A Thought Experiment - XVIII:

Divergence and convergence in mandalic geometry:

6- Axis of potentiality and probability plane

 

image

(continued from here)

We have previously broached the subject of square root as it applies to mandalic geometry and touched briefly upon the related subjects of imaginary and complex numbers (123). Next we will delve more deeply into these topics, considering first how the axis of potentiality is defined and used in mandalic geometry of any chosen number of dimensions. The number of dimensions may be as few as one though the resulting mandala in this case will be a primitive one, in essence equivalent to the coordinate system of 1-dimensional mandalic geometry.*

We will look first at this simplest case, that of a single real dimension positioned in a complex plane along with the non-ordinary dimension of potentiality plotted along a vertical axis (p) perpendicular to the x-axis. In doing this we are making use of a newly defined xp-plane which mandalic geometry labels the probability plane.

Although the xp-plane looks somewhat similar to the xy-plane keep in mind that the extra dimension here has no relation to the ordinary y-axis dimension. It is a newly defined extraordinary dimension of potentiality, distinct from and independent of the ordinary y-axis dimension. This new dimension of potentiality is related to the x-axis dimension and possesses its own axis of reference, the p-axis or axis of potentiality, also newly defined and unique to itself.

There is, of course, precedence for doing something of this sort. The configuration used here is similar to that used in describing the complex plane based on real and imaginary numbers. There are however a number of important differences. Some of the more signficant will be examined in the following posts. These differences will eventually prove invaluable in that they make mandalic geometry ideal for analysis and description of quantum structures and processes.**

(to be continued)

Image: Adaptation of a round dartboard mounted on a square piece of wood. The dartboard has a radius of one unit. The piece of wood is two units square so the round board fits perfectly inside the square.

Authors: Brad Miller, David Ranum 

Source: How to Think Like a Computer Scientist

As used here this is conceptually identical to a unit circle inscribed within the square of side length 2 formed by four unit squares, one in each of the four quadrants and all meeting at the center of the unit circle or origin of the coordinate system, here based upon the complex probability plane.

 

*Note that the diagram above is not yet this fundamental coordinate system of 1-dimensional mandalic geometry. It is the Cartesian analogue of it, prior to insertion of the Taoist “equivalent” notation and complete translation to the effective mandalic form. The resulting diagram of that holo-competent mandalic configuration will be presented in a subsequent post. Here simplistic minimalism is used for the purpose of introducing this unfamiliar concept. Understand that what is seen here is still a degenerate form not yet possessing the full potency of the mandalic form.

**One of the primary goals of mandalic geometry is to demonstrate that it can provide quantum theory an alternative approach to accomplish the same things the complex plane and quaternions (1, 2) do without resorting to the difficult and arcane procedures involved in the use of imaginary and complex numbers. It is, for instance, a commutative system (one which addresses whether a × b = b × a) and, unlike the complex number system and related quaternions, allows this symmetric relationship thus making computation much easier and reflecting an important aspect of physical reality, symmetry, a result the complex number system fails to achieve completely.

© 2014 Martin Hauser

A Thought Experiment - XVII:

Divergence and convergence in mandalic geometry:

5- Restitution of mandalic coordinates

 

King Wen (I Ching)

(continued from here)

We’ve noted previously how Cartesian coordinates are derived from mandalic coordinates. Here we will consider briefly the manner in which mandalic coordinates can be restored from Cartesian coordinates. Unlike the earlier situation where each 6-dimensional mandalic coordinate or hexagram yielded in 3-dimensional space a single Cartesian coordinate triplet*, each Cartesian coordinate triplet can reconstitute 1, 2, 4, or 8 hexagrams depending upon the number of zeros** it contains.

To reaffirm, this asymmetric relation arises because all 3-dimensional Cartesian coordinates are degenerate forms of the 6-dimensional mandalic coordinates described by the corresponding hexagrams. Put another way, the sharing hexagrams together contain more information than the single Cartesian point to which they refer. There is an inevitable information loss in translation from mandalic to Cartesian coordinates. This important fact has far-reaching consequences in any attempt to understand quantum structures and processes.

