#quantum logic

Toward a new geometry

Definitions and guiding principles

• Dimension is a primitive concept which for purposes here can be defined as any linearly independent parameter or variable.
• Mathematics is most fundamentally about measurement.
• Rules of mathematics are conceptual tools developed by the human mind and are subject to change if and when such should prove necessary or utilitarian.
• Points, lines and planes are products of the human mind. Neither they nor the realities they represent have continuous existence in space or time but can recur intermittently at intervals which may be either regular or irregular. As a matter of convenience for purposes of description, however, any or all of these may be represented as existing in some continuous abstract sense.
• The whole is greater than the part and has emergent properties not found in any single part or any number of subdivisions of the whole.
• The consequences of number structure and interaction cannot be explained entirely on a local basis but emerge from holo-interactive dynamic processs throughout an entire system or any portion thereof in conceptual focus.
• It is always possible to express a reflection as a rotation and a rotation as a reflection. The two are isomorphic via properly chosen operations.
• Symmetry is of great importance but not always obvious.
• Two foundational guiding principles of reality are continuity and change. These then provide the basis and primary focus of mandalic geometry.
• Of the two types of change, cyclic and sequential, cyclic change is the more fundamental.
• Neither coordinates nor coordinate systems are a feature of nature. They are man-made devices which are pragmatic and utilitarian means by which to grasp a reality which itself has no need of them. Geometry nevertheless requires these crutches to exist and execute its functions.
• Geometry frequently also requires conventions of expression to promote widespread understanding of content but should strive to be as convention -free as possible.
• Ambiguity is a permissible feature, in fact a necessary feature, of mandalic geometry. This is related to its probabilistic nature and multiple-valued logic which reflect what we, from our limited vantage point, misunderstand to be paradoxes of nature.

Axioms

• Numbers can be characterized by dimension of context.
• A number may be embedded in multiple dimensions concurrently, existing in variant forms specific to the dimension(s) of context.
• The same number may function differently in different dimensional contexts. The dimension of context of a “point” described by a number or subsidary number delimits its expression. In other words, the expression of a number in spatiotemporal terms is determined by its dimensional context. In referring to a number or subsidiary number in any of their variants, therefore, the dimension of context must always be specified.
• It may not always be possible to identify the full dimensional context of a number. It is sufficient to determine and elucidate those contexts essential to elaboration of the specific operation(s) under present consideration.
• Numbers are not necessarily elemental. They may consist of parts or subsidiary numbers which refer to various different dimensional contexts.
• A point is not dimensionless extension in space. It is an emergent feature of the system-as-a-whole which appears intermittently at the common intersection of three or more dimensions. Points, lines and planes are evanescent occurrences in terms of geometry and spacetime. They, and the things they represent, are fleeting events which come and go. Repetition is possible (in a conceptual sense certainly; possibly in a material/energetic sense as well) but not inevitable.

Rejected definitions

• The definition of a point found in Euclid’s Elements: A point is that which has no part. [This definition may or may not have been in Euclid’s original Elements.]
• The definition of a line found in Euclid’s Elements: A line is breadthless length.
• The definition of a surface found in Euclid’s Elements: A surface is that which has length and breadth only.

Rejected axioms

• 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 has been described as the Cantor–Dedekind axiom. The Cartesian coordinate system implicitly assumes this axiom which then becomes the cornerstone of analytic geometry.

Rejected notions

• Things which coincide with one another equal one another. [Euclid’s Common notion 4]

Also important to note:

• Euclid’s first postulate states that any two points can be joined by a straight line segment. It does not say that there is only one such line; it merely says that a straight line can be drawn between any two points.

Image credit: James Gyre-Naked Geometry

© 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 314-

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

Imaginary numbers arose in the history of mathematics as a result of misunderstanding the dimensional character of numbers.  There was a failure to acknowledge that numbers exist in a context of dimension. This has earlier been addressed at length.^[1]  Simply put, numbers exist always in a particular dimensional context.  Square numbers pertain to a context of two dimensions and therefore to a plane,  not a line.  Square roots then ought justly reference a two-dimensional geometrical context rather than the linear one mathematics has maintained ever since mathematicians of the Age of Enlightenment decreed it so.  Square roots contrary to the way mathematics would have it can neither exist in nor be found in any single line segment,  because they do not originate in the number line but in the two-dimensional square.

Algebra, not geometry, provided the breeding ground for imaginary numbers.  They were given a geometric interpretation as an afterthought only, long after the fact of their invention. Rationalist algebraists, feeling compelled to give meaning to equations of the form b² = -4 came up with the fantastic notion of imaginary numbers. Only indirectly did these grow out of nature, by way of minds of men obsessed with reason.^[2]

Descartes knew of the recently introduced square roots of negative numbers. He thought them preposterous and was first to refer to the new numbers by the mocking name imaginary, a label which stuck and which continues to inform posterity of the exact manner in which he viewed the oddities.  It is one of the ironies of history that when at last a geometrical interpretation of square root of negative numbers was offered it involved swallowing up Descartes’ own y-axis. Poetic justice? Or ultimate folly?

Had the essential dimensional nature of numbers been recognized there would have been no need to inquire what the square root of -1 was. It would have been clear that there was no square root of -1 nor any need for such as +1 also has no square root.  As linear numbers,  neither -1 nor +1 can legitimately be said to have a square root.  Both, though, have two-dimensional analogues and these do have square roots, not recognized as such unfortunately by the mathematics hegemony.^[3]

In the next post we will look at a comparison between imaginary numbers,  which were formulated in accordance with this misconstrual about how numbers relate to dimensions,  and probable numbers which grow organically out of a consideration of how numbers and dimensions actually relate to one another in nature.^[4]  The first of these approaches can be thought of as rational planning by a central authority; the second, as the holistic manner in which nature attends to everything, all at once, and without rational forethought.

(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] See the series of about nine posts that begins here.

[2] The Rationalists missed here a golden opportunity to relate number and dimension by defining square root much too narrowly. They seem to have been so mesmerized by their algebraic equations that they failed to pursue the search into deeper significance pertaining to essential linkages between dimension and number that intuition and imagination might have bestowed.

[3] As Shakespeare correctly pointed out, a rose by any name would smell as sweet. Plus one times plus one certainly equals plus one but that has nothing to do with actual square root really, just with algebraic linear multiplication.  Note has often been made in these pages of the difference between mathematical truth and scientific truth. Whereas mathematics demands only adherence to its axioms and consistency,  science requires empirical proof.  Mathematics defined square root in a certain manner centuries ago, and has since been devoutly consistent in its adherence to that definition.  In so doing it has preserved a cherished doctrine of mathematical truth, as though in formaldehyde.  It has also for many centuries contrived to be consistently scientifically incorrect.  The problem lies in the fact it has converted physicists and near everyone else to its own insular worldview.

[4] For an early discussion of the probable plane, potential dimensions, and probable numbers see here.

Neo-Boolean - II: Logic Gates
Thinking Inside the Lines

(continued from here)

We have already looked briefly at three of the more important Boolean operators or logic gates:  AND, OR,andXOR.NOT just toggles  any two Boolean truth values  (true/false; on/off; yes/no).  Here we introduce two new logic gates which do not occur in Boolean algebra. Both play an important role in mandalic geometry though.

We’ll refer to the first of these new operators or logic gates as INV standing for  inversionorinvert.  This is similar to Boole’s NOT except that it produces toggling betweeen  yang/+ and yin/- instead of 1 and 0. Because it is based on binary arithmetic, Boole’s NOT has been thought of as referring to inversion also (as in ONorOFF). Although both ANDandINV act as toggling logic gates they have very different results in the greater scheme of things,  since nature has created a  prepotent disparity between a  -/+ toggle  and a  0/1 toggle  in basic parameters of geometry, spacetime, and being itself. This makes Boole’s AND just a statement of logical opposition, notinversion.

Recognition of this important difference is built into mandalic geometry structurally and functionally,  as it is also into Cartesian coordinate dynamics and the logic of the I Ching,  but lacking in  Boole’s symbolic logic. This is necessarily so, as there is no true negative domain in Boolean algebra.  The OFF state of electronics and computers, though it may sometimes be thought of in terms of a negative state, is in fact not. It relates to the  Western zero (0), not the  minus one  of the number line. Where Boolean algebra speaks of  NOT 1  it refers specifically to zero and only to zero. When mandalic geometry asserts  INV 1  it refers specifically to  -1  and only to  -1 . The inversion of yang then is yin and the inversion of yinisyang.^[1]  In the I Ching,  Taoist thought,  and mandalic geometry the two are not opposites but complements and, as such, interdependent.

The second added logic gate that will be introduced now is the REV operator standing for reversionorrevert. This operator produces no change in what it acts upon.  It is the multiplicative identity element (also called the neutral elementorunit element),  as INV is the inverse element. In ordinary algebra the inverse element is -1, while the identity element is 1. In mandalic geometry and the I Ching the counterparts are yinandyang, respectively. If Boolean algebra lacks a dedicated identity operator, it nonetheless has its Laws of Identity which accomplish much the same in a different way:

• A = A
• NOT A = NOT A

Again, Boolean algebra has no true correlate to the INV operator. There can be no  sign inversion formulation  as it lacks negatives entirely. Although Boolean algebra may have served analog and digital electronics and digital computers quite well for decades now,  it is incapable of doing the same for any quantum logic applications in the future, if only because it lacks a negative domain.^[2]  It offers up bits readily but qubits only with extreme difficulty and those it does are like tears shed by crocodiles while feeding.

