#real numbers

Can a number system be both the new kid on the block and older than written history?

The real number system as it exists today has been with us for a few centuries.  In foundation it is monovalent,  monophasic,  and sequential.

The probable number system dates to prehistory but was lost in the mists of time until recently rediscovered and resurrected.  In contrast to the real number system it is foundationally bivalent, biphasic, and cyclic.

The probable number system has considerably more structure than the real number system and is therefore more robust.  In this sense, it is similar to the complex number system.

In contrast to the complex number system,  the probable number system in its foundation presupposes that numbers can assume wavelike forms capable of  constructive and destructive interference  operationally through the compositing of higher to lower dimension.

By means of compositing of dimension probable numbers are able to  distribute  throughout the entire  mandalic unit vector cube  (which is structurally a  superposition  of  the 6-dimensional unit vector hypercube on the 3-dimensional unit vector cube) a function analogous in important ways  to that performed in the complex number system by the centralized imaginary unit i.

Another important way in which the probable number system differs from both the real number system and the complex number system is the absence of  nothingness  and the zero representing it.  In its place we find the concepts of  balance and equilibrium.  Nullification still exists in form of annihilation and its opposite in the form of creation.  But the Cartesian coordinate system  of ordered pairs and ordered triads  is transformed by this approach to handling number and dimension  from a ring into a field of hyperdimensional numbers over real numbers in three dimensions.

(to be continued)

© 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 315-

Magic Theatre: For Madmen Only

If we describe a Cartesian ordered triad by x,y,z we can describe an analogous 6-dimensional ordered sextuplet  or 6-tuple  by x_a,y_a,z_a,x_b,y_b,z_b

The definitions that translate a 6-dimensional ordered sextuplet (hexagram in Taoist terminology) into a 3-dimensional ordered triad (trigram in Taoist terminology) are:^[1]

• (x_a + x_b) / 2 = x
• (y_a + y_b) / 2 = y
• (z_a + z_b) / 2 = z

I think the methodology will work for all scalar quantities. But as currently formulated,  mandalic geometry (MG) is a discrete geometry based entirely on unit vectors.  We are talking about the line segments between -1 and +1 in the various dimensions and only points -1, 0, and +1 in each line segment in Cartesian terms.

In essence we are not yet particularly concerned with scalars here but only with vectors :  -, +, and neutral (0).

Mathematically √−1 is important because by adding it to the real number field, as we have done, we create the algebraically complete field of complex numbers. In mathematics, a complete field is a field equipped with a metric and complete with respect to that metric. The real numbers and complex numbers are both complete fields. Cartesian coordinates- - - ordered pairs and ordered triads- - - although based on real numbers,  do not form a field. This has important implications, implications which can be ignored only at peril to the particular conceptual system involved..

The definitions above all give three possible results in Cartesian terms: -1, 0, +1.  Remember though MG hybridizes six dimensions with three dimensions and represents them superimposed. Wherever one or more zeros occurs in Cartesian coordinates we have also corresponding 6-dimensional forms,  composed of just +1s and -1s,  of which there are always two for each Cartesian zero.  A Cartesian ordered triad with one zero is associated with two such 6-dimensional forms; an ordered triad with two zeros, with four; an ordered triad with three zeros (the origin), with eight. An ordered triad without zeros will have only one associated 6-dimensional form.  This constitutes the mandalic pattern, which is an essential feature of the 6D/3D formulation of this geometric system and isomorphism naturally comes into play here as well.

Andthat is how and why all numbers in this coordinate system based on higher-dimensional extensions of the real numbers “square” to numbers which can be  either positive or negative  and then reduce or "collapse" to corresponding Cartesian forms that preserve the same sign. This is a necessary result of the fact that a primary “zero form” in
6-dimensional terms is lacking,  only +1s and -1s exist.  These can then interfere constructively and destructively as number waves, to produce a  "secondary zero"  by destructive interference  whenever linked forms differ in sign in one or more paired dimensions. Since the two linked 6-dimensional numbers are always inverse to one another, any Cartesian zero then can be substituted with two such 6-dimensional forms. This is the process that makes imaginary numbers unnecessary, replacing them with two inversely related probable numbers which behave in most ways like real numbers  and  are  distributed  throughout the entire geometric system.

“Hybridization” is probably not the best term here but will be used until I can think of a better descriptor. What I intend is not actual joining and unification,  but rather  a superposition and conceptual commingling in three-dimensional terms. Such a representational mapping substitutes for all Cartesian forms  "equivalent" forms  containing only 1s and -1s, no zeros.  In so doing, it effectively converts the Cartesian coordinate system from just a ring to a field as well, properly interpreted. Basically then, the probable numbers do for the real numbers much the same as the complex numbers do,  but with even greater and more utilitarian results which are also more easily managed.

In operational terms, complex numbers perform two rather simple binary operations: a scaling and a rotation. Scaling capability is clearly inherited through its real number lineage; rotational capacity, from its imaginary number lineage.  Together,  scaling and rotation combine to augment or diminish an axis of growth and produce vector ambulation in a circular path about a central origin point of reference.  The scaling factor  could be said to detemine the  radius of revolution;  the rotation factor, the angle of revolution. And that’s pretty much all there is to the “great mystery”  of complex numbers.  Their importance  resides in the great number of fields of endeavor where the combination of these two superpowers is necessary and/or convenient.

