#scalar
Mandalic geometry, Cartesian coordinates and Boolean algebra: Relationships - I
(continued from here)
In attempting to understand the logic of the I Ching it is important to know the differences between ordinary algebra and Boolean algebra and how Boolean algebra is related to the binary number system.[1]
In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted
1 and 0 respectively. Instead of elementary algebra where the values of the variables are numbers, and the main operations are addition and multiplication, the main
operations of Boolean algebra are the conjunctionand, denoted ∧, the disjunctionor, denoted ∨, and the negationnot, denoted ¬. It is thus a formalism for describing logical relations in the same way that ordinary algebra describes
numeric relations. [Wikipedia]
Whereas in elementary algebra expressions denote mainly numbers, in Boolean algebra they denote the truth values false and true. These values are represented with the bits (or binary digits), namely 0 and 1. They do not behave like the integers 0 and 1, for which
1 + 1 = 2, but may be identified with the elements of the two-element field GF(2), that is, integer arithmetic modulo 2, for which 1 + 1 = 0. Addition and multiplication then play the Boolean roles of XOR (exclusive-or) and AND (conjunction) respectively, with disjunction x∨y (inclusive-or) definable as x + y + xy. [Wikipedia][2]
Mandalic logic already occurs fully in the structure and manner of divinatory practice of the I Ching, if some of it only implicitly. Although mandalic geometry does not originate from either Boolean algebra or the Cartesian coordinate system but from the primal I Ching which predates them by millennia, it does combine and augment aspects of both of these conceptual systems. It extends Boole’s system of symbolic logic to include an additional logic value represented by the number -1. This necessitates modification of some of Boole’s postulates and rules, and increases their total number through introduction of some new ones. The hexagrams or native six-dimensional mandalic coordinates of the I Ching are related to Cartesian triads composed of the numbers -1, 0, and 1, making these two geometric systems commensurate by means of composite dimension, a 6D/3D hybridization or mandalic coordination of structure and function (or space and time).[3]
The introduction of composite dimension produces four distinct dimensional amplitudes and is solely responsible for the mandalic form. For anyone reading this who might be down on sacred geometry, itself a subject which I respect and admire, let it be known that I am talking here about genuine mathematics and symbolic logic, and my suspicion is that there is some genuine physics involved as well.
The mandalic number system, then, is a quasi-modular number system, different from Leibniz’s binary number system which is fully modular. Boole’s rule 1 AND 1 = 1 still holds true in mandalic logic. However we must add to this the new logic rule that -1 AND -1 = -1. Individually the two rules are modular, based on a clock arithmetic using a modulo-3 number system rather than Leibniz’s modulo-2 or binary number system, but with yet another added twist.
Together the two rules prescribe a compound system, one which is not singly modular but doubly modular. The two components, yinandyang, are complementary and are inversely related to one another in this unified system. This logic organization appears based on the figure 8 or sine wave and its negative, allowing for periodicity, for recursive periods of interminably repeating duration, and, perhaps most importantly, for wave interference, of constructive and destructive varieties. These two geometric figures also engender an unexpected decussation of dimension not recognized by Western mathematics. This is so because 1 AND -1 = 0 and -1 AND 1 = 0. The surprise here is that there are two distinct zeros: 0a and 0b.[4] In two- or three-dimensional Cartesian terms there exists no difference between these two zeros. However, in terms of 6-dimensional aspects of mandalic geometry and the hexagrams of the I Ching, the two are clearly distinct structurally and functionally.[5]
This arithmetic system is the basis of the logic encoded in the hexagrams of the I Ching. Each hexagram uniquely references a single 6- dimensional discretized point, of which there are 64 total. These 64 6- dimensional points of the mandalic cube are distributed among the 27 discretized points of the ordinary 3-dimensional cube through the compositing of dimensions in such manner that a mandala is formed which positions 1, 2, 4 or 8 hexagrams at each 3-dimensional point according to the dimensional amplitude of the particular point. This necessarily creates a concurrent probability distribution of hexagrams through each of the three Cartesian dimensions.
TheI Chinguses a dual or composite three-valued logic system. In place of truth values, the variables used are yin, yang and the two in conjunction. These fundamentally represent vector directions. Yin is represented by -1, yang by 1, and their conjunction, using Cartesian or Western number terminology, by zero (0). This symbol does not occur natively in the I Ching though where the representation used is simply a combination of yin and yang symbols, most often in form of a bigram containing both and regarded as representing a composite dimension, namely 0[1] or 0[2].[6]
The two bigrams that satisfy the requirement are
young yang
for 0[1]
and
young yin
for 0[2].