(to be continued)

Image: Boolean Search Operators. [Source]

Notes

[1] Leibniz’s binary number system, on which Boole based his logic, escapes this criticism, as Leibniz uses 0 and 1 simply as notational symbols in a modular arithmetic and not as  contrasting functional elements in an algebraic context  of either the Boolean or ordinary kind.

In the field of computers and electronics,  Boolean refers to a data type that has two possible values representing true and false.  It is generally used in context to a deductive logical system known as Boolean Algebra. Binary in mathematics and computers, refers to a base 2 numerical notation. It consists of two values 0 and 1. The digits are combined using a place value structure to generate equivalent numerical values. Thus, both are based on the same underlying concept but used in context to different systems. [Source]

[2] Moreover,  I expect physics will soon enough discover that what it now calls antimatter  is in some sense and to some degree a necessary constituent of  ordinary matter.  I can already hear  the loudly objecting voices  declaring matter  and  antimatter  in contact  necessarily annihilate one another,  but that need not invalidate the thesis just proposed.  My supposition revolves around the meaning of “contact” at Planck scale and the light speed velocity at which subatomic particles are born, interact and decay only to be revived again in an eternal dance of creation and re-creation. Material particles exist in some kind of structural and functional  homeostasis,  not all that unlike the  anabolic  and catabolic mechanisms that by means of negative feedback maintain all entities of the biological persuasion in the  steady state  we understand as life. Physics has yet to  get a full grip  on  this  aspect of reality,  though moving ever closer with introduction of quarks and gluons to its menagerie of performing particles.

© 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 304-

Mandalic geometry, Cartesian coordinates and Boolean algebra: Relationships - I

(continued from here)

In attempting to understand the logic of the I Ching it is important to know the differences between ordinary algebra  and  Boolean algebra and how Boolean algebra is related to the binary number system.^[1]

In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted
1 and 0 respectively. Instead of elementary algebra where the values of the variables are  numbers,  and the  main operations  are  addition and multiplication,  the main
operations of Boolean algebra are the conjunctionand, denoted ∧, the disjunctionor, denoted ∨, and the negationnot, denoted ¬. It is thus a formalism for describing logical relations in the same way that ordinary algebra describes
numeric relations. [Wikipedia]

Whereas in elementary algebra expressions denote mainly numbers, in Boolean algebra they denote the truth values false and true. These values are represented with the bits (or binary digits), namely 0 and 1.  They do not behave like the integers  0 and 1,  for which
1 + 1 = 2, but may be identified with the elements of the two-element field GF(2), that is, integer arithmetic modulo 2,  for which 1 + 1 = 0.  Addition and multiplication then play the  Boolean roles  of  XOR  (exclusive-or)  and  AND  (conjunction)  respectively, with disjunction  x∨y  (inclusive-or)  definable as  x + y + xy. [Wikipedia]^[2]

Mandalic logic already occurs fully in the structure and manner of divinatory practice of the I Ching,  if some of it only implicitly.  Although mandalic geometry does not originate from either Boolean algebra or the Cartesian coordinate system but from the primal I Ching which predates them by millennia, it does combine and augment aspects of both of these conceptual systems. It extends Boole’s system of symbolic logic to include an additional logic value represented by the number -1.  This necessitates modification of some of Boole’s postulates and rules,  and increases their total number through introduction of some new ones.  The hexagrams or native six-dimensional mandalic coordinates of the I Ching are related to Cartesian triads composed of the numbers -1, 0, and 1,  making these two geometric systems  commensurate  by means of composite dimension,  a 6D/3D hybridization or mandalic coordination of structure and function (or space and time).^[3]

The introduction of composite dimension produces four distinct dimensional amplitudes  and  is solely responsible for the mandalic form. For anyone reading this who might be down on sacred geometry,  itself a subject which I respect and admire, let it be known that I am talking here about genuine mathematics and symbolic logic,  and my suspicion is that there is some genuine physics involved as well.

Kalachakra Mandala

The mandalic number system, then, is a quasi-modular number system, different from Leibniz’s binary number system which is fully modular.  Boole’s rule  1 AND 1 = 1  still holds true in mandalic logic.  However we must add to this the new logic rule that  -1 AND -1 = -1.  Individually the two rules are modular,  based on a clock arithmetic using a modulo-3 number system rather than Leibniz’s modulo-2 or binary number system, but with yet another added twist.

Together the two rules prescribe a compound system, one which is not singly modular but doubly modular.  The two components, yinandyang, are complementary and are inversely related to one another in this unified system.  This  logic organization  appears based on the figure 8 or sine wave and its negative,  allowing for periodicity, for recursive periods of interminably repeating duration,  and,  perhaps most importantly,  for wave interference,  of  constructive  and  destructive  varieties. These two geometric figures also engender an unexpected decussation of dimension not recognized by Western mathematics.  This is so because 1 AND -1 = 0 and  -1 AND 1 = 0.  The surprise here  is that  there are two distinct zeros: 0_a and 0_b.^[4] In two- or three-dimensional Cartesian terms there exists no difference between these two zeros.  However,  in terms of 6-dimensional aspects of mandalic geometry  and  the hexagrams of the I Ching, the two are clearly distinct structurally and functionally.^[5]

This arithmetic system is the basis of the logic encoded in the hexagrams of the I Ching. Each hexagram uniquely references a single 6- dimensional discretized point, of which there are 64 total. These 64 6- dimensional points of the mandalic cube are distributed among the 27 discretized points  of the ordinary 3-dimensional cube  through the compositing of dimensions  in such manner  that a mandala is formed which positions  1,  2,  4  or  8 hexagrams at each 3-dimensional point according to the   dimensional amplitude  of the particular point.  This necessarily creates a concurrent probability distribution of hexagrams through each of the three Cartesian dimensions.

TheI Chinguses a dual or composite three-valued logic system.  In place of truth values,  the variables used are yin,  yang  and the two in conjunction.  These fundamentally represent vector directions.  Yin is represented by -1, yang by 1, and their conjunction, using Cartesian or Western number terminology, by zero (0). This symbol does not occur natively in the I Ching though where the representation used is simply a combination of yin and yang symbols, most often in form of a bigram containing both  and  regarded as representing a composite dimension, namely 0_[1]  or  0_[2].^[6]

The two bigrams that satisfy the requirement are

young yang

for 0_[1]

and

young yin

for 0_[2].

Although mandalic logic is in Cartesian terms a 3-valued system, in native terms it is 4-valued.  It is not a simple modulo-3  or  modulo-4 number system, but two interrelated modulo-3 systems combined.  The best way to think about this geometric arrangement is possibly to view it as a single composite dimension having four distinct vector directions: a negative direction represented by mandalic composite yin (Cartesian -1); positive direction represented by mandalic composite yang (Cartesian 1); and two decussating relatively undifferentiated directions in some sort of equilibrium, represented by mandalic 0_[1] (composite yin/yang) and 0_[2] (composite yang/yin).  both of which  devolve  to  Cartesian 0  (balanced vector direction of the origin or center).^[7]

So we’ve seen that the number system used in the I Ching is not binary as Leibniz believed but instead doubly trinary with the two halves, in simplest terms,  inversely related and intertwined.  Still, it was an easy mistake to make because the notation used is binary.  We’ve seen too that all trigrams and hexagrams in the system can be rendered commensurate with the Cartesian coordinate system:  trigrams by simple transliteration, hexagrams by dimensional compositing. What, then, of George Boole and his eponymous logic?  How do they fit in the logic scheme of the I Ching? I’m glad you asked. Stay tuned to find out.

(continuedhere)

Images: Upper: TRANSFORMATION OF THE SYMBOL OF YIN (LINE split in two) AND YANG (STRAIGHT-LINE). BLEND: 4 bigrams, THEN 8 trigrams. (MORAN, E. ET AL. 2002: 77). Found here. Lower: Modified from an animation showing how the taijitu (yin-yang diagram) may be drawn using circles, then erasing half of each of the smaller circles. O'Dea at WikiCommons [CC BY-SA 3.0orGFDL],via Wikimedia Commons

Notes

[1] Boole’s algebra predated the modern developmentsinabstract algebra and  mathematical logic  but is seen as connected to the origins of both fields. Similarly to elementary algebra, the pure equational part of the theory can be formulated without regard to explicit values for the variables.

[2] If you are new to Boolean algebra these definitions may be confusing because in some ways they seem to fly in the face of ordinary algebra.  I’ll admit, I find them somewhat daunting.  Let me see if I can clarify the three examples given in this quote. Those of you more familiar with the language of Boolean algebra might kindly correct me in the event I err.  I’m growing more comfortable with being wrong at times.  And this is after all a work in progress.