Nature uses this combination of scaling and rotation in many of its processes.  Atomic and subatomic proceedings  are probably not among these.  How then did it come about that  quantum mechanics  arrived at the notion that  rotation and scaling  could be applicable to modeling of discontinuos states of being?  Both refer to changes through continuous space. I think it was an accident of history. In 1925, Erwin Schrödinger, in his search for a way to explain  certain mysteries then perplexing the greatest physicists of the day,  hit upon his  eponymous equation  which appeared to do the trick.  So well,  in fact,  that quantum mechanics has been  justly considered  the single most successful description of reality ever devised. And the equation that basically accomplished this success involves the imaginary number i and complex numbers.^[2]

An important aspect of the operation of rotation, one which may have bearing on the Schrödinger equation and its huge success, has been largely overlooked. The result of a rotation can often mimic the result of inversion (reflection through a point), making the two indistinguishable by measurement alone. To someone wearing a blindfold there is no way to tell whether i has by the operations of squaring and rotation changed itself into  -1  or  -1,  the inversion element of multiplication,  has simply reflected  +1,  the identity element of multiplication,  through the origin point to  -1.  Explaining away a 90° rotation with a right angle reflection will no doubt prove more difficult but let’s not just yet deny that it might be doable.

Could there be a way to reformulate the Schrödinger equation then so it contains no imaginary or complex numbers?  Many have tried to do that very thing and failed. No one has succeeded in nearly a century. Still, we might wonder if the time is ripe now to remove the blindfold. Perhaps we might do well to inquire whether quantum physics is, in some manner we don’t quite understand, a victim of its own success.

In theory, circumventing use of complex numbers in a defining equation of quantum mechanics should be possible. On what basis do I say this?  The equation we have now relies on complex numbers.  These in turn derive an ability to produce rotation from the imaginary number √−1 .  But there are  other mathematical means  to accomplish the same. Trigonometry comes most immediately to mind. The circle and cyclicity it models have a very long and distinguished history. Complex numbers as we’ve noted can also produce scaling.  But so can real numbers.  And close examination reveals  that complex numbers inherit their ability to scale from the two real numbers they contain. The hard truth ultimately is  there is nothing all that special  about  complex numbers  or complex plane. Possibly it is their utilitarian ease of use that positions them as an attractive methodology. Other routes to ease of use exist as well. There is always more than one way to skin the proverbial cat  (even a cat residing only in the mind of a physicist named Schrödinger.)

Consider also, how great is the actual need for scaling in quantum mechanics?  The distance from  centermost part of the atom  to the outer reaches of electron orbital space is in fact quite small.  Furthermore,  the elements of this universe of discourse are quantized,  so actual distances involved are moot.  In the extreme,  the question persists  as to  whether “distance” is a concept even applicable  in this context  of quantum logic. Quantum numbers  themselves  range between 0 and 2.  I can count the allowed values on the fingers of one hand.

Regarding rotation, where exactly does that come into play in the quantum realm?  Electrons do not orbit the nucleus of the atom.  They jump from orbital to orbital by discretized changes in energy involving photon exchange. In the nucleus it seems such discretized instanteous changes take place as well,  obviating any need for rotation.  Obviously physics misguided here by labeling one of the quantum numbers “spin”. Sometimes a rose is best referred to as a rose. The problem here is that we don’t really know what it is that “spin” refers to.

The quintessential equation of quantum mechanics was formulated by a physicist, not a mathematician. It is not a simple algebraic equation, but in general a linear partial differential equation,  describing the time-evolution of the system’s wave function (“state function”). “Derivations” of the Schrödinger equation  do generally demonstrate its mathematical plausibility for describing wave-particle duality. To date, however, there are  no universally accepted derivations  of Schrödinger’s equation from appropriate axioms.  Nor is there any  general agreement  as to what the equation actually signifies.  Moreover, some authors have demonstrated that certain properties  emerging from Schrödinger’s equation  can even be deduced from symmetry principles alone.  This would appear to be a worthwhile direction of investigation to pursue.  Quantum mechanics is most fundamentally about symmetry.  Let’s make Emmy Noether proud by giving her the recognition she deserves.

Finally, it was not without considerabledifficulty that Schrödinger developed his equation.  In the end,  it almost seems  he pulled it out of a hat,  as a magician might a rabbit.^[3]   Part of the  Zeitgeist  of the physics community  in the early 1920s  revolved around  the peculiar notion  that particles  behaved as waves.  Schrödinger decided to follow this direction of thought  and  find an appropriate 3-dimensional wave equation for the electron. His equation succeeded beyond his wildest dreams.  Adopted in the canon of  the new physics,  it became the cornerstone of that radically different physics, changed forever. Physics has never looked back since.

Still, one startling and haunting fact persists: nowhere else in all of physics  has it ever been found necessary to invoke complex numbers.

Once,  quite a long time ago,  I believed  imaginary numbers  were wrong. I was the one that was wrong. Later, having grown a little more clever, I came to think that √−1 was a necessary evil- - -correct but not validly applicable to quantum physics. Wrong again. Currently it is my belief that imaginary numbers are guilty of an even worse offense: both true from the mathematical standpoint and partly applicable to physics. The worst of both worlds.  Yielding results that are in large part correct, imaginary and complex numbers have managed to lead us all down the garden path for the better part of a century. Have we then gone past the point of no return?  My contention  is  that it is possible to complete the ring that Cartesian coordinates present  and  transform it to a field over the real numbers, with appeal only to higher-dimensional analogues of the reals and no need for imaginary or complex numbers,  an approach which, if actually possible, would offer certain undeniable advantages.^[4]

Essentially the method of composite dimension does away with i and complex numbers by distributing an operation analogous to that of i throughout six dimensions or three in Cartesian terms and then working with same by means of reflections (inversions) only. So an algebra based on the system necessitates use of only the real numbers and their higher dimension extensions that I have called probable numbers.  Only simple addition and multiplication  are required.  For those in the audience who are "sufficiently mad”, there is the added bonus that a kind of division by zero becomes possible. We’ll find out soon enough whether you qualify.