Although mandalic logic is in Cartesian terms a 3-valued system, in native terms it is 4-valued. It is not a simple modulo-3 or modulo-4 number system, but two interrelated modulo-3 systems combined. The best way to think about this geometric arrangement is possibly to view it as a single composite dimension having four distinct vector directions: a negative direction represented by mandalic composite yin (Cartesian -1); positive direction represented by mandalic composite yang (Cartesian 1); and two decussating relatively undifferentiated directions in some sort of equilibrium, represented by mandalic 0[1] (composite yin/yang) and 0[2] (composite yang/yin). both of which devolve to Cartesian 0 (balanced vector direction of the origin or center).[7]
So we’ve seen that the number system used in the I Ching is not binary as Leibniz believed but instead doubly trinary with the two halves, in simplest terms, inversely related and intertwined. Still, it was an easy mistake to make because the notation used is binary. We’ve seen too that all trigrams and hexagrams in the system can be rendered commensurate with the Cartesian coordinate system: trigrams by simple transliteration, hexagrams by dimensional compositing. What, then, of George Boole and his eponymous logic? How do they fit in the logic scheme of the I Ching? I’m glad you asked. Stay tuned to find out.
(continuedhere)
Images: Upper: TRANSFORMATION OF THE SYMBOL OF YIN (LINE split in two) AND YANG (STRAIGHT-LINE). BLEND: 4 bigrams, THEN 8 trigrams. (MORAN, E. ET AL. 2002: 77). Found here. Lower: Modified from an animation showing how the taijitu (yin-yang diagram) may be drawn using circles, then erasing half of each of the smaller circles. O'Dea at WikiCommons [CC BY-SA 3.0orGFDL],via Wikimedia Commons
Notes
[1] Boole’s algebra predated the modern developmentsinabstract algebra and mathematical logic but is seen as connected to the origins of both fields. Similarly to elementary algebra, the pure equational part of the theory can be formulated without regard to explicit values for the variables.
[2] If you are new to Boolean algebra these definitions may be confusing because in some ways they seem to fly in the face of ordinary algebra. I’ll admit, I find them somewhat daunting. Let me see if I can clarify the three examples given in this quote. Those of you more familiar with the language of Boolean algebra might kindly correct me in the event I err. I’m growing more comfortable with being wrong at times. And this is after all a work in progress.
Boolean XOR (exclusive-or) allows that a statement of the form (x XOR y) is TRUE
if either x or y is TRUE but FALSE if both are TRUE or if both are FALSE. Since Boolean algebra uses binary numbers and represents TRUE by 1, FALSE by 0, then
for x = TRUE, y = TRUE x + y = 1 + 1 = 0 , so FALSE
for x = FALSE, y = FALSE x + y = 0 + 0 = 0 , so FALSE
for x = TRUE, y = FALSE x + y = 1 + 0 = 1 , so TRUE
for x = FALSE, y = TRUE x + y = 0 + 1 = 1 , so TRUEBoolean AND (conjunction) allows that a statement of the form (x AND y) is TRUE
only if both x is TRUE and y is TRUE. If either x or y is FALSE or both are FALSE
then x AND y is FALSE. Here algebraic multiplication of binary 1s and 0s plays the
role of Boolean AND. (Incidentally, binary multiplication works exactly the same
way as algebraic multiplication. There’s a gift!)
for x = TRUE, y = TRUE xy = 1(1) = 1, so TRUE
for x = FALSE, y = FALSE xy = 0(0) = 0, so FALSE
for x = TRUE, y = FALSE xy = 1(0) = 0 , so FALSE
for x = FALSE, y = TRUE xy = 0(1) = 0 , so FALSEBoolean OR (inclusive-or) is the truth-functional operator of (inclusive) disjunction,
also known as alternation. The OR of a set of operands is true if and only if one or
more of its operands is true. The logical connective that represents this operator is
generally written as ∨ or +. As stated in the Wikipedia article logical disjunction x∨y
(inclusive-or) is definable as x + y + xy [(x OR y) OR (x AND y)] as shown below.
