#discrete geometry
Quantum Naughts and Crosses Revisited - VIII
The Cube Sliced and Diced
Transliteration Series: Section FH(n)
(continued from here)
We come now to the Taoist/Cartesian transliteration sections of the three-dimensional cube.[1] The frontal FH section seen below 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 FHE plane which we’ll be viewing in a future post.)
The diagram shown here relates changing and unchanging trigrams of the I Ching to corresponding Cartesian ordered triads. Descartes views each of his ordered triads as referring to a single point having substantive reality in Cartesian geometric space. The I Ching and mandalic geometry, on the other hand, regard the trigrams as evanescent composite states of being in a spacetime which is ever-changing. They are relational elements in some ways analagous to the subatomic entities of particle physics.
Accordingly, it should be further understood each “point” here, though shown as a flat “square”, has a third dimension implied, and is therefore actually a “cube”, only one face of which is seen.[2] Mandalic geometry considers the point a fictional device which actually refers to a common intersection of three or more planes in a three-dimensional context, or two or more lines in a two-dimensional context. Moreover, mandalic geometry is a discretized geometry, and the trigram must be considered as having a distributed domain of action. This is illustrated in all the Cartesian transliteration points by distributing eight copies of trigrams with appropriate changing and unchanging lines among eight vertex-analogues of each Cartesian point.
The key to labeling of points in this section[3] and all those to follow can be found here. Additionally, by tradition, adding an “x” to a yin line indicates it is a changing line and adding an “o” to a yang line indicates it is a changing line. A changing yin line is considered an old yin line which is changing to a yang line; a changing yang line, an old yang line that is changing to a yinline.
Vector addition of two or more yinlines yields a yin line as result. Vector addition of two or more yang lines gives a yang line as the result. Vector addition of an unequal number of yin lines and yang lines yields as result that vector (yinoryang) in excess. Vector addition of an equal number of yin lines and yang lines gives as result Cartesian zero which, in mandalic systematics is to be considered a vector (direction) rather than a scalar (magnitude). This goes far in explaining how the I Ching and Taoism managed without an explicit zero.
Thezero was implicit or understood without using a special symbol of designation. Moreover, it was conceived as representative of an order of reality entirely different from that distinguished by the Western zero. It is, however, fully commensurate with Cartesian coordinate dynamics. It is this alternative zero, with its extraordinary capacities, that provides access to potential dimensions and to different amplitudes of dimension. This will be further elaborated in a future post where we will address how Boolean logic impacts what we’ve covered here.
For now simply note that the changing yin Line and changing yang Line in the horizontal first dimension (x) in each “point” shown in the middle column add to zero, not the zero of scalar magnitude though, but the zero of vector equilibrium.
Section FH(n)
In this section of the cube, as in all frontal sections, the third Line/dimension (z) never changes; the second Line/dimension (y) changes only in columns, as one progresses up or down; the first Line/dimension changes only in the rows, progressing left or right. This is just a consequence of viewing a two-dimensional Cartesian
xy-plane in context of a section of the three-dimensional Cartesian
xyz-cube. Although not the manner in which we are accustomed to viewing the plane, it is nonetheless fully compatible with ordinary Cartesian coordinates. It is simply an alternative perspective, one more suited for analysis/demonstration of trigram relationships in a Cartesian setting.
(continuedhere)
Notes
[1] This should be viewed as a work in progress. I’m still feeling my way with this so the content and/or format may change in the future. What is demonstrated here does not yet take into account the manner in which Boolean logic relates to the distribution of changing and unchanging trigrams nor does this series of cube sections include the all-important geometric method of composite dimension. As described, this is simply a Taoist notation transliteration of Cartesian coordinate structure. The meat and potatoes of the matter is yet to come. Of particular note here, though, is the fact that even at this early stage of translation to a version of mandalic geometry that can be considered comprehensive, what is possibly best described as a decussationbetweenyinandyang lines is already evident at every Cartesian triad point containing a “Cartesian zero”. Worth mentioning here, this will be a key feature addressed in future posts.