A few additional explanatory remarks are in order here:

Depending on the variant,  Cartesian geometry (CG),  represents space in two or three dimensions. Points in the former are referenced to two pairwise perpendicular axes; in the latter, to three.

Because Descartes assumes as axiomatic a 1:1 correspondence of number to spatial location each of his three axes becomes a facsimile of the number line, only in different dimensions.

Mandalic geometry (MG) approaches representation of space differently, using a hybrid coordinate system which relates a higher dimension space to a lower dimension space  with a 2:1 correlation.

Itcan be represented entirely commensurate with CG, but in so doing a “glass slipper effect” occurs. Just as Cinderella’s stepsisters can manage to force a too fat foot into her glass slipper, the results leave something to be desired.  In our context here,  the  "something to be desired"  is a clear and full understanding of six-dimensional reality in its own right. We end up interpreting it in time-sharing terms of probabilities and randomness.

What Descartes refers to as an ordered pair requires two higher dimension ordered pairs to represent in MG; a Cartesian ordered triad requires three higher dimension ordered pairs to represent in MG.

In Taoist terminology the notational equivalent of a Cartesian ordered pair is a  "bigram",  a two-line symbol,  each line of which  can take one of two values. As a result there are four types of bigram. Two bigrams make up a tetragram; three, a hexagram.

Descartes views a point as having only two essential characteristics:

• It is dimensionless.
• It is just a location in space which can be uniquely represented
  by a single ordered pairorordered triad.

Mandalic geometry rejects both of these axioms. It regards a point, or a particle so represented, as an evanescent entity emerging from interaction of two higher dimensions expressed in our world of three dimensions in such limited manner.

Thiscan be represented in context of Cartesian space but in making mandalic coordinates commensurate with Cartesian coordinates it is no longer possible to represent every “point” in space uniquely with a single mapping of number to location.  What results instead is the probabilistic distribution pattern of the mandala, which we,  from our limited vantage in spacetime, misinterpret as something it is not.

MG is a discrete geometry. The result of the mapping formula used is a mandalic configuration in which the 3-dimensional cube composed of  unit vectors in Cartesian space  becomes a  "probability distribution"  in combined mandalic space.

I have placed the quotation marksaroundprobability distribution because this is a perspective that arises  from our inability to see all that is involved accurately. I suspect this has repercussions pertinent to a full comprehension or grokking of quantum mechanics and possibly of string theory as well.

Since the 64 discrete “points” of  the unit vector hypercube of six dimensions represented by the hexagrams cannot “fit” simultaneously in the 27 discrete points of the 3-dimensional unit vector cube  by any representational method available to our inherited bio-psychocultural mechanism, a sort of time-sharing process occurs in observations and measurements of reality which we interpret in terms of probability.

What has been described here occurs at enormous velocities close to that of light, and likely refers only to processes in the subatomic quantum realm. For MG, which is also a hybridization of mathematics and physics, context is always of the essence.

There is much more to be said in explanation of mandalic geometry. I see, though, this post has already run rather long, so we will end it here. Enough has already been said in way of introduction of basic material.

Notes

[1] Since the coordinate system is describing a cube with an n-hypercube superimposed,  there is an additional constraint placed on all coordinates in
the 6-tuples.  All scalar values must be identical for x, y and z values.  That constraint assures that all vectors though they may differ in sign (direction) maintain equal magnitudes.

When the 6-tuples are dimensionally reduced to 3-tuples by the method I’ve called “compositing of dimension”  the resulting geometric figure consists of four different dimensional amplitudes of 6-tuples collapsed.  The amplitudes of dimension correspond  in spatial terms  to the vertices,  edge centers,  face centers and cube center. The pattern that emerges is that of a mandala. This is a highly symmetric pattern though all symmetries aren’t necessarily apparent immediately, even using Taoist notation. The probability distribution of the 6-tuples allots the hexagrams in the following manner:  one to each vertex;  two to each edge center; four to each face center; one to the cube center. The result is  placement of 64 6-tuples  in 27 positions of discrete 3-tuples  in the specific mandalic distribution pattern described.

Think here of the analogy of a hydrogen atom confined within a cubic space of specified side length determined by the nuclear and atomic force fields. The single electron,  existing in such quantized energy levels that are possible,  can assume various different locations in different orbital shells,  but every location in a given orbital must be equidistant from the nuclear proton. Once reduced by dimensional compositing the 6-tuples described here fill four distinct shells that have different radii or distances from the center.  From center to periphery these distances can be described as zero;  one (or square root one);  square root 2; and square root 3. (Pythagorean theorem)

[2] Schrödinger was not entirely comfortable with the implications of quantum theory. About the probability interpretation of quantum mechanics that came out of Solvay ‘27 he wrote:  "I don’t like it,  and I’m sorry I ever had anything to do with it.“ ["A Quantum Sampler”. The New York Times. 26 December 2005.]

[3] In later years another great physicist, Richard Feynman, would remark, “Where did we get that (equation) from? Nowhere. It is not possible to derive it from anything you know. It came out of the mind of Schrödinger.”

[4] A different approach to avoiding the need for complex numbers from the one I am suggesting is described here. To my mind it offers little of value other than an interesting alternative explanation of what complex numbers are and do. A similar conclusion seems to have been reached by the author.