• Boolean XOR (exclusive-or) allows that a statement of the form (x XOR y) is TRUE
  if either x or y is TRUE but FALSE if both are TRUE or if both are FALSE.  Since Boolean algebra uses binary numbers and represents  TRUE by 1,  FALSE by 0,  then
            for  x = TRUE,   y = TRUE    x + y = 1 + 1 = 0 ,    so FALSE
            for  x = FALSE,  y = FALSE   x + y = 0 + 0 = 0 ,  so FALSE
            for  x = TRUE,    y = FALSE   x + y = 1 + 0 = 1 ,   so TRUE
            for  x = FALSE,   y = TRUE    x + y = 0 + 1 = 1 ,   so TRUE

• Boolean AND (conjunction) allows that a statement of the form (x AND y) is TRUE
  only if both x is TRUE and y is TRUE. If either x or y is FALSE or both are FALSE
  then x AND y is FALSE. Here algebraic multiplication of binary 1s and 0s plays the
  role of Boolean AND. (Incidentally, binary multiplication works exactly the same
  way as algebraic multiplication. There’s a gift!)
            for  x = TRUE,    y = TRUE      xy  =  1(1) = 1,    so TRUE
            for  x = FALSE,   y = FALSE     xy = 0(0) = 0,   so FALSE
            for  x = TRUE,    y = FALSE      xy = 1(0) = 0 ,  so FALSE
            for  x = FALSE,    y = TRUE      xy = 0(1) = 0 ,  so FALSE

• Boolean OR (inclusive-or) is the truth-functional operator of (inclusive) disjunction,
  also known as alternation. The OR of a set of operands is true if and only if one or
  more of its operands is true. The logical connective that represents this operator is
  generally written as ∨ or +. As stated in the Wikipedia article logical disjunction x∨y
  (inclusive-or) is definable as x + y + xy [(x OR y) OR (x AND y)] as shown below.
  [Note: x AND y is often written xy in Boolean algebra. So watch out whichalgebra
  is being referred to, ordinary or Boolean. Are we confused yet?]
            for  x = TRUE,    y = TRUE      x + y = 1 , xy = 1 ,    so TRUE
            for  x = FALSE,   y = FALSE     x + y = 0 , xy = 0 ,   so FALSE
            for  x = TRUE,     y = FALSE     x + y = 1 , xy = 0 ,   so TRUE
            for  x = FALSE,    y = TRUE      x + y = 1 , xy = 0 ,   so TRUE
  

[3] Fundamentally, though,  the  coordinates of mandalic geometry  refer to vector directions alone, rather than to both vectors and scalars (or direction and magnitude) as do Cartesian coordinates. Yin specifies actually the entire domain of negative numbers rather than just the scalar value -1. Yang similarly refers to the entire domain of positive numbers rather than the scalar value 1 alone. Their conjunction  through the compositing of dimensions,  though represented by the symbol zero (0)  in the format commensurate with Cartesian coordinates,  refers actually to a  state or condition  not found in Western thought  outside of certain forms of mysticism  and other outsider philosophies like alchemy;  equilibration of forces in physics; equilibrium reactions in chemistry; and the kindred concept of homeostasis mechanisms of living organisms found in biology.

[4] This is to Westerners counterintuitive. Our customary logic and arithmetic allows for but a single zero. That two different zeros might exist concurrently or consecutively is - to our minds - irrational and we wrestle mightily with the idea. To complicate matters still more,  neither of these zeros is  conveniently  like our familiar Western zero.  So which should win out here?  Rationality or reality?  In fact,  the decision is not ours.  In the end nature decides.  Nature always decides. It stuffs the ballot box  and  casts the deciding vote much to our chagrin,  leaving us powerless to contradict what we may interpret as a whim. Our votes count for bupkis.

[5] This calls to mind also the Möbius strip which involves a twist that looks very much like a decussation to me.  The decussation or  twist in space  we are talking about here though has a sort of wormhole at its center that connects two contiguous dimensional amplitudes. I can’t say more about this just now. I need to think on it still. It seems a promising subject for reflection. (1,2,3)

[6] It needs to be pointed out here that in mandalic geometry, and similarly in the primal I Ching as well,  a bigram can be formed from any two related Lines of  hexagrams,  trigrams,  and tetragrams. The two Lines need not be (and often are not) adjacent to one another. I would think such versatility might well prove useful for modeling and mapping quantum states and interactions.

[7] Note that yin and yang in composite dimension can each take the absolute values 0, 1, and 2  but when yin has absolute value 2, yang has absolute value 0; when yang has absolute value 2,  yin has absolute value 0.  This inverse relation in fact is what makes the arrangement here a superimposed, actually interwoven, dual modulo-3 number system. It also makes the center points of mandalic lines,squares,  and cubes  more protean and less differentiated  than their vertices and elicits the different amplitudes of dimension.

The composite dimension value at the origin points(centers) of all of these geometric figures is  always  zero  in  Cartesian  terms  since the values of the differing Lines  in  the  two entangled 6-dimensional hexagrams  located here add to zero. But neither of these 6-dimensional entities is in its ground state at the center.  Both  have absolute value 1  at Cartesian 0.  Let me say that again: composite dimension values at the center or origin are zero in Cartesian terms but the values of both individual constituents are non-zero.Yin is in its ground state when yang is at its maximum and vice versa. At the center, since the two are equal and opposite they interfere destructively. This results in a composite zero ground state.

So from the perspective of  Cartesian coordinate dynamics, which is after all the customary perspective in our subjective lives,  we encounter only emptiness. But it is this very emptiness that opens to a new dimension. In the hybrid 6D/3D mandalic cube  only line centers and the cube center  have direct access through change of one dimension to face centers and only the face centers have a similar direct access through a single dimension to the cube center and edge centers. All coexist in an ongoing harmony of tensegrity. There is method to all this madness then.

© 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 302-

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 Taoism - Part 5
A Vector-based Probabilistic
Number System
Part II

(continued from here)

Taoism and the primordial I Chingare in agreement that temporal changes have two different aspects: sequent and cyclic.  Western thought in general follows suit. The I Ching differs from the other two in asserting that  the direction of change - for both sequent and cyclic change - is fully reversible,  with the proviso  that sufficiently small units of measurement are involved.^[1]  The probability that reversal can be achieved  diminishes proportionately to the magnitude of change that has taken place.^[2]

Taoist appropriation of bigrams and trigrams of the I Ching to model such phenomena as change of seasons and phases of the moon  is plausible if not quite legitimate. The natural phenomena so modeled are macroscopic and vary continuouslyandinexorably throughout an ever-repeating cyclic spectrum. And there’s the rub.

As they occur and function in the I Ching bigrams and trigrams are dicontinuous discrete elements,  formed by other similarly discontinuous discretized entities,  and they follow evolutionary courses which are most often nonrepetitive. So the Taoist usage is misleading at best, annihilative at worst. Unfortunately, as the I Ching itself evolved through centuries of commentaries and reinterpretations,  it became  ever more contaminated and tainted by these Taoist corruptions of meaning, at the same time that it was being inundated by  Confucian sociological and ethical reworkings.  What we have today is an amalgam, the various parts of which do not sit well with one another.^[3]

Though it may in part be hyperbole to prove a point,  the stark difference between the two approaches,  that of Taoism and that of the I Ching, is epitomized by comparison of the Taoist diagram of the cycle of seasons with diagrams at the top and bottom of the page,  which are based on  the  number,  logic,  and coordinate systems of   The Book of Changes.^[4] The increased complexity of the latter diagrams should not prove a stumbling block, as they can be readily understood in time with focus and attention to detail.  The  important take-away  for now is that in the I Ching bigrams  exist within a larger dimensional context  than the Taoist diagram avows,  and this context makes all their interactions more variable,  conditional,  and complex. As well, the same can be said of trigrams and hexagrams.

One of the more important aspects of these differences has to do with the notion of equipotentiality.  As bigrams and trigrams function within  higher dimensional contexts  in the  I Ching,  this introduces a possibility of multiple alternative paths of movement and directions of change.  Put another way,  primordial I Ching logic encompasses many more  degrees of freedom  than does the logic of Taoism.^[5]  There is no one direction or path  invariably decreed or favored.  An all-important element of conditionality prevails.  And that might be the origin of what quantum mechanics has interpreted as indeterminism or chance.

Next up, a closer look at equipotentiality and its further implications.

Section F_H(n)^[6]

(continuedhere)

Notes

[1] There are exceptions. Taoist alchemy describes existence of certain changes that admit reversibility under special circumstances.  Other than the Second Law of Thermodynamics (which is macroscopic in origin,  not result of any internally irreversible microscopic properties of the bodies), the laws of physics neglect all distinction between forward-moving timeandbackward-moving time. Chemistry recognizes existence of certain states of equilibrium in which the rates of change in both directions are equal. Other exceptions likely occur as well.

[2] Since change is quantized in the I Ching, which is to say, it is divided into small discretized units,  which Line changes model,  the magnitude of change is determined by the number of Line changes that have occurred  between Point A and Point B in spacetime.  Reversal is far easier to achieve if only a single Line change has occurred than if three or four Lines have changed for example.

