  dynamic,  phase-shifting forms of nanomaterials,  see here.

[2] For a short history of the concept of zeroseeWho Invented Zero?

[3] The West, after all, derived its zero symbol ultimately via India.

[4] One might well speculate whether the significant root difference in world view between traditional Indian and Chinese thought lay in the fact that Indian mathematicians could have created a Zero out of nothingness (Śūnyatā),  a key term in Mahayana Buddhism and also some schools of Hindu philosophy while Taoist thought did not include a concept of nothingness. Instead it conceived of a formlessness prior to manifestation. In Taoist cosmology Taiji is a term for the “Supreme Ultimate” state of undifferentiated absolute and infinite potential,  the oneness before duality,  from which  yinandyang  originate.  So it might be that lacking a concept of nothingness forestalled invention of a zero symbol.  Still, it also allowed creation of an original,  unique holistic philosophy of reality, found perhaps nowhere else.

[5] The Russian philosopher, mathematician and authorPeter D. Ouspensky (1878-1947)  relates an apocryphal legend regarding the origin of the Tarot,  the moral of which has significance also to the history of the I Ching.

[6] In its emphasis on vector analysis and primacy of dimension the philosophy which underlies the I Ching and mandalic geometry  shares some characteristics of Clifford algebra.

[7] One of the important things with respect to physics I hope to show with mandalic geometry is that it is possible to construct an integrated geometrical / logical system which is self-sufficient and self-consistent, capable of modeling interactions of subatomic particles of the Standard Model and then some.  This goal is,  I believe,  approximated in mandalic geometry by meticulous coupling of the methodologies of composite dimension and trigram toggling,  although it quickly becomes apparent that a system based upon what is after all a relatively small number of dimensions - six in the case in point - becomes vastly complex and difficult to follow, at least initially.  One can’t help wondering how physics will be able to correlate all the intricate data resulting from its countless particle accelerator collisions and combine it into a consistent whole without some very fancy mental acrobatics on the part of theoretical physicists.  Without a suitable logical scaffold that might take an inordinately long time to achieve.

© 2015 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 277-