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Doing research for a drawing

#mathblr #discrete geometry #voronoi diagrams

If anyone is interested in learning mathematics!

First a great resource to start learning most fields but especially mathematics is Khan Academy It has courses in: Early math, arithmetic, pre-algebra, algebra, geometry, trigonometry, intro to statistics, precalculus, calculus, multivariable calculus, differential equations, and linear algebra.

Here also gives you a great set of resources for you to learn beginning mathematics Heres a nice calculator to check your work: Wolfram Alpha

However if you learn better by reading books I’ll link a few pdfs for a couple of fields, these are mostly introductory books and notes.

Pre-algebra: here

Algebra: here

Geometry: here

Trigonometry: here,

Calculus and Multivariable Calculus: here

Differential Equations: here

Linear Algebra: here, here,

Intro to Proof: here, here, here,

Set Theory: here,

Logic: here

Category Theory: here

Group Theory: here

Intro to Abstract Algebra: here

Commutative Ring Theory: here

Intro to Combinatorics and Graph Theory: here

Intro to Algebraic Topology: here

Intro to Topology: here, here,

Convex Geometry: here

Discrete Geometry: here here

Differential Geometry: here

Manifolds: here

Intro to Analysis: here

Intro to Real Analysis: here

Complex Analysis: here

Elementary Number Theory: here

Feel free to add to this with more pdfs for mathematics and more resources to learn!

Earlier to Later Heaven: Fugue V Alchemy Is Not a Dirty Word

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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.

(continued here)

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.

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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Earlier to Later Heaven: Fugue IV Bootstrapping Earlier Heaven - Part 3

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The Faces of the Cube diagram can be viewed as a kind of periodic chart of the trigrams, which is to say, as a graphic representation of all elements of three-dimensional space. The various elements are mapped by smaller cubes within a larger cube, smaller faces within larger faces, smaller edges contained within larger edges, vertices viewed free and in containment, yin and yang directions within bigrams within trigrams, and, inferentially, the sixty-four hexagrams, which are six dimensional elements as manifested in three-dimensional space. We have here a rich recursive panoply of fractals, housed within Chinese boxes ad infinitum. We are concerned now though with only these eight unit cubes arranged tangentially about a center which Descartes defined as (0,0,0).[1]

The key to the Faces of the Cube diagram is as follows:

The six faces are divided into two basic types: (1)identity type containing the trigram HEAVEN; (2)inversion type containing the trigram EARTH. There are three of each type. In the EARLIER HEAVEN arrangement the trigrams HEAVEN and EARTH never appear together in any of the faces. All three identity faces appear in the top row of the diagram. All three inversion faces appear in the lower row. Each column contains an identity face above and an inversion face below. The perspective intended in viewing the two faces in each column is always that of viewing the identity face and through it as though with x-ray vision also the inversion face. All identity faces are thus viewed as from without the cube and all inversion faces as from within the cube. By so doing the bigrams composing the trigrams of the paired identity and inversion faces are kept always in sync with one another. This cannot be achieved if both faces are viewed from outside the cube without x-ray vison.[2] Each of the eight vertices of the larger reference cube is defined by one of the trigrams. This trigram appears at the particular vertex it defines in the figure of the large cube, the one we've been referring to all along. Each of the eight trigrams also has as its principal domain the entire octant in which it occurs. Its exact location at any given time may be a matter of probability. The vertex is the point of greatest efficacy and most probable location. The large cube is partitioned into eight smaller cubes or octants. At discrete interfaces between octants dominion is divided between two or more trigrams: two at single dimensional interfaces along edges of the large cube; four at two-dimension interfaces at the center of each face of the large cube where four small squares mutually meet; eight at the three-dimension interface at the center of the large cube where eight small cubes mutually meet at Cartesian (0,0,0).[3]

(continued here)

Notes

[1] Each pair of faces (identity and inversion) encode eight trigrams and both the eight vertices and eight octants of the cube they represent. So the diagram is effectively a two-dimensional representation of all the structural and functional capacities of one-, two-, and three-dimensional space. The viewer is called upon to perform some mental gymnastics here, to mentally abstract certain one- and two-dimensional elements and to construct some three-dimensional elements as they exist within the cube as a whole. Although the recursive pattern continues outward to infinity, we tackle here only that basic modular unit of eight octants forming our cube of reference.