© 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 313-

A Recap of Some Important Ideas Regarding Mandalic Geometry

1. Mandalic geometry (MG) is a new kind of mathematical methodology based on a worldview having roots that predate written history.
2. It is a discrete geometry which currently consists of just a coordinate system but can be extended as Descartes did his to encompass an entire analytic geometry.
3. Mandalic geometry introduces and is based on a new number system, the probable number system (or probabilistic number system.)
4. Just as the complex number system combines real numbers and imaginary numbers and is more robust than either, the probable number system combines real numbers and probable numbers and is more robust than either.
5. The probable number system is also more robust than the complex number system. Complex numbers combine real numbers with imaginary numbers to form the single complex plane. Composite numbers combine real numbers with probable numbers to form six interdependent composite planes.
6. Axiomatic to the system is the contention that numbers can exist in different dimensions and therefore can be described as being of some particular dimension. Numbers are always viewed and treated within context of a stated dimension.
7. Probable numbers are an extension of the real numbers to higher dimensions and are independent of imaginary and complex numbers.
8. Mandalic geometry does not admit the existence of square root of -1 in the real world other than in mathematics invented by the human mind. In place of square root of negative numbers, MG introduces the new concept of contra-square root. In brief this involves substitution of a combination form of interactive two-dimensional analogues of +1 and -1 for -1 as currently used in imaginary number contexts. This is more fully explained elsewhere in the blog.
9. Put another way, in place of imaginary numbers MG posits the existence of probable numbers. These can be considered the result of what is essentially wavelike interactions of higher dimensional numbers to form the real numbers we know in the 3-dimensional world.
10. Higher dimensional numbers can interact with one another through wavelike constructive and destructive interference to generate ordinary
    3-dimensional numbers. Numbers are not viewed as constants to be acted upon as Descartes so views them but rather as being themselves active and changeable. They participate in process. This feature alone enables composite numbers to mediate between mathematics and physics better than either real or complex numbers can.
11. The interactions of higher dimensional numbers in the process of dimensional compositing to yield 3-dimensional numbers is a function of time and therefore probabilistic from our limited ordinary point of view. From this perspective, certain probablity distributions are the result of dimensional compositing and the consequent mandalic form. MG considers the probabilistic nature of quantum mechanics likely to be based on such.
12. The probabilistic nature in three dimensions of what are here called probable numbers is what gives rise to the mandalic form which can in a sense be considered the 3-dimensional evolution of 6-dimensional numbers from protean representations through progressive differentiation of form to the stage of maximal differentiation and back again to the undifferentiated state of greatest probability.
13. The mandalic form has a geometric progression of its line structures in the three Euclidean/Cartesian dimensions such that series of numbers of the form 1:2:1, 2:4:2, and 4:8:4 occur throughout all of those dimensions when a hybrid 6D/3D coordinate system results from performing 2:1 compositing from six to three dimensions.
14. Mandalic geometry views points and lines in three dimensions as convenient fictions that exist only as evanescent probabilistic concurrences of analogous entities in higher dimensions.
15. The probabilistic nature of MG makes it ideal for investigations and descriptions of quantum mechanics.
16. The exclusion of imaginary and complex numbers and substitution of probable and composite numbers which are easily reducible to ordinary algebraic/arithmetic forms and can be worked with using the same methods as those mathematical disciplines makes MG more utilitarian and appropriate to application to quantum mechanics than are complex numbers. All operations performed are based on simple inversion (reflection through a point) and on real numbers, maintaining all the usual rules and properties of ordinary arithmetic, including commutativity (which quaternions fail to preserve.)
17. MG is currently based on discrete numbers and is concerned mainly with the positive and negative integers. Fractions and irrational numbers are not excluded from the system but do not currently play a significant role. Future incarnations of MG will extend it outward beyond the unit vector cube to tile the geometric universe and inward to encompass fractional entities and fractals.
18. It is a hybrid geometry resulting from superposition of 6-dimensional numbers and 3-dimensional numbers and is fully commensurate with
    3-dimensional Cartesian geometry.
19. It describes a linear mapping of two dimensions to one dimension which forms a field of probable numbers over the field of real numbers, analogous to the field of complex numbers but constructed on a different principle and extending to the real numbers in all three Cartesian dimensions rather than just one. The two independent higher dimensions so mapped become dependent variables in the mandalic “line” that results from the compositing of the two. This is expressed, in a sense, as two sine waves 180 degrees out of phase that mutually intersect a common Cartesian axis (x,y or z) at Cartesian +1 and -1 and are maximally separated at Cartesian 0.
20. This phase difference produces wave interference of both constructive and destructive varieties. So-called “points” or “particles” they represent come into existence only discretely and intermittently at Cartesian -1, +1, and 0, the locations of intersection or confluence (-1 and +1) and maximum separation, the maxima/minima of the two entangled sine waves that occur at Cartesian 0.
21. As the unit vector cube corresponds to and describes only half of each of the two sine waves, two unit vector cubes are required for a full cycle. Mandalic geometry as currently formulated with a single unit cube then needs to be extended to at least two of these. Extension in both directions of all three Cartesian axes is easily accomplished by repeatedly inverting the current single unit vector cube.
22. This means that mandalic coordinates alternate positive and negative on both sides of Cartesian 0. The extensions can be continued to infinity in both directions, but not, properly speaking, positive and negative infinity since the manner of extension has created what is essentially a convention-free coordinate system which consists of repeated units of consecutively inverted unit vector cubes in which positive and negative alternate ad infinitum and every Cartesian even-numbered coordinate becomes a “zero equivalent” , or better, a neo-zero in this extended mandalic coordinate system.
23. The resulting geometry is a dynamic one with “points”, “lines”, and “planes” coming into and passing out of existence intermittently in a time-sharing of corresponding Cartesian entities. It “persists” in time and space by means of continuous creation, destruction and re-creation and is “held together” by “force fields” produced and maintained by means of tensegrity which is based ultimately on dimension and number, and by a process that.might best be described as a “weaving of reality” with warp and woof.
24. The 2:1 compositing of dimension involved creates a new number system the members of which are like the real integers in all ways except that they map differently to a Cartesian geometric space. Whereas Decartes assumes that one number maps to one point, MG does not make this assumption which is just an unproved axiom that Descartes makes implicit use of.
25. The method of dimensional compositing automatically results in a mandalic formation having a geometric progression through three Euclidean/Cartesian dimensions from periphery to center (origin).
26. Currently MG is limited to a description of unit vectors in a composite hybrid 6D/3D geometry but can be extended to include all scalar values and any even number of dimensions.
27. The notation system used is borrowed from Taoism and foreign to most Western mathematicians. It is, however, basically equivalent to Cartesian coordinate signs (yin=minus; yang=plus); ordered pairs (=bigrams); and ordered triads (=trigrams); and extends these concepts to include ordered quads (=tetragrams) and ordered sextuplets (=hexagrams).
28. This notation system is used rather than the usual Cartesian notation because it is much easier for the mind to manipulate dimensional numbers using it. It takes only a little practice to become accustomed to using it. Without its use, understanding of mandalic geometry becomes extremely difficult, if not impossible.
29. As MG views a point as a concurrence of various different dimensions, it interprets Cartesian ordered pairs and triads, and their extensions to higher dimensions, as tensors and treats them as such. This makes it possible to apply operations of addition and multiplication to these mathematical entities in a manner analogous to the way William Rowan Hamilton applied these operations to complex numbers by way of what he called “algebraic couples”.
30. The probabilistic mandalic form that is the hallmark of MG conveys and necessitates a new interpretation of zero(0). In MG “zero” is not the empty null that it is in Cartesian geometry and Western mathematics generally, but rather a fount of being, so to speak, and a logic gate spanning dimensions. Wherever a zero occurs in Cartesian coordinates two Cartesian-equivalent forms are found in mandalic coordinates. So in the mandalic cube based on unit vectors the twelve edge centers, having a single Cartesian zero, have two Cartesian-equivalent forms (hexagrams); the six face centers, having two Cartesian zeros, have four Cartesian-equivalent forms; and the single cube center, the Cartesian origin point with three zeros, has eight Cartesian-equivalent forms.
31. Thisalternative zero and the mandalic structure it inhabits force the creation of four different amplitudes of dimension in the 6-dimensional unit vector cube. These are not independent but all mutually dependent and holo-interactive within the composite 6D/3D coordinate system. All of this occurs in a context reminiscent of the one inhabited by nuclear particles. The mapping proposed by MG may in fact model the elementary force fields, electromagnetism and quantum chromodynamics. It suggests a possible mechanism for formation of the state of matter known as a quark-gluon plasma. Hidden within it may even be the secret of quantum gravity.