[3] Ironically, Taoism itself has pointed out the perils of popularity. Had the I Ching been less popular, less appealing to members of all strata of society, it would have traveled through time more intact.  Unless,  of course,  it ended up buried or burned. What is fortunate here is that much of the primordial logic of the I Ching can be reconstructed by focusing our attention on the diagrammatic figures and ignoring most of the attached commentary.

[4] These diagrams do not occur explicitly in the I Ching. The logic they are based on, though, is fully present implicitly in the diagramatic structural forms of hexagrams, trigrams, and bigrams and the manner of their usage in  I Ching divinatory practices.

[5] Or, for that matter, than does the logic of Cartesian coordinate space if we take into account the degrees of freedom of six dimensional hexagrams mapped by composite dimensional methodology to model mandalic space. (See Note [4] here for important related remarks.)

[6] This is the closest frontal section to the viewer through the 3-dimensional cube using Taoist notation.  See here for further explanation.  Keep in mind this graph barely hints at the complexity of relationships found in the 6-dimensional hypercube which has in total 4096 distinct changing and unchanging hexagrams in contrast to the 16 changing and unchanging trigrams we see here. Though this model may be simple by comparison,  it will nevertheless serve us well as a key to deciphering the number system on which I Ching logic is based as well as the structure and context of the geometric line that can be derived by application of reductionist thought to the associated mandalic coordinate system of the I Ching hexagrams. We will refer back to this figure for that purpose in the near future.

© 2016 Martin Hauser

Please note:  The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent).  This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added.  To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering.  To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 300-

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-

Beyond Taoism - Part 2
Number System of the I Ching

(continued from here)

Many different number systems exist in the world today. Others have existed in times past. The number system we are most familiar with is base 10 or radix 10,  which makes use of ten digits,  numbered  0  to  9. Beyond the number 9, the numbers recapitulate, beginning again with 0 and shifting a new “1” to the 10s position, in a positional number system. Using this conventional technique all integers and decimals can be easily and uniquely expressed.  This familiar  numeral system  is also known as the decimal system.^[1]

Another number system we are familiar with and use every day is the modular numeral system, particularly in its manisfestation of modulo 12, better known as clock arithmetic.  This is a system of arithmetic in which integers “wrap around” and begin again upon reaching a set value, called the modulus. For clock arithmetic, the modulus used is 12. On the typical 12-hour clock,  the day is divided into two equal periods of 12 hours each. The 24 hour / day cycle starts at 12 midnight  (often indicated as 12 a.m.), runs through 12 noon  (often indicated as 12 p.m.),  and  continues  to the midnight at the end of the day. The numbers used are 1 through 11 and 12 (the modulus,  acting as zero).  Military time is similar,  only is based on a 24-hour clock with modulus-24 rather than modulus-12. The modulus-24 system is the most commonly used time notation in the world today.

Binary arithmetic is similar to clock arithmetic, but is modulo-2 instead of modulo-12.  The only integers used in this system are  0 and 1, with the “wrap around” back to zero occurring each time the number 1 is reached.  Computers, in particular, handle this arithmetic system,  which we owe to Leibniz, with remarkable acumen. George Boole also based his true/false logic on binary arithmetic.  This, in itself, accounts for some of its strange, counterintuitive aspects,  like the fact that in Boolean algebra the sum of 1 + 1 equals 0.  Not your father’s arithmetic.  But both Leibniz and Boole found profound uses for it. As did the entire digital revolution.

When we come to consideration of the number system and arithmetic used in the I Ching we can anticipate encountering equal difficulty in comprehension, possibly more. The system employed is a modular one - sort of.  However,  it uses negative 1 (yin) as well as positive 1 (yang) whereas zero (0) is nowhere to be seen, at least not in guise of  an explicit dedicated symbol  earmarked for the purpose. The "wrap around" appears to occur at both -1 (yin)  and  +1 (yang). Something different and quite extraordinary is going on here. This is no simple modular numeral system, though it may be masquerading as one.

Thus far the number system of the I Ching sounds much like that of Taoism. It is not, though. We have some big surprises in store for us.

Section F_H(n)^[2]

(continuedhere)

Notes

[1] See here for a list/description of numeral systems having other bases. A more comprehensive list of numeral systems can be found here.

[2] For explanation of this diagram see here.

© 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 297-

Beyond Descartes - Part 10
Taoism Meets Boolean Logic: Introduction

Logic gate symbols

(continued from here)

Before we can hope to comprehend Taoist arithmetic and geometry we need to take a short detour through Boolean logic. First and foremost, we need to see how Boolean logic^[1] relates to Cartesian coordinates. That will provide what may be the best foundation available for understanding the Taoist approach to mapping of spacetime and the methodology which mandalic geometry derived from it.^[2]

For Descartes, his coordinate system is one thing,  his coordinate geometry another.  For Taoism, the coordinate system is the geometry.^[3] Boolean logic helps to explain how the two perspectives are similar,  how different. Cartesian coordinates are static and passive. Taoist coordinates and the derivative mandalic coordinates are active and dynamic.  In brief, the latter are changeable and self-changeable, a feat carried out by means of a brand of Boolean logic intrinsic to the system. Although it is true that Descartes’ coordinates do encode much the same information,  that is not where their focus of interest lies. Accordingly they turn our own attention elsewhere and we overlook those inherent possibilities.^[4]

Descartes’ geometric system is one based on vectors, that is, on both  magnitude and direction.  But in the scheme of things,  the former has somehow eclipsed the preeminence of the latter in the Western hive mind.  The opposite is true of Taoist thought and of mandalic geometry. Direction is uniformly revered as primary and prepotent. Magnitude, or scale,  is viewed as secondary and subordinate.  This mindset allows the Boolean nuances inherent in the system to come to the fore, where they are more easily recognized and deployed.

From such small and seemingly insignificant differences ensue entirely disparate worldviews.

(continuedhere)

Notes

[1] George Boole’s monumental contribution to symbolic logic was published in 1854 but was viewed as only an interesting academic novelty until the second decade of the twentieth century,  when it was at last exhumed as a mathematical masterpiece by Whitehead and Russell in their Principia Mathematica.

[2] In Boolean logic (Boolean algebra) logical propositions are represented by algebraic equations in which  multiplication  and  addition  (and negation) are replaced with ‘and’ and 'or’ (and 'not’),  and where the numbers  '0’ and '1’ represent 'false’ and 'true’ respectively. Boolean logic has played a significant role in the development of computer programming and continues to do so.

[3] This is true also of mandalic geometry in its current formulation.

[4] This might be a proper place to proclaim that nature has little use for Descartes’ breed of coordinates,  finding them far too stagnant and limiting for her purposes. Fortuitously, she devised her own choice coordinate stock long before Descartes thought to invent his.

© 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 293-

Quantum Naughts and Crosses Revisited - IV
The Cube Sliced and Diced
Cartesian Series: Section F_HE

(continued from here)

Below we have the second of three frontal sections through the 3-cube, labeled with the Cartesian coordinates of each point. This “slice” is through a plane that lies between an identity face, which contains the trigram  HEAVEN,  and an inversion face with the trigram  EARTH.  As such it does not belong fully to either the one or the other,  but it shares some characteristics of both. It is a plane, then, of mediation.  Again we see here nine Cartesian ordered triads. Due to an artifact of the “slicing” procedure,  the four edge centers deceptively appear as though vertices, and the four face centers could be taken as edge centers. Make note that these appearances are illusory.  At the center of this section we have the origin point of the cube, Cartesian (0,0,0).^[1]

The key to labeling of points in this section^[2] and all those to follow can be found here.

Section F_HE

(continuedhere)

Notes

[1] It might be well to note here that the origin point of the coordinate system never appears in either an identity plane or an inversion plane of any of the three section types.  All of the planes in which it appears are mediation planes of three dimensions in the case of the Cartesian 3-cube,  or of six dimensions in the case of the hybrid mandalic 6D/3D hypercube.  This is likely the rationale for why in the  I Ching  a change involving passage through this central point  is referred to as  "crossing the Great Water.“  There must be more than coincidence in the fact that Western thought refers to this point as the "origin” and Taoist thought views it as the source and beginning of all things. It’s not that something important was lost in translation.  The two notions arose independently, from two very different worldviews. Somehow in the scheme of things, the West came to equate “origin” with  "zero"  whereas the East came to equate  "origin"  with “the beginning and end of all things.”  Taoism, in particular, sees in this a focus of both creation and dissolution. As we shall soon enough discover,  this alternative perspective leads to a different species of arithmetic,  one of great antiquity though long lost in the sands of time.  Mandalic geometry has unearthed it and will reveal it here, in this blog, for the first time in millennia.  As a teaser,  it involves a different treatment of what the West calls “zero”. It is an arithmetic more in line with Boolean logic.

[2] The 2-dimensional version of this section is obtained from the  x and y coordinates, which by convention are the first and second, respectively, in the Cartesian ordered triads seen here. So the only difference between this section and the F_Hsectionpreviously viewed is the fact that the z-coordinates here are all zero (0) instead of +1.  In our next section, F_E,  the x and y coordinates will again be as seen here but all z-coordinates will be -1.  I believe I detect a trend developing here.