[2] This is similar in principle to the approach of computed tomography to mapping the human body. The slices are all made from the same perspective, front to back, in the frontal projection.

[3] This portioning of dominion prepares the way for full definition of the methodology of composite dimension and prescribes location of the sixty-four hexagrams in twenty-seven (3x3x3) discretized points of the reference cube, alloted in such a way as to determine the mandalic pattern that thereby appears in 3-space. Felicitously, it also indicates how the ancients may have arrived at the mandalic patterning of hexagrams without requiring a coordinate system of the type invented by Descartes.

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Earlier to Later Heaven: Fugue III Bootstrapping Earlier Heaven - Part 2

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It is curious that pure mathematics has paid so little attention to how patterns of lower dimensionality operate within patterns of higher dimensionality. They do so with some degree of independence, but are required still to adhere to the structural and functional demands of the greater whole. When mathematics does attend to such matters it seems almost as though an afterthought, not really the meat of the matter. In fact though it is this very contingency that is the crux of the matter when it comes to resolving how matter and energy operate in the real world.[1]

Here we are looking at the varieties of partial independence of two-dimensional structures (the bigrams) within a greater dimensional reality of three dimensions (the trigrams) and how both fit into the structure of the cube. What we determine to be true here will naturally apply also to Cartesian coordinate cubic structure, but the Taoist symbols are so much easier for the mind to manipulate once familiar with them.[2] Don't give up on them too soon.

In the next post we'll look at a full key to the diagram above. Just to get our feet wet here though, we can say that the figure dissects the cube into its six component faces explicitly and its eight component octants implicitly. The trigrams are apportioned to their respective loci from all rectilinear perspectives. The eight trigrams are based upon four bigrams and they in turn upon yin and yang. Let's run with that for now. More to follow. Stay tuned.

(continued here)

Notes

[1] Oh, that's right. I forgot there for a moment that pure mathematics is not much concerned with the real world because it need not be. Physics though does have that requirement, and because of that must carefully choose which specific formulations of mathematics it pays homage to and follows. It could be that the calculations of both quantum mechanics and string theory have been made more complex than they need be due to poor choice of mathematical formulation. Just a thought. I'm not claiming that this is necessarily so.

[2] Recall how long it took European scholars of the Middle Ages to become comfortable using Indo-Arabic numerals. Centuries, wasn't it? It didn't come about overnight.

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Earlier to Later Heaven: Fugue II Bootstrapping Earlier Heaven - Part 1

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Here we are entering unknown territory, and I'm not entirely sure where it leads. [Where's that infernal Cheshire Cat when you need him?] I've found recently though that when I commit ideas to writing they often develop faster and with more ease than they do otherwise. I'll allow here a week or two after publishing to make changes. After that, the ideas stay put in this post, right or wrong. Corrections, if needed thereafter, will be made in future posts. So let's give it a go, but be patient with the material and yourself. Take whatever time necessary to scout out these unfamiliar new ideas. This venture is not without its hidden dangers. We're not yet 100% certain we're tracking a snark here and not a boojum. Discretion is still the better part of valor.