In order to reinstate the hexagrams or mandalic coordinates from their Cartesian counterparts use must be made of combinatorial mathematics to fill in the informational gaps.*** The formula is as follows: Any x-, y- or z-coordinate that is +1 becomes a solid line and any x-, y- or z-coordinate that is -1 becomes a broken line. In both cases doubling of the resulting line occurs in the hexagram. The x-coordinate becomes lines 1 and 4; the y-coordinate, lines 2 and 5; and the z-coordinate, lines 3 and 6. If on the other hand a zero coordinate is encountered, whether in the x-, y-, or z-coordinate or any combination of these it is translated into two lines in the hexagram, one solid, the second broken and in all combinatorial possibilities (two for one zero, four for two zeroes, eight for three zeros.) 

Image: The 64 hexagrams of the I Ching (King Wen arrangement). By TarcísioTS at pt.wikipedia [Public domain], from Wikimedia Commons.

Note that this is a 2-dimensional presentation of the I Ching hexagrams not the 6-dimensional arrangement applied in mandalic geometry. For the 6-dimensional patterning and a description of how it is generated see the series of blog posts beginning here.

(to be continued)

*In 3-dimensional Cartesian space every point is uniquely defined by a triplet of Cartesian coordinates. From the standpoint of 6-dimensional mandalic geometry though the same point more often than not cannot be uniquely defined by the Cartesian triplet, as the triplet is the degenerate form of a mandalic sextuplet or hexagram, so comprising less information.

**In situations where a Cartesian triplet contains one or more zeros the corresponding Cartesian coordinate location will be shared by two or more hexagrams, the total number of sharing hexagrams being equal to 2 raised to the nth power where n equals the number of zeros in the 3-D Cartesian triplet. In other words, since a Cartesian triplet may contain 0, 1, 2, or 3 zeros it can translate to 1, 2, 4, or 8 hexagrams in accordance with a specific and unique distribution pattern which itself constitutes the mandalic form and which has been described previously in this blog. That distribution pattern can be likened to a probability wave in six dimensions and would have implications important to quantum theory.

***Though only making use of an elementary range of combinatorial members, numbering 1, 2, 4, or 8 depending as noted on the number of zeros present in the specific Cartesian triplet in question.

© 2014 Martin Hauser

A Thought Experiment - XVI:

Divergence and convergence in mandalic geometry:

4- Sources and sinks

 

Roof hafez tomb 

(continued from here)

The universe is an emergent process (12), an exercise in geometry, at once repetitive, recurring in cycles, and non-repetitive, aperiodic. Its evolution is non-linear and not entirely a forward motion. It backtracks at times and, recanting what it has once achieved, sometimes repeats itself and other times not. Its power is in undoing as well as in doing. So this is a quite special kind of emergent process about which we speak here.

Mandalic geometry is similarly based upon emergent processes of a special variety, unusual in that it is imbued concurrently with the yoked capabilities of reversibility and irreversibility by virtue of its combination of cyclicity and aperiodicity. In this emergent system sources* can become sinks and sinks in turn, sources.

A self-sufficient universe demands a geometry which allows for both filling and emptying, especially at the smallest scale of spacetime, what we call Planck scale. It must master through trial and error the primary trick of self-replication: creation alternating with destruction, at certain stages with emphasis ironically more on error and elimination than on ascendancy and success.

Paradoxical as this may seem it is necessary because it sets the scene for the greater success a sustainable universe requires. Unless it is able to reclaim itself from the midst of its errors it cannot be self-sustaining. The true higher sustenance requires, in addition to processes of accumulation and aggregation, those of decomposition and elimination for maintenance and preservation.**

Image: Roof of the tomb of Persian poet Hafez at Shiraz, Iran, Province of Fars. By Pentocelo (Own work) [CC-BY-3.0], via Wikimedia Commons

 

*In information theory any process that generates successive messages can be considered a source of information.