© 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 312-

Beyond the Enlightenment Rationalists:
From imaginary to probable numbers - II

(continued from here)

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.

Mandalic Line Segments,
Entanglement and Quantum Gravity
Part I

(continued from here)

We are going to consider once again now geometric line segments of mandalic geometry  and  their relation to Cartesian line segments and the Western number line. Yes,  this is sort of a detour from what I stated we would look at next. But this is not unrelated and lies at the very heart of mandalic geometry, and I’m not yet ready to address what I projected in the last remark of my previous post.

I keep returning to this subject because of its extreme importance. Beyond its significance to understanding the logic encoded in mandalic geometry and the I Ching, I believe it may also hold the key to quantum entanglement and quantum gravity.  Despite the fact that mandalic line segments are really fundamentally mental constructs,  a fiction of sorts, it is still important to understand how they are composed and how their components interact.  Though they may themselves be fictions,  the line segments and the points that compose them do in fact map a number of physical entities,  realities that may be related to quantum numbers and quantum particles and states.

When Descartes invented his coordinate system, with its points and line segments,  he based his system on the number line extended to two or three dimensions. In modeling it on the number line the space he described was imagined to bear a  necessary  one to one correspondence to the real numbers.^[1]  However this  1:1 mapping  of geometric space to the real numbers was a premise implicitly assumed by Descartes.  It was in fact axiomatic,^[2]  but apparently not stated as such.^[3]  As a result, the presumed relation has become a blind spot^[4] in Western thought,  never proved nor disproved, at least not at subatomic scales.^[5]

Neither mandalic geometry nor the primal I Ching make such an assumption. In place of Descartes’ 1:1 correspondence of geometric space and the numbers on the number line, we find a mandalic arrangement in which there are different categories of spatial location which can host one or more discrete numbers in a probabilistic manner.  This creates various dimensional amplitudes and a multidimensional waveform of component entities.^[6]

To my mind these characteristics of the mandalic coordinate system in combination with others described elsewhere make it more relevant to investigation and interpretation of many quantum phenomena which are as yet poorly understood than Cartesian coordinate dynamics may be and without need for recourse to imaginary numbers and complex plane.

(continuedhere)

Image: 6 steps of the Sierpinski carpet, animated. By KarocksOrkav (Own work) [CC BY-SA 3.0],via Wikimedia Commons

Notes

[1] Real numbers are numbers that can be found on the number line. This includes both the rational and irrational numbers.

[2] That is to say, taken for granted as self-evident.

[3] See Note [4] here.

[4] We have lived with this unproved premise so long that we no longer even question it,  or imagine that there might be an alternative which conforms better to reality at certain scales, notably subatomic scales.  The I Ching also seems to suggest  that a complete true description of complex relationships that involve a large number of dimensions,  including complex societal relationships,  requires more than a simple 1:1 correspondence between the notational symbols involved and the realities they represent.

[5] And from what I can see, no one seems to have much interest in proving or disproving this assumption.