I said earlier the original significance of Earlier Heaven was lost in the mists of time. That might not be entirely true though. To decipher its intent we need to do two things. First, really listen to what Taoists have said about it. Second, resolve the Earlier Heaven presentation as a cube into its constituent parts which can naturally be accomplished in various different ways. Two of the more important are: 1) the square faces - this may be much a matter of the way two-dimensional structure functions in three-dimensional context; 2) grouping of EARTH with son trigrams and HEAVEN with daughter trigrams. These two groups each form a pyramid having base an equilateral triangle, formed by diagonals linking the three sons or daughters (length = 2 x sqrt 2). Sides of the pyramids are formed by edges of the cube (length=2) which link son trigrams with Earth at the apex of one pyramid and daughter trigrams with HEAVEN at the apex of the other. The two triangles forming the bases of these two pyramids are rotated 60 degrees with respect to one another, and complementary son and daughter trigrams are distanced as great as possible from each other. We will look more closely at these two methods of grouping trigrams and others as well in posts coming up soon.[1]

One of the important clues the Taoists have left us is the remark that Earlier Heaven is not intended as a sequence of eight trigrams. What they say, rather emphatically, (and I've previously not paid sufficient attention to this) is that Earlier Heaven consists of four pairs of polar trigrams, the members of a pair acting in concordance as complements and together all four pairs interrelated, connected in such a way that all interact with and affect the others. That is precisely how they appear in the Earlier Heaven arrangement. Although I have at times referred to this arrangement as a sequence, that is actually a misnomer. In reality it is a pattern which can be viewed at times, depending upon one's perspective and requirements, as two or more patterns variously that meld into one. One such limited specific perspective is to regard the eight trigrams as two or more related trigram cycles, based in a particular way on the cyclicity of the bigrams.[2]

The plot thickens. I can hardly wait to see where this all leads.

(continued here)

Images: The upper figure is our now familiar cubic representation of the Earlier Heaven arrangement of the eight trigrams. The lower image then represents a tabula rasa of sorts, the as yet unknown territory, the six faces of the Earlier Heaven cube which we are charged to fill in with the defining conceptualizations that will relate these six faces of the cube to one another and to the whole by means of some suprarational logic.

Notes

[1] As we go forward with this two-dimensions-in-context-of-three-dimensions perspective, the principal idea to keep in mind is that the plane is not completely independent but exists within a greater context which may be considered a three dimensional or higher context, and that the dimensions transcending two are for some particular purpose, and from some limited perspective, for the time being temporarily suppressed. In our present context (or that of the next post actually) it is the third dimension of the cube which is suppressed.

[2] I recently stated that "Later Heaven was not intended as a straightforward single sequence in the manner of Earlier Heaven, but rather as a codification of possible types of transformation between trigrams of Earlier Heaven." Presently I view two of three claims in this statement to be incorrect. Later Heaven was in fact meant as a represention of the natural order of events after manifestation in physical form has been achieved and as such is certainly a sequence or rather an archetypal sequence. EARLIER HEAVEN on the other hand was never intended as a single sequence nor as a sequence of any kind. Its characteristic structure is that of a pattern rather than an order (an arrangement of items in sequence). The important distinction here is that the terms order and sequence refer to structures of one dimension (linear structures) whereas the term pattern refers to structures of two or more dimensions.

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Earlier to Later Heaven: Fugue I In quest of hidden symmetries

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We've been investigating the EARLIER HEAVEN arrangement of trigrams for a long while now, but we've just begun to scratch the surface. We need to hone our grasp of this arrangement in its three-dimensional incarnation before we can begin to comprehend LATER HEAVEN and its relation to EARLIER HEAVEN. We need to find where the symmetries of EARLIER HEAVEN reside. They lead a mostly clandestine subterranean existence. The surface appearance here is deceptive, consisting largely of asymmetries while the symmetries lie veiled in some deeper stratum yet to be uncovered. We'll have to do some digging to get at them.

With this in mind we are going to call attention to what appear to be symmetries, of whatever kind, as they occur to us in our peregrinations through this still unfamiliar territory. At a later time, we can assemble them entire within a single post. Meanwhile we can't afford to pass any by unclaimed.