© 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 289-

Quantum Naughts and Crosses Revisited - III
The Cube Sliced and Diced
Cartesian Series: Section F_H

(continued from here)

The first slice through the cube, shown below, the F_H section,  is the Cartesian xy-plane we’re all familiar with from the 2-dimensional version of the Cartesian coordinate system with the third Cartesian dimension (z) added to the labeling of points.  This gives us nine distinct Cartesian triad points: four vertices, four edge centers, and one face center.  For all of the points, the third Cartesian dimension (z) is constant in this slice,  and the vector value is positive (located toward the viewer with respect to the z=0 value of the z-axis or F_HE plane which we’ll be viewing in the next post.)

The key to labeling of points in this section^[1] and all those to follow can be found here.

Section F_H

(continuedhere)

Notes

[1] This is the frontal section through the cube nearest a viewer. It is Descartes’ xy-plane with label of the third dimension (z) added so each point label shown is a Cartesian ordered triad rather than an ordered pair as textbooks generally show the plane. Why the difference?  Because the geometry texts are interested only in demonstrating the two-dimensional plane in isolation,  whereas we want to see it as it exists in the context of three or more dimensions. Cartesian triads are shown by convention as (x,y,z),  so the xy-plane  emerges from the first two coordinates of the points in this section, and all the z-coordinates seen here are positive (+1). The F_E plane, which we’ll be viewing in the post after next, has all of its x and y coordinates identical to those seen here but its z-coordinates are all negative (-1).

© 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 288-

Quantum Naughts and Crosses Revisited - II

(continued from here)

This post builds on orientational material offered in the previous post.  An explanation of the procedural method of graphic demonstration used in this post and those following can be found there,  and it would be helpful to review that earlier post, if not already done,  before proceeding further.

Due in part to the challenging subject matter,  in part to arduous graphic demonstration,  we’ll approach this investigation in three stages of progressive difficulty. In the first stage we’ll just dangle our feet in the water by looking at how the  "slicing methodology"  works with ordinary three-dimensional  Cartesian coordinates.  In the second stage,  we’ll go waist-deep, and consider the same Cartesian coordinates in their Taoist notation transliteration equivalents.  And in the final stage,  we’ll go for full immersion,  with graphic representation of true mandalic geometry, that is,  plotting all 64 hexagrams  in a hybrid 6D/3D coordinate system using the methodology of composite dimension which, of course, has no analogue in purely Cartesian terms.

At each stage - Cartesian, Taoist transliteration, and mandalic - we’ll look at the respective cube in  frontal,transverse, and sagittal slices, always in that order and always progressing from identity face containing Cartesian (1,1,1),  trigram HEAVEN,  or hexagram HEAVEN  to inversion face, containing Cartesian (-1,-1,-1),  trigram EARTH, or hexagram Earth, as the case may be.

To accomplish our purpose we will require an effective, consistent way to refer to the individual “slices” and each of the 27 Cartesian points. There are three “slices” for each type of sectioning of the “cube”, so a total of nine. I propose that we uniquely identify each “slice” by labeling it with the first letter of the section type  (frontal, transverse, or sagittal)  and the subscript letters “H” for planes containing trigram or hexagram HEAVEN but not Earth, “E” for planes containing trigram or hexagram EARTH but not HEAVEN, and “HE” for planes containing both trigram forms.^[1]

The labels of the sections, then, will be:

• F_H     frontal section containing HEAVEN but not EARTH
• F_HE   frontal section containing both HEAVEN and EARTH
• F_E     frontal section containing EARTH but not HEAVEN
• T_H    transverse section containing HEAVEN but not EARTH
• T_HE   transverse section containing both HEAVEN and EARTH
• T_E     transverse section containing EARTH but not HEAVEN
• S_H     sagittal section containing HEAVEN but not EARTH
• S_HE   sagittal section containing both HEAVEN and EARTH
• S_E      sagittal section containing EARTH but not HEAVEN

For the 27 individual discretized Cartesian points, I propose the following labeling convention:

Each point is to be first identified as to type.  There are four point types: vertex(V), edge center(E), face center(F), and cube center(O).  The cube center corresponds to the Cartesian triad (0,0,0), the origin point of the Cartesian coordinate system. In the Cartesian/Euclidean cube there are 8 vertices, 12 edge centers, 6 face centers, and a single cube center.  The higher dimensional mandalic cube has many more of each of these.

Vertices

Having identified the point type, each point is then further identified by a subscript consisting of the first letter of the name of  trigram or hexagram that is resident at the point.  The single exception to this will be  WATER. To differentiate between  WATER  and  WIND,  I propose using the letter “A” (first letter of “aqua”, Latin for “water”) to specify WATER.  This plan allows us, then, to discriminate among the various vertex points, and also to distinguish them from the other point types.  Accordingly,  we arrive at these labels for the 8 vertex points:

• V_H  HEAVEN
• V_E   EARTH
• V_T  THUNDER
• V_W WIND
• V_A  WATER
• V_F   FIRE
• V_M  MOUNTAIN
• V_L   LAKE

Edge centers

Edge centers will be labeled “E” along with a subscript consisting of the first letter of its two vertices, “A” being used instead of “W” for WATER. Though this may initially seem excessively complicated,  the reasons for setting things up this way will soon be made clear, and it will all become second nature. The 12 edge centers will be labeled as below:

• E_HW
• E_HF
• E_HL
• E_ET
• E_EA
• E_EM
• E_TF
• E_TL
• E_AW
• E_AL
• E_MW
• E_MF

Face centers

There are six face centers.  Three occur in  identity faces  of the cube that contain the trigram or hexagram HEAVEN; three, in inversion faces that contain the trigram or hexagram EARTH. Labeling will be with the letter “F” and a subscript consisting of either “E” for EARTH along with one of its companion diagonal vertices, “W” for WIND, “F”, FIRE, “L”, LAKE or “H” for HEAVEN,  along with one of its companion diagonal vertices, “T” for THUNDER, “A”, WATER, “M”, MOUNTAIN.  So these six face center labels are:

• F_EW
• F_EF
• F_EL
• F_HT
• F_HA
• F_HM

Cube center

The cube center, which is singular in Cartesian terms but a multiple composite in terms of mandalic geometry, will be labeled as:

• O

identifying it as the origin of the coordinate system, that is to say, of both the Cartesian coordinate system and the mandalic coordinate system.

With that, let the games begin!

(continuedhere)

Notes

[1] There are no sections among those described that include both the hexagram HEAVEN and the hexagram EARTH.

© 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 287-

Quantum Naughts and Crosses Revisited - I

(continued from here)

Because nature is ever playful, grokking mandalic geometry is much like a game.  We view it as a largely serious one, though, one that involves combinatorics, Boolean logic, and magic squares and cubes. Groundwork for what appears in this post, and several to follow, was laid in May, 2014 in a series titled “Quantum Naughts and Crosses” which began here.

The game is played on a board or field made of three-dimensional coordinates of the Cartesian variety upon which are superimposed the six additional extraordinary dimensions unique to mandalic coordinates. For convenience and ease of representation,  the board will be displayed here in two dimensional sections abstracted from the Cartesian cube and from the superimposed mandalic hypercube in a manner analogous to the way computed tomography renders sections of the human body.

The sections commonly used^[1] in computed tomographyandmagnetic resonance imaging (MRI) are

• Frontal
• Transverse
• Sagittal

For our purposes here, these can be thought of as

• Planes perpendicular to the z-axis viewed from front to back of cube
• Planes perpendicular to the y-axis viewed from top to bottom of cube
• Planes perpendicular to the x-axis viewed from side right to left of cube

These “cuts” will produce square sections through xy-, xz-, and yz-planes, respectively,  of the Cartesian cube and,  in the case of the mandalic cube, analogous sections of higher dimension.

These choices of sections are made largely for convenience and ease of communicability. They are mainly of a conventional nature.^[2]  On the other hand,  there is special significance in the fact that all three section types progress from identity faces of the cube, containing the trigram or hexagram HEAVEN, to inversion faces, containing trigram or hexagram EARTH.  Some manner of consistency of this sort is necessary.  The one chosen here will make things easier as we progress.

Ourgameboard has 27 discretized Cartesian points,  centered in 3 amplitude levels about the Cartesian origin (0,0,0).^[3] Each point in the figure on the right above is represented by a single small cube,  but in the two-dimensional sections we’ll be using for elaboration,  they will appear as small squares.  So the gameboard is “composed of” 27 cubes arranged in a 3x3x3 pattern. But in descriptions of sections, we will view 9 squares in a 3x3 pattern. This configuration will appear as

But keep in mind each small square in this figure is actually a small cube representing one of the 27 discretized Cartesian points we’ve described.

Until next time, then.

(continuedhere)

Notes

[1] The origin of the word  "tomography"  is from the Greek word “tomos” meaning “slice” or “section” and “graphe” meaning “drawing.” A CT imaging system produces cross-sectional images or “slices” of anatomy,  like the slices in a loaf of bread.  The “slices” made are transverse  (cross-sections from head to toes or, more often, a portion thereof), but reconstructions of the other types of sections described above are sometimes made,  and MRI generates all three types natively.