**Compare here the physiologic parallels regarding homeostasis set forth by Walter Cannon in his book The Wisdom of the Body, first published in 1932.

© 2014 Martin Hauser

A Thought Experiment - XV:

Divergence and convergence in mandalic geometry:

3- Nature knows neither lines nor naughts

 

image

(continued from here)

Euclid, Heraclitus and William Blake enter a bar together. As they approach the bartender Heraclitus addresses him first. “I’ll have six beers not twice from the same tap.” Happy to comply, the bartender saunters along his row of taps, stopping at each long enough to fill just one glass. Returning to where Heraclitus is seated at the bar, the bartender sets the six beers before him in a single long row.

Euclid speaks up next. Not to be outdone by a pre-Classical Era Greek, he looks the bartender straight in the eye and orders. “Make mine twelve, six from each of two rows of parallel taps and take care they don’t meet at infinity.”

With the perplexed bartender still pondering Euclid’s unusual request, Blake produces a small thimble from his pocket, tops it up himself from the closest tap, carries the filled thimble to an empty table in a corner of the bar and sets it on the table before him. Staring off into empty space with an altogether vacant look he proceeds then to recite the first stanza of his Auguries of Innocence

image

Seated in another corner of the room, unseen until now in the dimly lit bar, a wizened man with a long white beard, stands up and, approaching the center of the room, joins the fray. First lifting his cup of beer high in the air for all to see he then downs its contents entirely in a single draft, following this with the sublimely assured remark, 

"The usefulness of a vessel comes from its emptiness."

(continued here)

 

Image: Lao Tzu Remove Things Quotation. By Environmental Illness Network(All sizes of this photo are available for download under a Creative Commons license.) LicenseAttributionNoncommercialNo Derivative Works Some rights reserved by Environmental Illness Network

© 2014 Martin Hauser

A Thought Experiment - XIV:

Divergence and convergence in mandalic geometry:

2- Taoism saw it differently

 

Sammelpunkt

(continued from here)

The first way convergence and divergence act as organizing principles with respect to the mandalas of mandalic geometry is in the very structure of those mandalas and in the way they function through time. As is true of all higher order mandalas the I Ching mandalas are intricately structured, having a well-defined network of symmetries and asymmetries relating the center to the periphery* and the periphery to the center.

Centripetal movement toward the center of the mandalic square or mandalic cube involves increase in potentiality and also a corresponding decrease in actuality. It also involves a progressive increase in density of occupation at all points in each passing orbital** or group of similar point categories. The greatest density occurs at the center of the mandala***, the point of origin, which ironically in Western thought is labeled the zero or null point. Taoism, it seems, valued the potential rather more than the already actualized, this being a significant thrust of distinction in the two varying world views.

There is a sense in which the center and the periphery are one. They are related through the twin aspects of potentiality and actualization by the organizing principles of convergence and divergence. Despite the very strict manner in which all the points**** or hexagrams are defined within the network which is the mandala, the diverse avenues of actualization possible through time are myriad as is so in the real universe of which the mandala is a microcosm.

(to be continued)

Image: German standardized pictogram for a fire safety assembly point (a geographically defined safe place where people meet.) By Xavax.[Public domain], via Wikimedia Commons

 

*The term “periphery” here is used to refer not solely to the outer boundary or circumference of the mandalic structure but to the entire structure residing outside or beyond the center, the single point from which divergence and convergence in their various actualization forms emanate and return. From the point of view of convergence the center corresponds to a “black hole”; from the point of view of divergence it corresponds to a “white hole”. Structurally and functionally, then, the center operates as a combined black hole/white hole.

**The different geometric shells can be thought of as being in some way analogous to electron orbitals though the term “orbital” has been used here in a more or less generic manner.

***This is reminiscent of the manner in which Chinese cartographers prior to introduction of Western influences in the 19th and 20th centuries often represented spatial distance independent of our modern concern for mathematical scale, with density sometimes increasing as the center was approached and with greatest density of space shown at the very center of the map where the revered emperor resided.