We know, for example, HEAVEN and EARTH are complementary trigrams. This is so in context of the cube as a whole.[1] In context of the six faces of the cube, which is to say in terms of its two-dimensional elements, HEAVEN and Earth never occur together in any of the square faces. In those two-dimensional contexts some other trigram must serve as complement. In the three faces where HEAVEN occurs, one of the son trigrams, each of which contains two yin Lines, is found opposite in what I've described as meta-position. It is these meta-trigrams that serve as complements to one another in these planar contexts. In the three faces where EARTH occurs, one of the daughter trigrams, which have a single yin Line and two yang Lines, lies opposite EARTH in meta-position and serves as complement to EARTH in these contexts.[2][3]

In all six faces of the cube, the trigram in meta-position to EARTH or HEAVEN as the case may be has its two yin or yang Lines in dimensions, points of which vary in that particular cube face. The third dimension has constant point value for all four trigrams of the cube face. This is either a symmetry of a kind or it has the makings of one. Such incidents are, in any case, of great significance. Interactions between and among trigrams in two-dimensional contexts within the cube occur with at least as great a frequency as do those involving all three dimensions.[4] This would seem to be a necessary logical condition of some kind. It seems likely that some similar condition may apply to particle interactions within the nucleus, a matter possibly of two quantum numbers changing in a given interaction that involves two dimensions rather than three. This is just one possible example, remember. Any of a number of other possible interactions may prevail.

To summarize, HEAVEN and EARTH are complementary trigrams in context of the entire cube. In the context of the faces of the cube, which are two-dimensional squares with third dimension of the cube ordained constant, both have alternative complementary trigrams. These are those trigrams in meta-position to HEAVEN or EARTH, as the case may be, in the face of the cube in question. For HEAVEN this will be a son trigram having two yin Lines; for EARTH, a daughter trigram with one yin Line.

This brings us a little closer to explaining why Heaven becomes FIRE and Earth becomes WATER in LATER HEAVEN. But why that particular daughter and son rather than the others? My thinking here is the reason may have something to do with the fact that, among all the daughter and son trigrams, those are the only ones that have a palindrome structure in a manner similar to the trigrams HEAVEN and EARTH. Since they read the same from top to bottom as they do from bottom to top, they image a kind of reversibility the other son and daughter trigrams cannot. Recall that the I Ching regards movement from the bottom Line to top Line of trigram or hexagram as movement into the future. So it is possible we are dealing here with a statement of a sort regarding time and the passage of time from past to future and reciprocally from future to past in a manner that can be configured as symmetrical.[5]

(continued here)

Notes

[1] That is to say, in a context of three dimensions.

[2] In context of the three dimensional cube, complementary trigrams are separated through three dimensions. In the planar context of the cube faces, however, complement trigrams are separated through only two dimensions. Each planar face of the cube has four vertices, arranged in two pairs of two each, the members of each pair in meta-relation to one another. The two vertices of each of these pairs generate one of the two diagonals of the face. The vertices of each diagonal of a face are always complementary to each other in that planar context, and the product of the multiplication of the two vertices of both diagonals is identical. This is true for each of the six faces of the cube. This product is alway one of the son trigrams. For the identity planes containing HEAVEN it is the son trigram located in that plane which is in meta-position to HEAVEN. For inversion planes containing EARTH it is the son trigram that is the complement of the daughter trigram in meta-position to EARTH in the particular face of the cube. This means then that it is the same son in opposite faces. These relationships likely involve some of the hidden symmetries that we are seeking. [Happily this will all make much more sense on the tenth or twelfth reading. At least it did for me.]

[3] The basic premise here is that there exist within three dimensional structure elements of two dimensions that contribute to structural and functional integrity of the whole while also engaging in their own quasi-independent converse with other such limited parts. I might call attention here specifically to the manner in which a trigram exhibits divergent modes of operation of its component bigrams in context of the three different faces of the cube in which it resides, which has been previously covered in detail in a series of posts starting here. If one thinks about it, this finding is not so strange and is intimately related to the mechanisms of Cartesian coordinates as well as to those of Taoist symbolic representation. It is a natural and necessary instrumentality of three-dimensional space but one we tend most often to overlook, not without serious repercussion.

[4] In terms of particle physics this could give rise to a host of different quantum numbers or states, at least some of which may not be currently recognized by physicists and the Standard Model.