[2] Admittedly, I’ve chosen the convention here myself and to date it is shared by no one else.  Perhaps at some future time it will be a shared convention.  One can only hope.

[3] These three discrete amplitude levels of potentiality in the mandalic 9-cube correspond geometrically to face centers, edge centers and vertices of the 3-cube of Cartesian coordinates.  They are encoded by the six new potential dimensions interacting with the three ordinary Cartesian dimensions in context of the hybrid 6D/3D mandalic cube. They are a feature of the manner of interaction of all nine temporospatial dimensions acting together in holistic fashion. This should begin to give an idea why there is no Taoist line that can generate a 9-cube in a fashion analogous to the way the Western number line is used to generate the Cartesian / Euclidean 3-cube. The 9-dimensional entity is primeval and a variety of different types of  "line"  can be derived  from it.  Similarly,  the  mandala  of the  I Ching  hexagrams cannot be derived from the logic encoded in any linear structure.  An overarching perspective is required to derive first the mandala of hexagrams and then  from it,  a variety of  Taoist line types.  Nature may be playful,  but it is not nearly as simplistic  as our Western science, mathematics, and philosophy would have 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 286-

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 8:
A Good Convention Gone Bad,
An Opportunity Missed

Composite Dimension and
Amplitudes of Potentiality
Episode 2

(continued from here)

We cannot blame Descartes for imaginary numbers. It was he, after all,  who christened these numbers “imaginary” due to his disdain for them.  We can,  however,  fault him  for his lack of insight  into how his coordinate system could be extended to create a viable substitute to show that imaginary numbers and the complex plane were nonsensical and make them unnecessary. Alas, that was not to be. Certain powerful forces of history decreed that imaginary numbers were here to stay and we seem stuck with them still, nearly five centuries later.

Not all would agree that imaginary numbers are a bad convention. We should all,  however,  be able to agree that they are  a convention and nothing more. They were invented by humanity.^[1]  Mathematics may not have taken to them at first - but did eventually welcome them into its fold for better or worse. The real damage was done when physics did the same without first subjecting the mathematical concepts involved to the kind of scrutiny and empirical review it demands of its own theories.

Where is the proof that imaginary numbers and complex plane in fact apply to the real world and particularly to the subatomic realm?  It is lacking in the main, and though the geometric concepts have indeed been successfully applied to a number of branches of physics  and explanations of  a variety of physical phenomena,  the reconciliation is incomplete,  the fit an uncomfortable one, and too many mysteries remain unexplained.

The term imaginary unit refers to a solution to the equation  x² = -1. By convention, the solution is usually denoted i. As no real number exists with this property,  the imaginary number i extends the real numbers and creates an entirely new and different category of numbers.  And crucially, at this point an assumption is made,  a rather sweeping assumption.  It is assumed that the properties of addition and multiplication we’re familiar with - (closure, associativity, commutativity and distributivity) - continue to hold true for this new species of number, or I should say, for this newly derived artificial species of number.  That may fly in the ivory tower^[2]  of pure mathematics,  but it lacks the wings and propelling force required to maneuver effectively in the real world that physics investigates.  Still,  the complex plane,  generated by mathematically motivated minds,  was soon adopted by physicists the world over.^[3]

Mandalic geometry offers an alternative solution in the effective combination of  dimensional numbers,  composite dimension,  and plane of potentiality. We’ll take a close look at potential numbers first. Let’s see how they stack up against  the imaginary numbers,  how  and where  they differ. Distinctions between complex plane and potential plane are subtle but they make for a world - a universe, actually - of difference. When next we meet, kindly check all preconceptions at the door.  Entirely untrodden paths await.

(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] Let those who suppose differently, who believe them to be an indelible part of nature itself, prove their case. Until they do, I will see fit to call such numbers manmade inventions.

[2] I use the term ivory tower without malice of any kind in this context, rather judiciously, because mathematics demands no more than internal consistency for its particular brand of truth. It is not much interested in examining its definitions and axioms to determine how they shape up against hard reality. Mathematicians leave that  "sordid work"  to physicists and philosophers, both of whom are more willing to dig in  the mire of nature,  seeking its actual relics.  Enthusiastically to persist in such a real world-oblivious manner as pure mathematicians do, I think, requires a very special type of mind, one I don’t fully understand myself.

[3] In some circles this would be considered no less than a monumental leap of faith, particularly in view of the many unproved assumptions made in creation of imaginary and complex numbers. This was, in fact,  the New Faith  promulgated by Descartes’ contemporaries, the rationalists of the Age of Reason,  to supplant the Old Faiths of Religion and Scholasticism.

© 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 284-

Beyond Descartes - Part 7

Composite Dimension and
Amplitudes of Potentiality
Episode 1

(continued from here)

Having frightened away all the cognitive wusses with my remark in that last post about the complexity of composite dimension and of the mandalic coordinate system  based on it,  I have a confession to make to those followers who remain. Although understanding the ideas involved requires a step back and viewing them from a different perspective alien to our Western modes of thought, composite dimension and the plane of potentiality are at once  more natural  and  far less complicated  than are imaginary numbers and the complex plane. Stay with me here. There is a light at the end of the tunnel growing ever brighter.

The 6D/3D mandalic cube is a hybrid structure having four levels of amplitude potentiality represented geometrically by 27 3D points which correspond to Cartesian points centered about Cartesian (0,0,0) and 64 6D points,  corresponding to the 64 hexagrams,  similarly centered and distributed among the 27 Cartesian points  in such a way  as to create a probability distribution through all three Cartesian dimensions,  that is with geometric progression of the number of hexagrams resident in the different amplitudes or orbitals. This gives rise to the mandalic form of the coordinate system. There are  four well-defined orbitals or shells  in this unique geometric arrangement of hexagrams and,  parenthetically, whatever it is they represent in physical terms.^[1]

We can conceptually abstract and decompose the 3D moiety of this concept entity, the part corresponding to Cartesian space. In doing so we identify a cube having a single center and eight vertices, all points by Euclidean/Cartesian reckoning, twelve edges (lines), each having an edge center (points), and six faces (planes), each having a center (point) equidistant from its four vertices. Each vertex point is shared equally by three faces or planes of the cube and each edge, by two adjacent faces or planes. We have  previously analyzed in detail  how the six planes of the 3D cube dovetail with one another and the repercussions involved. (See hereandhere.) One of the most important consequences we find is that each face center coordinates in a special way all four vertices of the face. This becomes particularly significant  in consideration of the composite dimension-derived hypercube faces of mandalic geometry.

The 6D moiety follows an analogous but more complex plan and has been formulated so as to be commensurate with the convention of the Cartesian coordinate system.  It also introduces measurement of a discretized time  to the coordinates,  thus rendering the geometry one of spacetime.  The hybrid 6D/3D configuration introduces probability as well through its bell curve/normal distribution (1, 2) of hexagrams; and also,  the two new directions,  manifestation (differentiation) and potentialization (dedifferentiation).^[2] These unfamiliar directions are unique to mandalic geometry and the I Ching upon which it is based.

In the lower diagram above, the figure on the right represents the skeletal structure of the hybrid 6D/3D coordinate system;  the figure on the left, the skeletal structure of the corresponding 3D Cartesian moiety. The  27 discretized points  of the cube on the left have become 64 points of the 6D hypercube on the right.  In the next post we will begin to flesh these two skeletons out.^[3] The end results are nothing short of amazing.

(continuedhere)

Notes

[1] With this remark I am avowing that mandalic geometry is intended not just as an abstract pure mathematical formulation,  but rather as a logical/geometrical mapping of energetic relationships that exist at some scale of subatomic physics, Planck scale or other. I maintain the possibility that this is so despite the obvious and unfortunate truth  that we cannot now ascertain just what it is the hexagrams represent, and may, in fact, never be able to.

[2] Manifestation/differentiation corresponds to the direction of divergence; potentialization/dedifferentiation, to the direction of convergence. The former is motion away from a center; the latter, motion toward a center. Convergenceanddivergence are the two directions found in every Taoist line that do not occur in Cartesian space, at least not explicitly as such.  There are functions in Cartesian geometry that converge toward zero as a limit. To reach zero in Cartesian space however is to become ineffective. That is quite different from gaining increased potential, potential which can then be used subsequently in new differentiations. (See also the series of posts beginning here.)  Both the terms differentiationanddedifferentiation  were  brazenly borrowed  from the field of biology,  while the designations manifestandunmanifest  have been shamelessly appropriated from Kantian philosophy, though similar concepts also occur in different terminology in deBroglie-Bohmian pilot-wave theoryasexplicitandimplicit.

[3] In the figure of the cube on the lower left above there is a single Cartesian triad (point) identifying each vertex (V),  edge center (E),  face center (F),  and cube center C.  In the right figure, the  hybrid 6D/3D hypercube  at each vertex has one resident hexagram identifying it,  two resident hexagrams at each edge center, four resident hexagrams identifying each face center, and eight resident hexagrams identifying the hypercube center. This brings the total of hexagrams to 64, the number found in the I Ching and the total possible number (2⁶ = 64). This geometric progression of hexagram distribution,  through three Cartesian dimensions constitutes the mandalic form. It is entirely the result of composite dimension.

