#probability distribution

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A Recap of Some Important Ideas Regarding Mandalic Geometry

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

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. :)

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#mandalic geometry #dimensional numbers #dimension #probability distribution #probable numbers #composite numbers #composite dimension #conplex numbers #complex plane #imaginary numbers #real numbers #algebra over a field #tensor #mandalic form #mandala

@mandalicgeometry

Sat, 30 Apr 2016 15:51:11

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.

-Page 302-

@mandalicgeometry

Tue, 23 Feb 2016 17:46:48

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.

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@mandalicgeometry

Fri, 19 Feb 2016 18:47:32

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.

-Page 283-

@mandalicgeometry

Wed, 04 Nov 2015 21:49:27

#probability distribution

A Recap of Some Important Ideas Regarding Mandalic Geometry

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

Notes

Mandalic Line Segments, Entanglement and Quantum Gravity Part I

Notes

Beyond Descartes - Part 7

Composite Dimension and Amplitudes of Potentiality Episode 1

Notes

Mandalic Line Segments,
Entanglement and Quantum Gravity
Part I

Composite Dimension and
Amplitudes of Potentiality
Episode 1