The artifactual record suggests that scale mapping was not the primary concern of Chinese mapmakers, although they certainly understood its principles .  .  .

Mapmakers could express their concern for measurement in different ways, and they had available different methods of presentation for different purposes  .  .  .

[Cartography in China, p.64]

Regarding one nautical map described the author of this article states:

.  .  . without the lengthy notes on the map itself, the map would be of little use for navigational purposes. The orientation and scale of the image vary across the map, and the notes make one aware of the changes.

[Cartography in China, p.64]

****Note that the terms “point” and “hexagram” are being used interchangeably in this context. Both are complex geometric entities arising at and defining a particular region of spacetime, resulting from intricate interweaving of different dimensions and having evanescent existence. A point in mandalic geometry is not the simple fundamental primitive entity it is in Euclidean geometry, far from it.

The advantage here is that a point, which Euclidean geometry leaves undefined, stating that it has a location but no dimension, is in mandalic geometry fully defined in terms of dimensions and their interactions and is, in spite of its apparently infinitesimal size, both a composite of multiple dimensions and a wholly conditional transitory entity, the location of which may be variable through time with change of its dimensional components. These differences between Euclidean and Cartesian geometry on the one hand and mandalic geometry on the other hand would, of course, have enormously significant ramifications in any application involving quantum processes and structure.

© 2014 Martin Hauser

A Thought Experiment - XIII:

Divergence and convergence in mandalic geometry:

1- A change of heart 

 

image

(continued from here)

Before proceeding to a description of the composite two-dimensional geometry stage I think it proper even at this early “linear” stage to say a few words about divergence* and convergence* as general organizing principles and about their importance to mandalic geometry. This part of the story touches necessarily upon the quite closely related concepts of white holes and black holes.

I know I stated that my purpose in creating this blog was to separate the geometric aspects from the physical aspects of blindmen6.tumblr.com but truth be told I’ve found it impossible to do so fully. I think once we admit time to the mix, and mandalic geometry is very much a geometry of time as well as of space, it becomes a question of how does one divorce pattern from process.

The answer is: it can’t be done. At least not completely. When the process (time-related) aspects of mandalic geometry are removed what remains is dead (functionless) pattern. I envisage mandalic geometry as imbued with a kind of life of its own which involves the full unification of structure and function. I do not aspire to a lifeless geometry.

(to be continued)

Image: A 2D vector field pointing inward. By Duane Q. Nykamp. Licensed under a Creative Commons Attribution-Noncommercial-ShareAlike 3.0 License.

 

*See for example here (1,2,3,4,5,6,7,8) to start.

© 2014 Martin Hauser

A Thought Experiment - XII:

Dimensions of  potentiality and actuality

 

image

(continued from here)

The three dimensions of Cartesian 3D coordinate space which we perceive as our ordinary everyday dimensions are for mandalic geometry composite (or diploid*) dimensions whereas the two extra dimensions of which each is the resultant are singular (or haploid*) dimensions.

The composite dimensions, from our limited human point of view, can be considered “dimensions of actuality”; the singular extraordinary dimensions can be considered “dimensions of potentiality”.** The geometry based upon this supposition postulates the interchangeability of these different varieties of dimension. Mandalic geometry should be viewed as a geometry of spacetime, which is to say one of time as well as of space, and as such is in continual flux.

If this postulate is accepted one consequence of it is that each of the sixty-four hexagrams describes a potential set of circumstances.*** The Cartesian “equivalent” of a hexagram is a point in ordinary 3-dimensional space. As such it does not represent the actualization of all possiblilities inherent in the hexagram but is simply our limited way of viewing space and is a degenerate form**** of the total information contained in the hexagram and its relationships with all other hexagrams or points in the   higher, more comprehensive 6-dimensional space.

Cartesian space is, simply put, a degenerate space actualized in a manner our minds are best able to view and comprehend real potential space of higher dimensions. The process of actualization necessarily always involves the giving up of some potentiality. Such is the human condition and the way of the natural world.