[6] When speaking about hexagrams the number of dimensions involved is six as each Line of the hexagram encodes a value for a single distinct dimension in a 6-dimensional space.  In a hybrid 6D/3D compositing of dimensions though, two such Lines in relation reference a single Cartesian dimension in 2- or 3-space.  A concept not to be missed here is that  interactions of quantum particles  may well involve such  integration of dimension,  of dimensions  we are not even aware of beyond the unsettling fact  they upset the neat applecart of customary conceptual categories.

© 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 301-

Beyond Descartes - Part 9:
The Potential Plane
and Probable States of Change

Composite Dimension and
Amplitudes of Potentiality
Episode 3

(continued from here)

We have seen that an imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit i,  which is defined by its property ixi =−1.  The square of an imaginary number bi is −b².  For example,  6i is an imaginary number,  and its square is −36.^[1] Other than 0,  imaginary numbers yield negative real numbers when they are squared.^[2]

Turning now to potential numbers, we can similarly define a unit of potentiality p by the property p x -p = -1. [Long pause here waiting for the other shoe to drop.] Just a minute, you say, that’s just like 1 x -1 = -1.  Yes, it is. And that is just the point. All real numbers. Nothing to imagine. And Descartes finally vindicated after all these years - imaginary numbers just imaginary after all.  But how does this work? Or does it even work?  What exactly is the point? Is this a joke? It’s no joke, I assure you.  It’s an easier and better way to achieve the same ends - - - and more. Muchmore.

The secret is in the sauce, I say slyly. Really? Well, yes - in a way. Though imaginaries use a sauce with nearly identical ingredients.  The recipe is p + (-p) = 0. And, of course, i + (-i) = 0 as well.  The trick is in how - - - and where - - - the sauce is applied.  In the potential plane the sauce is applied more liberally in more locations for greater lubrication.

Levity aside. (This is after all a TST^[3].) The complex plane uses a single axis.  This axis represents a new dimension, wholly distinct from the x, y and z dimensions.  Strangely,  we’re never informed where this axis/dimension might be located,  just that it is somewhere other than where x, y and z are located. Stranger still, the complex plane allocates the y-axis of the Cartesian plane for its own use in location of its points. Although never specifically mentioned, to my knowledge, I surmise the imaginary dimension exists in what mathematics and physics both call phase space.^[4]

The mandalic or potential plane uses no such underhanded plan. It openly posits the existence of six new dimensions, allocated equally with two accompanying each of the Cartesian dimensions,  all overtly evident. (All nine spatial dimensions in plain sight together, that is.)  Nothing left to the imagination. As the new dimensions are made commensurate with the old in a hybrid geometric display,  no imaginary dimension is needed. Coordinates of  all potential dimensions  are  readily communicable  with the real number system through all of the ordinary Cartesian dimensions concurrently along with the Cartesian coordinates.  Moreover,  mandalic geometry conjectures that the ordinary Cartesian dimensions may in fact originate in  interactions among number species  of potential dimensions filtered through impacts on inherited biological sensory mechanisms.^[5] This raises yet another interesting possibility.^[6]

In the long convoluted history of mathematics, the imaginary numbers were introduced as a correlative to the number line with its real numbers. That meant, among other things, that they were linear, consisting of a single dimension.  The  complex plane  related the two
in a kind of hybrid geometry that consisted of one real dimension and one imaginary dimension.  Mathematician  William Rowan Hamilton in 1843 proffered the  quaternions,  a number system that extends the complex numbers to three dimensions, whereupon things went, to my mind, from bad, to very much worse.

Quaternions came with certain dysfunctional characteristics, among them,  the fact that multiplication of two quaternions is noncommutative. This is problematic.  The imaginary and complex numbers,  at least,  had both been commutative.  Nevertheless, physics endorsed the quaternions as it earlier had imaginary and complex numbers.

Why? Because the quaternions do in fact give partly correct results, and when investigating a dimly illuminated region of reality, such as the subatomic world still is today, even partial results are heartily welcomed if that is all that can be had.  The sad consequence of this, is that physics has been led astray in its quest for truth for over a century now,  because partial truths can be much more misleading than complete errors. Total error is often uncovered much sooner than partial truth, which can pass undiscovered, depending upon circumstances, for a very long time.

Mandalic geometry will be shown to be free of the difficulty posed by noncommutative multiplication. It is fully commutative throughout its nine dimensions (three ordinary, six extraordinary). It was not composed that way from a number line,  with elements that could be commutatively multiplied with one another. It came that way fully formed from the start, in its primeval embodiment  as a multidimensional structure,  expressing behavior intrinsic to holistic nature.

Next time around, we’ll begin to look under the hood of the mandalic approach to geometry and see if we can grokit.

(continuedhere)

Image: (lower left) Imaginary unit i in the complex or Cartesian plane. Real numbers lie on horizontal axis, imaginary numbers on the vertical axis.  By Loadmaster  (David R. Tribble), (Own work) [CC BY-SA 3.0orGFDL], via Wikimedia Commons; (lower right) A diagram of the complex plane. The imaginary numbers are on the vertical axis, the real numbers on the horizontal axis. By Oleg Alexandrov [GFDLorCC-BY-SA-3.0],via Wikimedia Commons

Notes

[1] 6²xi² = 36 x (-1) = -36.

[2] Zero (0) is considered both real and imaginary, and both the real part and the imaginary part are defined as real numbers. (If that makes little sense to you, don’t blame me. I’m just the messenger here, reporting what the mathematicians have stated to be the case.) This seems to me to be purely an arbitrary definition, and it confuses me as much as it probably does you.  Could it be they did this to avoid the situation where 0² x (-1) = -0?  I think I would find that definition less disturbing, welcome even.

[3] Newly coined Internet acronym for Truly Serious Topic. (Not to be confused with TSR Totally Stupid Rules.)