[2]Point, square, and cube, have all been placed in quotation marks to indicate that what is being referred to here is actually a different category of objects or elements which should in some sense be understood as relating to fractals or fractal structure and of a different dimensionality entirely than are those ordinary geometric objects. The admittedly deficient terminology used here is necessitated by the fact that sufficiently appropriate vocabulary terms to describe the reality intended do not currently exist, or if they do are not as yet known to me. Since we are representing a Cartesian point (ordered triad) as a quasi-cubic structure here, it must have a near face (n) and a far face (f) in each section with respect to the viewer. The chart displayed details the near face (n) of Section FH.
[3] 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 FE plane has all of its x and y coordinates identical to those seen here but its z-coordinates are all negative (-1). The FHE plane has all the x and y coordinates identical to those seen here but its z-coordinates are all zero (0).
© 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 296-
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-
Earlier to Later Heaven: Fugue V Alchemy Is Not a Dirty Word
(continued from here)
As is the case with all great mysteries, many different explanations for the origins of both the EARLIER and LATER HEAVEN arrangements of the trigrams have been proposed. In seeking plausible explanations, it is well to keep in mind what the worldview contextual origin of these two different arrangements might have been. The dates of their origins seem to be separated by thousands of years.[1] They grew out of very different worldviews. Still there was a clear attempt to correlate LATER HEAVEN with EARLIER HEAVEN. At the time LATER HEAVEN appeared on the scene, there was a longstanding tradition in place which was not entirely discarded. A kind of conceptual amalgamation took place.[2] The mystery lies in how and why that was accomplished.
Alchemy in its most fundamental aspect has to do with all kinds of relationships and from such a perspective both Earlier Heaven and Later Heaven are alchemical in nature. But differently so. They are reflections of different ways of seeing reality. Both, however, are debilitated, in the sense that they are missing a third dimension. The trigram, having three Lines, is a combination form which maps three dimensions. It cannot be adequately represented by figures of one or two dimensions. Any attempt to do so inevitably provokes injustice to reality by misrepresenting all the combinatoric relationships possible among the eight trigrams.
Placing Earlier Heaven in context of three dimensions enables it to express the full range of relationships and changes that can occur among the eight trigrams, thereby creating a combinatoric system resembling a Boolean lattice. This is very likely the form in which Earlier Heaven was originally understood, whether explicitly or implicitly. At some stage of the development of Chinese philosophy, it lost its clear connection to the third dimension, possibly as a result of the new method of encoding and storing ideas in writing, which demanded linear text, displacing the older oral tradition. And that suggests a possible clue to the mystery of the two trigram arrangements.
(continuedhere)
Notes
[1] Both arrangements of the trigrams are of great antiquity, Earlier Heaven possibly dating to around 3000 BCE and Later Heaven to the 8th century BCE. The point here is that the two arose in widely different cultural contexts having differing worldviews with all that entails. Yet they were related one to the other in the Chinese mind. Imagine, if you will, say, the unlikely amalgamation of the phlogiston theory with the scientific oxygen theory of combustion or with modern thermodynamics in the West. How likely is it that such amalgamations could have occurred though the ideas involved were separated not by millennia but only a few centuries. Something very strange and unexplained happened in the history of Chinese thought when these two widely different arrangements of trigrams were entangled with one another. Things are not as they appear on the surface here. Something hermetic and profound is going on at a deeper level.
[2] Something similar did occur in the West when chemistry took over from alchemy. Though chemistry owes a debt to alchemy for its very existence it is somewhat in denial regarding its origins or at least about this particular aspect of its origins. Having alchemy in its family tree, however, is not something for chemistry to be ashamed about. Alchemy had been practiced in many parts of the world for several millennia before chemistry appeared and can boast many important accomplishments in the history of human cognition. It could be said chemistry threw out the baby with the bath water when it conclusively broke with alchemy, except that it never did do that, not completely. Alchemy in the broadest and best sense has to do with relationships of objects, one to another. Much the same can be said about chemistry.
Modern astronomy also pays homage to an earlier form of ideation when it invokes the notion of a constellation, a term first used in astrology which was considered a scholarly tradition throughout most of its history. Constellations in most cases are composed of stars which, though visible in the same general area, are often located at very different distances from Earth. Nonetheless the tradition of referring to constellations is still in use by astronomers today. Its expediency is considerable as any given point in a celestial coordinate system can easily and unambiguously be assigned to a constellation, 88 of which are officially recognized and used in modern astronomy. The past lives on, in the history of ideas, but changed.
© 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 276-