(continued here)

 

*to borrow a term from the science of genetics.

**This is, in fact, quite analogous to the situation in reproductive biology existing between the number of sets of chromosomes in diploid cells (zygotes and body cells) and haploid cells (germ cells or gametes). Read all about ploidy here. This is, I think, not simply an interesting aside and coincidence. It points to how nature in managing its economy tends to make use of similar patterns of structure and function at very different operational scales.      

"Simplicity is the ultimate sophistication."

                               - Leonardo da Vinci

 ***This is consistent with the manner in which the I Ching has been viewed and used in traditional Chinese culture for hundreds, possibly even thousands of years.

****Degeneracy in mathematics refers to a special case derived from some more general case and lacking one or more of the potencies present in the original.

In mathematics, a degenerate case is a limiting case in which an element of a class of objects is qualitatively different from the rest of the class and hence belongs to another, usually simpler, class. Degeneracy is the condition of being a degenerate case.

A degenerate case thus has special features, which depart from the properties that are generic in the wider class, and which would be lost under an appropriate small perturbation. [Wikipedia]

See also article on Degenerate energy levels with reference to quantum states.

© 2014 Martin Hauser

A Thought Experiment - XI:

Not just for Flatlanders

 

image

(continued from here)

The geometry of one composite dimension is in one sense built from the bigrams. Its four component points (three in ordinary 1-dimensional space) indeed look exactly like the bigrams. There is, however, at least one important difference. The bigrams as shown heretofore refer only to the four quadrants of the xy-plane in Cartesian geometry. The points of all the dimension-composite planes (xy, xz and yz) require an enhanced notation derived from the hexagram notation for consistency and clarity. The significant idea here is that the context of a bigram determines its unique signature. More on this presently.*

It is important to understand that as used above the bigrams refer only to the x-axis, not to the xy-plane. The second (upper) line here is not a y-axis coordinate but rather a second x-axis coordinate in a new kind of geometry derived from composite dimensions. At this point it would be a good idea to review the differences between the real number line and the mandalic number line.

The simple rule for translating the composite coordinates of mandalic geometry into the ordinary coordinates of Cartesian plane geometry is to add the two composite coordinates and divide by 2. This operation will yield one of only three possible results: +1, -1, or 0. The two bigrams that share the zero coordinate of Cartesian space both yield Cartesian x=0 when the stated operation is performed upon them.

Note that this mechanism obviates any need for the vacuous zero of Western mathematics, replacing it with the fully functional and capacious “zero” of Taoist notation. Although the two alternate forms of “zero” here are necessarily shown side by side it should be understood that they are in actuality superpositions of one another at the origin and may be properly shown with either to the right of the other.

(continued here)

 

*For the time being, for simplicity and preciseness in introduction of the subject we will continue to use the bigram notation as before, here in reference to the x-axis rather than the xy-plane. We will not need to use the enhanced bigram notation until we arrive at the description of the geometry of two composite dimensions to follow in subsequent posts.

Nevertheless it would be good to keep in mind that every point in mandalic space consists of three ordinary dimensions determined by six extraordinary dimensions and therefore every point requires a six-line designation for full characterization.

Ergo, the four higher dimension points shown in the diagram above if placed in context of three ordinary (or six extraordinary) dimensions would require six lines for unique delineation and appear as hexagrams. Moreover, the bigrams shown refer to the xy-plane and are derived from the first and fourth lines of the hexagram, not from the first and second lines as might be initially intuitively thought proper.

The short explanation for this is that hexagrams are composed of two trigrams, an upper trigram and a lower trigram Each trigram is composed of three lines referring to three dimensions of space, the first to the x-axis dimension, the second to the y-axis dimension and the third to the z-axis dimension. Therefore the two x-axis coordinates of mandalic geometry notation come from the first (lowest) line of the lower trigram and the first (lowest) line of the upper trigram (or put another way as we have, from the first and fourth lines of the hexagram.)

 

© 2014 Martin Hauser