[5] This is not wholly wild speculation. The Secret of the Golden Flower, a Taoist text of the early Qing dynasty first published circa 1668-1692, dealing with meditation and inner alchemy, refers to a "backward-flowing movement" which I interpret pertains to reversibility in time as well as process in general. The specific potencies of the palindrome structure is not just a matter of logic or aesthetics. It possesses a more general efficacy. Consider, for example, the palindromic structure of the water molecule (HOH) which determines in part the remarkable capacities of this important chemical.

Earlier Heaven to Later Heaven: tracking a relationship - Part 4

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Continuing along similar lines of thought, we are told by Taoism that the trigram MOUNTAIN in EARLIER HEAVEN becomes trigram HEAVEN in LATER HEAVEN. We're told also that the trigram WIND in EARLIER HEAVEN becomes EARTH in LATER HEAVEN. Again the symmetries involved, if such in fact exist, are obscure and seem not to be located within the transforms suggested in any explicit manner. We need constantly to remind ourselves that over and above the real world sequence Taoism tells us LATER HEAVEN represents, it may also be a conceptualization of possible distinctive modes of transformation in a manner that is related to but different from that of EARLIER HEAVEN arrangement. Why else would the Taoists admonish us to superimpose the two arrangements to understand them fully.

With that in mind we note that the son trigram MOUNTAIN toggles between trigrams HEAVEN and MOUNTAIN through a combination of x- and y-dimensions; also, that the trigram THUNDER toggles EARTH and WIND through a combination of y- and z-dimensions. Further, we know these two son trigrams are in turn themselves toggled by the third son trigram WATER through a combination of x- and z-dimensions, so THUNDER in EARLIER HEAVEN is toggled by WATER to MOUNTAIN in LATER HEAVEN. The daughter trigram LAKE which is located above THUNDER in EARLIER HEAVEN is likewise transformed to WIND in LATER HEAVEN through the toggling action of WATER. These various transformations appear to give us another possible clue to translation of Earlier to Later Heaven. Importantly too, a pattern may be beginning to emerge here, hopefully one that will reveal a hidden symmetry.[1]

The LATER HEAVEN arrangement of trigrams is unlikely to be one without intentional meaning in symbolic terms. Just as Cartesian coordinates offer a key to unlocking interpretation of how trigrams are related in the EARLIER HEAVEN arrangement it seems there ought be another key, yet to be found, that would accomplish something similar for LATER HEAVEN trigrams. It would be a mistake, though, to think that the EARLIER HEAVEN arrangement grew out of anything like the method of Cartesian coordinates. Descartes' coordinate system is only an ad hoc means to present EH in such manner as to make it coherent and acceptable to the modern, mostly rationally motivated mind, a sort of Rosetta Stone for a breed of pure mathematicians and other thinkers for whom earlier holistic forms of logical relationship are anathema. In antiquity, however, the Taoist emblematic arrangement of trigrams in EARLIER HEAVEN grew out of a very different worldview.[2]

An interesting alternative approach to the rational one of Cartesian coordinates involves a more intuitional focus on what might be termed alchemical relationship. Of note is the fact that LATER HEAVEN places the father and three son trigrams on the lower tier or bottom face of the cube and places mother and three daughter trigrams on the upper tier or face of the cube which are the respective relative positions that Taoism views as open to approach and access. For example, to clarify here, since Taoism considers the natural movements of HEAVEN to be upward and EARTH to be downward, positioning of HEAVEN below EARTH is, as a result, conducive to their interaction, whereas the opposing positioning, EARTH below HEAVEN, is conducive to their going their separate ways. So from this relational standpoint it makes sense in LATER HEAVEN to have HEAVEN below, EARTH above if a universe of continuous process is the goal.[3][4]

Remarkably, this provisional account of the relationship between Earlier Heaven and Later Heaven arrangements presented here borders on being a rational one and is certainly far more so than any justification heretofore presented by sinology. New symmetries are involved here not as yet revealed through Western thought and which hold the promise, I believe, of new physics. Taoism claims that the Earlier Heaven paired arrangement of trigrams represents the fundamental relationships of pristine cyclical nature[5] and the Later Heaven arrangement represents the sequential order of trigrams which tracks changes after they become manifest in the physical universe. Radical stuff. At the present time still beyond my comprehension but here at least is a start: the noumenal and the phenomenal.