© 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 283-

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 Descartes - Part 4
Directional Locatives

(continued from here)

Descartes derives his directional locatives from considerations of human anatomy, as does most of Western culture. The descriptive terms generally used for orientation purposes include left/right;up/down; and forward/backward.^[1] The first two sets have been extended also to refer to the cardinal directions, North/South and East/West.

To the degree that they conform to Cartesian coordinates, mandalic coordinates adhere to this schema as well.  However, mandalic geometry and the Taoist I Ching upon which it is largely based constitute a system of combinatorial relationships that is rooted mainly in  radial symmetry rather than bilateral symmetry. For mandalic coordinates, the principal directional locatives can be characterized as  divergentandconvergent, and the principal movements or changes in position, as centrifugalandcentripetal.^[2]

One of the important consequences of this alternative geometric perspective is that the frame of reference as well as the complex pattern produced are more integrative than in the method of Descartes. Looked at another way, Descartes is most enamored by specification of location of individual points whereas mandalic geometry is more concerned with relationships of parts - and the overall unification of the entire complex holistic system.^[3]

From this one seemingly small difference an enormous disparity grows in a manner reminiscent of chaos theory.^[4] Cartesian coordinates and mandalic coordinates can be made commensurate, but remain after all two exclusive systems of spatial awareness,  leading to very disparate results arising out of what seem small initial differences.^[5]

(continuedhere)

Image (bottom): Animation of a double compound pendulum showing chaotic behaviour. By Catslash (Own work). [Public domain], via Wikimedia Commons.^[6]

Notes

[1] Such terminology is of little use, despite its biological origins, to an amoeba or octopus,  not to mention those  extraterrestrials  who have been blessed with a second set of eyes at the back of their heads. (We wuz cheated.)

[2] To be more correct, the radial symmetry involved is of a special type. It is not simple planar radial symmetry, nor even the three-dimensional symmetry of a cube and its circumscribed and inscribed spheres. It is all of those but also the symmetry involved in all the different faces of a six-dimensional hypercube and the many relationships among them.

[3] To be fair, Descartes eventually gets around to relating his points in a systematic whole we now know as analytic geometry (1,2).  But as great an achievement though it might be,  Cartesian geometry  lacks the overarching cosmographical implications which characterize mandalic geometry and the I Ching. Descartes’ system is purposed differently, arising as it does out of a very different world view. To paraphrase George Orwell,

“All geometries are sacred, but some geometries are more sacred than others.”

[4] Chaos theory was summarizedbyEdward Lorenzas:

“When the present determines the future, but the approximate present does not approximately determine the future.”

[5] An example of one unique result of mandalic coordination of space is the generation of a geometric/logical probability wave of all combinatorial elements that occur in the 6D/3D hybrid composite dimension specification of the system. I envision this as offering a possible model at least,  if not an actual explanation, of the  probabilistic nature  of quantum mechanics.  Extrapolating this thought to its uttermost conclusion, it is not entirely inconceivable, to my mind at least, that probability itself might be the result of composite dimensioning. (And for such a brash remark I would almost surely be excommunicated from the fold were I but a member.)

[6] Starting the pendulum from a slightly different initial condition would result in a completely different trajectory.  The double rod pendulum is one of the simplest dynamical systems that has chaotic solutions. [Wikipedia]

© 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 280-

Beyond Descartes - Part 3
Logic Gates and Switches: Introduction

(continued from here)

It has been often noted throughout this work that mandalic geometry does not view points as fundamental geometrical elements in the manner Descartes and Euclid do. It considers them to be evanescent communions of two or more dimensions.  This  alternative perspective  conveys further the insight  that such conjoint formative interface locations both separate and connect. They are both boundaries and tipping points between all the participating dimensions,  what I have whimsically referred to  previously as dimension interchange lanes.  This is a far cry from the way Descartes regards and handles hispoints.

Descartes’points are locations, pure and simple, defining occupants of a uniform geometrical space. They don’t really doorattempt anything; they simply are.  They do not act,  but are acted upon by the equations of Cartesian geometry.  The  points themselves,  for all the reality Descartes attempts to imbue them with, turn out,  when the curtain is drawn,  to be no more capable of mustering an original thought  than is  the Scarecrow in  L. Frank Baum’s  The Wonderful Wizard of Oz.  Being of feeble mind themselves,  they just sit there awaiting brainy algebra to act upon them. In and of themselves,  beyond determining location,  they are essentially impotent.^[1]

A useful way to apprehendpoint locations of mandalic coordinates is to  interpret them  as  logic gates  which can handle  transition operations in a variety of different ways  depending upon the  dimension amplitudes verged on.  Passage through such locations is potentially bidirectional,  in theory if not always in actuality at a given moment, so they accommodate both  convergent and divergent flows  throughout varied amplitude levels of the mandalic structure.  To wit,  they can promote both  differentiationandpotentialization  phases of an evolving process.  Because these points arise through confluence of dimensions,  they bear within their transitory being information imparted by the participating dimensions.  Contrary to Descartes’ simpleminded points, these points have the capacity to encode an intelligence derived from their parent dimensions.^[2]

In electrical engineering,aswitch is an electrical component that can control an electrical circuit  by initiating or interrupting the current  or by diverting it from one conductor to another.  The most usual configuration consists of  a manually operated electromechanical device  having  one or more sets of electrical contacts.  These contacts are connected to external circuits. Each set of contacts can be in either of two states: either “closed” meaning the contacts are touching and electricity can flow between them, or “open”, meaning the contacts are separated in which case the switch is nonconducting. The mechanism that brings about the transition between these two states - openorclosed - can be either a “toggle”  (flip switch for continuous “on” or “off”)  or  “momentary”  (depress and hold for “on” or “off”) type.

Understand that logic gates don’t apply only to electronic devices nor are they controlled only by such devices. The concepts and methodologies involved go far beyond simple electronics.

• Logic gates are primarily implemented using diodes or transistors acting as electronic switches, but can also be constructed using vacuum tubes, electromagnetic relays (relay logic), fluidic logic, pneumatic logic, optics, molecules, or even mechanical elements. With amplification, logic gates can be cascaded in the same way that Boolean functions can be composed, allowing construction of a physical model of all of Boolean logic, and therefore, all of the algorithms and mathematics that can be described with Boolean logic. Wikipedia

For our purposes here and now, we need only mention that scalar numbers and vectors can be implemented in the context of Boolean logic as well.  Indeed, the incessant complex cotillion performed by subatomic particles can likely be subjected to such an analysis or one similar.^[3] And, of course, also digital circuits and computer architecture.

This has been just an introductory teaser to the topic of logic gates in mandalic geometry.  I’m getting my feet wet now myself. This is all still quite new to me so we’ve barely scratched the surface here.  An upcoming post will survey the logic gates and switches identifiable among groups of transliteration Cartesian coordinates and mandalic coordinates. This may take a while to materialize, but I think will be worth the wait.  And in case I forget to bring up the subject of how fractals fit into all this sometime in the next month or two, remind me please that I intended to.

(continuedhere)

Notes

[1] This could be a mathematician’s beautiful dream, but a physicist’s abhorrent nightmare.

[2] Although this statement pertains especially to composite dimension points, it is true, to a degree, of ordinary three-dimensional points as well when viewed in a manner similar to that using trigram tranliterations of Cartesian triads.  This means then that Cartesian coordinates could do the same and to the same degree, if  they were handled in the same manner as trigram coordinates are. The point is they are not and presumably never were.

[3] With that last remark I likely committed quantum mechanical heresy. If I in fact did, so be it. If it doesn’t quite hit the intended mark we can refer to it as steampunk mechanics.

Image (lower): Boolean lattice of subsets. KSmrq. Licensed under CC BY-SA 3.0viaCommons.

© 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 279-

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
n = x + 1 - p. :)

-Page 278-

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-

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 - V

(continued from here)

The four Cartesian quadrants provide the two-dimensional analogue of the number line and its graphic representation in Cartesian coordinate space.  This is the true native habitat of the square and, by implication, of square root.  Because  Enlightenment mathematicians  found fit to define square root in a different context inadvertently  -that of the number line- we will find it necessary to devise a different name for what ought rightly to have been called square root,  but wasn’t.  I propose that we retain the existent definition of tradition and refer to the new relationship between opposite numbers in the square,  that is to say,  opposite vertices through two dimensions or antipodal numbers, as contra-square root.^[1]

Modified from image found here.

Given this fresh context - one of greater dimension than the number line - it soon becomes clear with little effort that a unit number^[2]ofany dimension multiplied by itself gives as result the identity element of that express dimension. For the native two-dimensional context of the square the identity element is OLD YANG,  the bigram composed of two stacked yang (+) Lines,  which corresponds to yang (+1),  the identity element in the  one-dimensional context  of the number line. In a three-dimensional context,  the identity element is the trigram HEAVEN which is composed of three stacked yang (+) Lines.  The crucial idea here is that the identity element differs for each dimensional context,  and whatever that context might be,  it produces no change when in the operation of multiplication it acts as operator on any operand within the stated dimension.^[3]

As a corollary it can be stated that any number in any dimension n composed of  any combination  of  yang Lines (+1) and yin Lines (-1) if multiplied by itself (i.e., squared) produces the identity element for that dimension.  In concrete terms this means, for example, that any bigram multiplied by itself equals the bigram OLD YANG; any of eight trigrams multiplied by itself  equals the trigram HEAVEN;  and  any of the sixty-four hexagrams multiplied by itself  equals the hexagram HEAVEN; etc. (valid for any and all dimensions without exception). Consequently, the number of roots the identity element has in any dimension n is equal to the number 2ⁿ, these all being real roots in that particular dimension.

