**A Thought Experiment - XIII:**

**Divergence and convergence in mandalic geometry:**

**1- A change of heart **

(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**

(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.

(to be continued)

***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 caseis 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.Degeneracyis 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**

(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.

(to be continued)

***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

**A Thought Experiment - X:**

**An introduction to composite dimension**

(continued from here)

**Let’s admit from the outset** that no one knows what a dimension really is. It has been defined in a number of different ways with various intents and with somewhat limited success. For mandalic geometry dimension has to do with the ways in which parts relate to wholes and to one another at the Planck scale. Which is to say mandalic geometry views dimension as related to combinatorial capacity of quantum entities***** and as well to the associated symmetries and asymmetries so engendered.******

**So we are free to treat dimension** differently than in the strict manner in which Descartes treats it in his coordinate system and geometry. For the purposes of mandalic geometry we will consider dimensions as viewed in our 3-dimensional perception as being composites of two or more higher dimensions which always occur together in our perception of reality.******* In the 6-dimensional geometry developed from the I Ching each of the ordinary three dimensions of space is a composite resulting from explicit interaction of two higher dimensions.

(to be continued)

**Image:** John Tenniel (1, 2) illustration of Tweedledum (centre) and Tweedledee (right) and Alice (left). [Public domain], via Wikimedia Commons

***And in some manner to quantum numbers** and the ways in which they relate to one another.

****The view here is one of dimension as a relational network.**

In physics and philosophy, a

relational theoryis a framework to understand reality or a physical system in such a way that the positions and other properties of objects are only meaningful relative to other objects. In a relational spacetime theory, space does not exist unless there are objects in it; nor does time exist without events. [Wikipedia]

See also** Combinatorics** and **Boolean lattice.**

** ***A more general notion of composite perception** has been advanced by some who suggest that there is evidence that the brain in some ways operates on a slight “delay”, to allow nerve impulses from distant parts of the body to be integrated into simultaneous signals.

^{}

© 2014 Martin Hauser

Rene Descartes (1596 -1650)

Image: René Descartes.

Source: Extrait de “L’Homme et la Terre”. Author: Élisée Reclus (1830-1905) [Public domain], via Wikimedia Commons

(via eternal-hero)

**Euclid meets Descartes**

**Having completed our detour*** through some pertinent aspects of the history of mathematics we are now ready and better equipped to position Euclid’s square in Descartes’ coordinate system.

**If we take four identically sized squares** and place one in each of the four quadrants of coordinate geometry in such manner that all four meet at the origin and none overlap we will take note of a number of interesting features. Though the squares have been chosen so as to be all of one size and therefore congruent by Euclid’s reckoning, still no two squares can be considered identical. They differ not only in position but also in the varying geometric properties their respective positions confer upon them.

**The differences observed from one quadrant** to another are basically attributable to differences involving dimension and sign. Just as the sign structures of the four quadrants differ resulting in uniqueness of the four quadrants, so the four squares occupying the four quadrants are similarly unique.

**In Quadrant I the square has two roots** which are both positive; in Quadrant II the square has a positive vertical root and a negative horizontal root; in Quadrant III the square has two negative roots; in Quadrant IV the square has a positive horizontal root and a negative vertical root. We will soon have to reckon with the (possibly unsavory) mathematical consequence of this situation: that some squares (those in Quadrants II and IV) have negative areas and square roots of opposite sign.******

**Image:** The four quadrants of a Cartesian coordinate system. Created by Gustavb using en:PSTricks. This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.

***Which began here.**

** **This assertion has other **strict dependencies as well. It is important always to remember that mandalic geometry was conceived as a form of geometry incorporating aspects of quantum logic and is most appropriately applied to quantum orders of magnitude. That said, if negative numbers of the linear sort can find meaning and application in human endeavors, say in banking and tallying of debt, there would seem no essential reason why negative planar numbers could not similarly assume pragmatic meaning at the scale of human activity, for instance in referring to matters of land usage, ownership and inheritance.

The projections of arable land lost due to climate change throughout the rest of this century would in some ways be better thought of in terms of negative areas gained rather than positive areas lost. It is much a matter of perspective from the human standpoint. In the future I think it rather likely that inundated lands (seawater gain = land lost) will be viewed as negative areas rather than positive.

Also negative lengths in a planar context, used as vectors, as opposed to scalar(1, 2) one-dimensional quantities, in conjunction with positive lengths can usefully model making and breaking of connections of many different sorts, whether at the quantum scale or otherwise. This is, I think, one subject area ripe for exploration into the true significance of square roots of opposite sign and their relation to, for instance, quantum numbers and Planck-scale symmetries.

© 2014 Martin Hauser