(continued here)

Images: The circular presentations of Earlier Heaven and Later Heaven are the traditional forms that have been handed down from antiquity, as they occur in the I Ching and other historical texts. The presentations in cubic form are as reconstituted in the manner I envisage them to have originally existed in a still earlier oral tradition which has long been lost to history.

Notes

[1] Surely it must be of some significance here that the trigram FIRE and its brother complement trigram WATER are the only trigrams that act as toggles to transform two of eight trigrams each. It is interesting too that WATER and FIRE are the only two trigrams that are both self-identical when inverted and consist of both negative (yin) and positive (yang) Lines. One of our major difficulties is that we are beneficiaries of thousands of years seeing the world in very different terms than did the Taoists. Is it even possible to reconstruct the manner in which they viewed the world? My suspicion is that the symmetries here fall onto one or more of our ingrained cognitive blind spots. They are spread out in spacetime in a manner and/or extent that eludes recognition even of their existence. If that is so, this will be a very tough nut to crack. Like attempting to resurrect the dead.

[2] The Taoist worldview of Chinese antiquity shares many characteristics of what the psychologist/philosopher Carl Jung describes as alchemy and which he has discussed at great length in much of his writing. Jung wrote the Forwards to Richard Wilhelm's translations of the I Ching and Secret of the Golden Flower, the latter an alchemical text of the early Qing dynasty based to a great extent on the eight trigrams. For both Jung and the Taoists, alchemy was not a matter of proto-chemistry but a profound study of all sorts of complex relationships closer effectively in spirit to an amalgam of what we Moderns call complexity theory and combinatorics. In consideration of this, alchemy deserves considerably more respect than it has heretofore received. Born of a very different worldview than modern rationalism/empiricism it was nonetheless one of the great achievements of documented human thought, not without its legitimate fruits. The great irony here is that given the appropriate mindset, Chinese philosophy could have quite easily derived a system of coordinates identical to that of Descartes thousands of years before it appeared in the West solely by application of the EARLIER HEAVEN arrangement of trigrams to a more rigidly defined geometry. That it failed to do so speaks to the vast differences in focus of worldviews of the East and West. An ample opportunity was abidingly lost here, to be sure, but a fully alternative mode of understanding reality was thereby gained.

[3] Although I've expressed all this in terms of alchemical relationships, recall I've also described nuclear interactions of physics as alchemical relationships of a kind as well. I'll leave you to draw your own conclusions here . . . but wouldn't it be nice if nuclear physics could be reduced to a matter of two signs and a few discrete integers interacting with one another. We'd be able to explain particles in an a priori manner without recourse to experimentation other than for the final proof. We might even finally be able to explain why quantum mechanics works and commit the failed Copenhagen interpretation at long last to the fiery grave where it belongs.

[4] Quantum mechanics is the most successful theory of physics ever devised, but there has never been any agreement on how to interpret it. The Schrödinger equation gives correct results. It is in the interpretation of the meaning of those results that disagreement persists. Schrödinger himself never agreed with Niels Bohr's interpretation which emerged victorious among the numerous belligerent contender theories at Solvay '27. His flagrant thought experiment was actually intended as a satirical response to Born's Copenhagen interpretation. Frequently misunderstood and misrepresented in the popular literature, his representation of a cat both dead and alive was in fact meant to spoof the flaws of the Copenhagen interpretation which insists a particle exists in all states at once until observed.

[5] A claim surely reminiscent of the claim of quantum mechanics that pair production, the creation of an elementary particle and its antiparticle from the vacuum, is an ongoing process of nature, one that allows both momentum and energy to be conserved while change flows ceaselessly permeating the universe.

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