Similar contextual analysis would show that the inversion element of any dimension n  has  2ⁿ roots of the kind we have agreed to refer to as contra-square roots in deference to the Mathematics Establishment.^[4]

That leads us to the possibly startling conclusion that in every dimension n  there is an  inversion element  that has the same number of roots as the identity elementandall of them are real roots.  For two dimensions the two pairs that satisfy the requirement are bigram pairs

For one dimension there is only a single pair that satisfies. That is (surprise, surprise)  yin(-1)/yang (+1).  What it comes down to is
this:

If we are going to continue to insist on referring to square root
in terms of the one-dimensional number line, then

• +1 has two real roots of the traditional variety, +1 and -1
• -1 has two real roots of the newly defined contra variety,
  +1/-1 and -1/+1

So where do imaginary numbers and quaternions fit in all this? The short answer is they don’t.  Imaginary numbers entered the annals of human thought through error.  There was a pivotal moment^[5]  in the history of mathematics and science, an opportunity to see that there are in every dimension two different kinds of roots - - - what has been called square root and what we are calling contra-square roots.  Enlightenment mathematicians and philosophers  essentially allowed the opportunity to slip through their fingers unnoticed.^[6]

Descartes at least saw through the veil.  He called the whole matter of imaginary numbers ‘preposterous’.  It seems his venerable opinion was overruled though. Isaac Newton had his say in the matter too. He claimed that roots of imaginary numbers “had to occur in pairs.” And yet another great mathematician, philosopher opined.  Gottfried Wilhelm Leibniz,  in 1702 characterized √−1 as  “that amphibian between being and non-being which we call the imaginary root of negative unity.” Had he but preserved such augury conspicuously in mind he might have elaborated the concept of probable numbers in the 18th century.  If only he had truly understood the I Ching,  instead of dismissing it as a primitive articulation of his own binary number system.

(continuedhere)

Image: The four quadrants of the Cartesian plane. By convention the quadrants are numbered in a counterclockwise direction.  It is as though two number lines were placed together, one going left-right, and the other going up-down to provide context for the two-dimensional plane. Sourced from Math Is Fun.

Notes

[1] My preference might be for square root to be redefined from the bottom up, but I don’t see that happening in our lifetimes. Then too this way could be better.

[2] By the term unit number,  I intend any number of a given dimension that consists entirely of variant elements of the number one (1) in either its positive or negative manifestation.  Stated differently,  these are vectors having various different directions within the dimension,  but all of scalar value -1 (yin) or +1 (yang). All emblems of I Ching symbolic logic satisfy this requirement. These include the Line, bigram, trigram, tetragram, and hexagram.  In any dimension n there exist 2ⁿ such emblems.  In sum, for our purposes here, a unit number is any of the set of numbers, within any dimension n, which when self-multiplied (squared) produces the multiplicative identity of that dimension which is itself, of course, a member of the set.

ADDENDUM (01 MAY 2016): I’ve since learned that mathematics has a much simpler way of describing this. It calls all these unit vectors. Simple, yes?

[3] I think it fair to presume that this might well have physical correlates in terms of quantum mechanical states or numbers. Here’s a thought: why would it be necessary that all subatomic particles exist in the same dimension at all times given that they have a playing field of multiple dimensions, - some of them near certainly beyond the three with which we are familiar? And why would it not be possible for two different particles to be stable and unchanging in their different dimensions,  yet become reactive and interact with one another when both enter the same dimension or same amplitude of dimension?

[4] Since in any contra-pair (antipodal opposites) of any dimension, either member of the pair must be regarded  once as operator  and  once as operand. So for the two-dimensional square, for example, there are two antipodal pairs (diagonals) and either vertex of each can be either operator or operand.  So in this case, 2 x 2 = 4.  For trigrams there are four antipodal pairs, and 2 x 4 = 8. For hexagrams there are thirty-two antipodal pairs and 2 x 32 = 64. In general, for any dimension n there are 2 x 2ⁿ/2 = 2ⁿ antipodal pairs or contra-roots.

[5] Actually lasting several centuries, from about the 16th to the 19th century. Long enough,  assuredly,  for the error  to have been  discovered and corrected. Instead,  the 20th century dawned with error still in place,  and physicists eager to explain the newly discovered bewildering quantum phenomena compounded the error  by latching onto  √−1 and quaternions  to assuage their confusion and discomfiture.  This probably took place in the early days of quantum mechanics when the Bohr model of the atom still featured electrons as traveling in circular orbits around the nucleus or soon thereafter, visions of minuscule solar systems still fresh in the mind. At that time rotations detailed by imaginary numbers and quaternions may have still made some sense. Such are the vagaries of history.

[6] I think an important point to consider is that imaginary and complex numbers were, -to mathematicians and physicists alike,- new toys of a sort that  enabled them  to  accomplish certain things  they could not otherwise. They were basically tools of empowerment which allowed manipulation of numbers and points on a graph more easily or conveniently.  They provided
their controllers a longed for power over symbols, if not over the real world itself. In the modern world ever more of what we humans do and want to do involves manipulation of symbols. Herein,  I think,  lies the rationale for our continued fascination with and dependence on these tools of the trade. They don’t need to actually apply to the world of nature,  the noumenal world,  so long as they satisfy human desire for domination  over the world of symbols it has created for itself and in which it increasingly dwells, to a considerable degree apart from the natural world’s sometimes seemingly too harsh laws.

© 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 310-

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.

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

Imaginary numbers arose in the history of mathematics as a result of misunderstanding the dimensional character of numbers.  There was a failure to acknowledge that numbers exist in a context of dimension. This has earlier been addressed at length.^[1]  Simply put, numbers exist always in a particular dimensional context.  Square numbers pertain to a context of two dimensions and therefore to a plane,  not a line.  Square roots then ought justly reference a two-dimensional geometrical context rather than the linear one mathematics has maintained ever since mathematicians of the Age of Enlightenment decreed it so.  Square roots contrary to the way mathematics would have it can neither exist in nor be found in any single line segment,  because they do not originate in the number line but in the two-dimensional square.

Algebra, not geometry, provided the breeding ground for imaginary numbers.  They were given a geometric interpretation as an afterthought only, long after the fact of their invention. Rationalist algebraists, feeling compelled to give meaning to equations of the form b² = -4 came up with the fantastic notion of imaginary numbers. Only indirectly did these grow out of nature, by way of minds of men obsessed with reason.^[2]

Descartes knew of the recently introduced square roots of negative numbers. He thought them preposterous and was first to refer to the new numbers by the mocking name imaginary, a label which stuck and which continues to inform posterity of the exact manner in which he viewed the oddities.  It is one of the ironies of history that when at last a geometrical interpretation of square root of negative numbers was offered it involved swallowing up Descartes’ own y-axis. Poetic justice? Or ultimate folly?

Had the essential dimensional nature of numbers been recognized there would have been no need to inquire what the square root of -1 was. It would have been clear that there was no square root of -1 nor any need for such as +1 also has no square root.  As linear numbers,  neither -1 nor +1 can legitimately be said to have a square root.  Both, though, have two-dimensional analogues and these do have square roots, not recognized as such unfortunately by the mathematics hegemony.^[3]

In the next post we will look at a comparison between imaginary numbers,  which were formulated in accordance with this misconstrual about how numbers relate to dimensions,  and probable numbers which grow organically out of a consideration of how numbers and dimensions actually relate to one another in nature.^[4]  The first of these approaches can be thought of as rational planning by a central authority; the second, as the holistic manner in which nature attends to everything, all at once, and without rational forethought.

(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] See the series of about nine posts that begins here.

[2] The Rationalists missed here a golden opportunity to relate number and dimension by defining square root much too narrowly. They seem to have been so mesmerized by their algebraic equations that they failed to pursue the search into deeper significance pertaining to essential linkages between dimension and number that intuition and imagination might have bestowed.

[3] As Shakespeare correctly pointed out, a rose by any name would smell as sweet. Plus one times plus one certainly equals plus one but that has nothing to do with actual square root really, just with algebraic linear multiplication.  Note has often been made in these pages of the difference between mathematical truth and scientific truth. Whereas mathematics demands only adherence to its axioms and consistency,  science requires empirical proof.  Mathematics defined square root in a certain manner centuries ago, and has since been devoutly consistent in its adherence to that definition.  In so doing it has preserved a cherished doctrine of mathematical truth, as though in formaldehyde.  It has also for many centuries contrived to be consistently scientifically incorrect.  The problem lies in the fact it has converted physicists and near everyone else to its own insular worldview.

[4] For an early discussion of the probable plane, potential dimensions, and probable numbers see here.