Speaking about “greater lubrication”(wewere a moment ago, remember?), I use the phrase not simply as  a figure of speech,  or a simile,  but rather,  as a metaphor.  "Spicing" of mandalic geometry with all those zeros of potentiality makes for a very “fluidic dish” which, I believe, reflects the changeable nature of reality far better than the stricter, strait-laced coordinates of Descartes or the complex plane are able to do. And it’s not just a matter of fluidity involved here. The mandalic form so begotten is, in fact, a probability distribution through the three Cartesian dimensions concurrently,  which feature alone  makes mandalic geometry an ideal candidate for application to quantum physics.

[4] A phase space of a dynamical system is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space. In a phase space every degree of freedom or parameter of the system is represented as an axis of a multidimensional space; a one-dimensional system is called a phase line, while a two-dimensional system is called a phase plane.  For every possible state of the system (that is to say, any allowed combination of values of the system’s parameters) a point is included in the multidimensional space. [Wikipedia]

[5] I am speaking here of the hybrid 6D/3D formulation of mandalic geometry which combines the features of  dimensional numbers,  potential numbers,  and composite dimension,  this being a fully open access geometric system that has nothing hidden, nothing held back. What you see is what you get. (WYSIWYG)

[6] It is tempting to wonder whether there might be a close connection between the composite dimensions/potential coordinates  proposed by mandalic geometry and the pilot wave theoryorde Broglie–Bohm theory of quantum mechanics. At least there seems to be a correlation  between  David Bohm’s implicate/explicate order and the manifest/unmanifest (potential) coordinates of mandalic geometry.

© 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 285-

Beyond Descartes - Part 5

Reciprocation, Alternation, Decussation
Imaginary and Complex Numbers

(continued from here)

Previously in this blog a number of attempts have been made to explicate the Taoist number line and contrast it with the Western version of the same.  It is essential to do this and to do it flawlessly,  first because different systems of arithmetic result from the two, and secondly because the mandalic coordinate system is based on the former perspective while the Cartesian coordinate system is based on the latter.^[1]

What has been offered earlier has been accurate to a degree, a good first approximation. Here we intend to present a more definitive account of the Taoist number line,  describing both how it is similar to and how it differs from the  Western number line  used by Descartes in formation of his coordinate system.  This will inevitably transport us  well beyond that comfort zone offered by the more accessible three-dimensional cubic box that has heretofore engaged us.

Both Taoist and Western number lines observe directional locative division of their single dimension into two major partitions:  positive and negative for the West;  yinandyang for Taoism.^[2]  There the similarities essentially end.  From its earliest beginnings Taoism recognized a second directional divisioning in its number line, that of manifest/unmanifestorbeingandbecoming.^[3]  The West never did such.  As a result, some time later the West found it necessary to invent imaginary numbers.^[4][5]

Animaginary number is a complex number that can be written as a real numbermultiplied by theimaginary uniti, which is defined by its property i² = −1. [Wikipedia]

Descartes knew of these numbers but was not particularly fond of them.  It was he, in fact, who first used the term “imaginary” describing them in a derogatory sense. [Wikipedia]  The term “imaginary number” now just denotes a complex number with a real part equal to 0,  that is, a number of the form bi. A complex number where the real part is other than 0 is represented by the form a + bi.

In place of the complex plane, Taoism has (and always has had from time immemorial)  a plane of potentiality.  An explanation of this alternative plane was attempted earlier in this blog,  but it can likely be improved. This post has simply been a broad brushstrokes overview. In the following posts we will look more closely at the specifics involved.^[6]

(continuedhere)

Image (lower): A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram representing the complex plane. “Re” is the real axis, “Im” is the imaginary axis, and i is the imaginary unit which satisfies i² = −1. Wolfkeeper at English Wikipedia [GFDL (http://www.gnu.org/copyleft/fdl.html) or CC BY-SA 3.0 (http://creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons

Notes

[1] The arithmetic system derived from the Taoist number line can perhaps best be understood as a  noumenal  one. It applies to the world of ideas rather than to our phenomenal world of the physical senses, but it may also apply to the real world, that is, the real real world which we can never fully access.

Much of modern philosophy has generally been skeptical of the possibility of knowledge independent of the physical senses, and Immanuel Kant gave this point of view its canonical expression: that the noumenal world may exist, but it is completely unknowable to humans. In Kantian philosophy, the unknowable noumenon is often linked to the unknowable “thing-in-itself” (Ding an sich, which could also be rendered as “thing as such” or “thing per se”), although how to characterize the nature of the relationship is a question yet open to some controversy. [Wikipedia]

[2] From the perspective of physics this involves a division into two major quanta of charge, negative and positive, which like yinandyang can be either complementary or opposing.  Like forces repel one another and unlike attract. This is the basis of electromagnetism, one of four forces of nature recognized by modern physics. But it is likely also the basis, though not fully recognized as such, of the strong and weak nuclear forces, possibly of the force of gravity as well. I would suspect that to be the case. The significant differences among the forces  (or force fields, the term physics now prefers to use)  lie mainly, as we shall see, in intricate twistings and turnings through various dimensions or directions that negative and positive charges undergo in particle interactions.

[3] It is this additional axis of probabilistic directional location, along with composite dimensioning, both of which are unique to mandalic geometry, that make it a geometry of spacetime,  in contrast to Descartes’ geometry which, in and of itself, is one of space alone. The inherent spatiotemporal dynamism that is characteristic of  mandalic coordinates  makes them altogether more relevant for descriptions of particle interactions than Cartesian coordinates, which often demand complicated external mathematical mechanisms to sufficiently enliven them to play even a partial descriptive role, however inadequate.

[4] In addition to their use in mathematics, complex numbers, once thought to be  "fictitious" and useless,  have found practical applications in many fields, including chemistry, biology, electrical engineering, statistics, economics,  and, most importantly perhaps, physics..

[5] The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers. He called them “fictitious” during his attempts to find solutions to cubic equations in the 16th century.  At the time, such numbers were poorly understood,  consequently regarded by many as fictitious or useless as negative numbers and zero once were. Many other mathematicians were slow to adopt use of imaginary numbers, including Descartes, who referred to them in his La Géométrie, in which he introduced the term imaginary,  that was intended to be derogatory. Imaginary numbers were not widely accepted until the work of Leonhard Euler (1707–1783) and Carl Friedrich Gauss (1777–1855).  Geometric interpretation of  complex numbers as points in a complex plane  was first stated by mathematician and cartographer Caspar Wessel in 1799. [Wikipedia]

[6] What I have called here the plane of potentiality occurs only implicitly in the Taoist I Ching but is fully developed in mandalic geometry. It may be related to  bicomplex numbers  or tessarines in abstract algebra, the existence of which I only just discovered. Unlike the quaternions first described by Hamilton in 1843, which extended the complex plane to three dimensions, but unfortunately are not commutative,  tesserines or bicomplex numbers  are hypercomplex numbers in a commutative,  associative  algebra over real numbers,  with two imaginary units (designated i and k). Reading further, I find the following fascinating remark,

The tessarines are now best known for their subalgebra of real tessarines t = w + y j, also called split-complex numbers, which express the parametrization of the unit hyperbola. [Wikipedia]

The rectangular hyperbola x2-y2 and its conjugate, having the same asymptotes. The Unit Hyperbola is blue, its conjugate is green, and the asymptotes are red. By Own work (Based on File:Drini-conjugatehyperbolas.png) [CC BY-SA 2.5],via Wikimedia Commons

Note to self:  Also investigate Cayley–Dickson constructionandzero divisor. Remember,  this is a work still in progress,  and if a  bona fide mathematician  believes division by zero is possible in some circumstances,  (as is avowed by mandalic geometry), I want to find out more about it.

© 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 281-

Beyond the Enlightenment Rationalists:
From imaginary to probable numbers - IV

(continued from here)

One of the notable things the Rationalists  failed to take into account in their analysis and codification of square roots  was  the significance of context. In so doing they assured that all related concepts they developed would eventually degenerate into a series of errors of conflation.  Do  not ever underestimate the importance of context.

Mathematicians, for example, can show that for any 3-dimensional cube  there exists  a  2-dimensional square,  the area of which equals the volume of the cube.^[1] And although that is true, something has been lost in translation. This is another of the sleights of hand mathematicians are so fond of.  Physicists cannot afford to participate in such parlor tricks as these, however mathematically true they might be.^[2]

We will begin now, then, to examine how the mandalic coordinate approach stacks up against that of imaginary numbers and quaternions. The former are holistic and respective of the natural order; the latter are irresponsibly rational, simplistic and, in final analysis, wrong about how nature works.^[3] Ambitious endeavor indeed, but let’s give it a go.

We’ve already looked at how the standard geometric interpretation of imaginary numbers in context of the complex plane is based on rotations through continuous Euclidean space.  You can brush up on that aspect of the story here if necessary. The mandalic approach to mapping of space is more complicated and far more interesting.  It involves multidimensional placement of elements in a discrete space, which is to say a discontinuous space,  but one fully commensurate with both Euclidean and Cartesian 3-dimensional space. The holo-interactive manner in which these elements relate to one another leads to a  probabilistic mathematical design  which preserves commutative multiplication,  unlike quaternions which forsake it.

Transformations between these elements are based on inversion (reflection through a point) rather than rotation which cannot in any case reasonably apply to discrete spaces.  The spaces that quantum mechanics inhabits are decidedly discrete.  They cannot be accurately detailed using imaginary and complex numbers or quaternions.  To discern the various, myriad transitions which can occur among mandalic coordinates requires some patience. I think it cannot be accomplished overnight but at least in the post next up we can make a start.^[4]

(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] If only in terms of scalar magnitude. Lost in translation are all the details relating to vectors and dimensions in the original.  Conflation does not itself in every case involve what might be termed ‘error’ but because it always involves loss or distortion of information,  it is nearly always guaranteed to eventuate in error somewhere down the line of argument. The point of all this in our context here is that, in the history of mathematics, something of this sort occurred when the Rationalists of the Enlightenment invented imaginary and complex numbers and again when quaternions were invented in 1843. These involved a disruption of vectors and dimensions as treated by nature. The loss of information involved goes a long way in explaining why no one has been able to explain whyandhow quantum mechanics works in a century or more.  These  misconstrued theses  of mathematics behave like a demon or ghost in the machine that misdirects,  albeit unintentionally, all related thought processes.  What we end up with is a plethora of confusion. The fault is not in quantum mechanics but in ourselves, that we are such unrelentingly rational creatures, that so persistently pursue an unsound path that leads to reiterative error.

[2] Because physicists actually care about the real world; mathematicians, not so much.

[3] It must be admitted though that it was not the mathematicians who ever claimed imaginary numbers had anything to do with nature and the real world. Why would they? Reality is not their concern or interest. No, it was physicists themselves who made the mistake. The lesson to be learned by physicists here I expect is to be careful whose petticoat they latch onto. Not all are fabricated substantially enough to sustain their thoughts about reality, though deceptively appearing to do just that for protracted periods of time.

[4] My apologies for not continuing with this here as originally intended. To do so would make this post too long and complicated. Not that transformations among mandalic coordinates are difficult to understand,  just that they are very convoluted. This is not a one-point-encodes-one-resident-number plan like that of Descartes we’re talking about here. This is mandala country.

© 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
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Beyond Taoism - Part 3
A Multidimensional Number System

(continued from here)

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-