[Note: x AND y is often written xy in Boolean algebra. So watch out whichalgebra
is being referred to, ordinary or Boolean. Are we confused yet?]
for x = TRUE, y = TRUE x + y = 1 , xy = 1 , so TRUE
for x = FALSE, y = FALSE x + y = 0 , xy = 0 , so FALSE
for x = TRUE, y = FALSE x + y = 1 , xy = 0 , so TRUE
for x = FALSE, y = TRUE x + y = 1 , xy = 0 , so TRUE
[3] Fundamentally, though, the coordinates of mandalic geometry refer to vector directions alone, rather than to both vectors and scalars (or direction and magnitude) as do Cartesian coordinates. Yin specifies actually the entire domain of negative numbers rather than just the scalar value -1. Yang similarly refers to the entire domain of positive numbers rather than the scalar value 1 alone. Their conjunction through the compositing of dimensions, though represented by the symbol zero (0) in the format commensurate with Cartesian coordinates, refers actually to a state or condition not found in Western thought outside of certain forms of mysticism and other outsider philosophies like alchemy; equilibration of forces in physics; equilibrium reactions in chemistry; and the kindred concept of homeostasis mechanisms of living organisms found in biology.
[4] This is to Westerners counterintuitive. Our customary logic and arithmetic allows for but a single zero. That two different zeros might exist concurrently or consecutively is - to our minds - irrational and we wrestle mightily with the idea. To complicate matters still more, neither of these zeros is conveniently like our familiar Western zero. So which should win out here? Rationality or reality? In fact, the decision is not ours. In the end nature decides. Nature always decides. It stuffs the ballot box and casts the deciding vote much to our chagrin, leaving us powerless to contradict what we may interpret as a whim. Our votes count for bupkis.
[5] This calls to mind also the Möbius strip which involves a twist that looks very much like a decussation to me. The decussation or twist in space we are talking about here though has a sort of wormhole at its center that connects two contiguous dimensional amplitudes. I can’t say more about this just now. I need to think on it still. It seems a promising subject for reflection. (1,2,3)
[6] It needs to be pointed out here that in mandalic geometry, and similarly in the primal I Ching as well, a bigram can be formed from any two related Lines of hexagrams, trigrams, and tetragrams. The two Lines need not be (and often are not) adjacent to one another. I would think such versatility might well prove useful for modeling and mapping quantum states and interactions.
[7] Note that yin and yang in composite dimension can each take the absolute values 0, 1, and 2 but when yin has absolute value 2, yang has absolute value 0; when yang has absolute value 2, yin has absolute value 0. This inverse relation in fact is what makes the arrangement here a superimposed, actually interwoven, dual modulo-3 number system. It also makes the center points of mandalic lines,squares, and cubes more protean and less differentiated than their vertices and elicits the different amplitudes of dimension.
The composite dimension value at the origin points(centers) of all of these geometric figures is always zero in Cartesian terms since the values of the differing Lines in the two entangled 6-dimensional hexagrams located here add to zero. But neither of these 6-dimensional entities is in its ground state at the center. Both have absolute value 1 at Cartesian 0. Let me say that again: composite dimension values at the center or origin are zero in Cartesian terms but the values of both individual constituents are non-zero.Yin is in its ground state when yang is at its maximum and vice versa. At the center, since the two are equal and opposite they interfere destructively. This results in a composite zero ground state.
So from the perspective of Cartesian coordinate dynamics, which is after all the customary perspective in our subjective lives, we encounter only emptiness. But it is this very emptiness that opens to a new dimension. In the hybrid 6D/3D mandalic cube only line centers and the cube center have direct access through change of one dimension to face centers and only the face centers have a similar direct access through a single dimension to the cube center and edge centers. All coexist in an ongoing harmony of tensegrity. There is method to all this madness then.
© 2016 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 302-
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-
This is my attempt at rewriting the infamous “I’m a weirdo” scene from the last Riverdale episode into Jughead’s coming out scene for Day Four of @jugheadweek. Because if Riverdale were ever to not be crap of course they’d make his coming out scene an overdramatic angst-fest.
“Why are you getting so upset? It’s just a party, Jug.”
“It’s not just a party! It’s the fact that you don’t know, or even care, that this is the last thing I would want.
In case you haven’t noticed, Betty, I’m different. Doesn’t it ever occur to you how different we are, like on a cellular DNA kind of level? For once, that’s not just more of my pretentious emo bullshit. I don’t fit in. I’m the damaged, loner outsider from the wrong side of the tracks. Betty, come on. I don’t know who we were kidding.
Have you ever seen me without this stupid hat on? You know what the design stands for? It’s a dot and a dash. That’s Morse code for the letter “A”. “A” as in asexual. Aromantic. As in I don’t do relationships. I tried to deny it, Betty. I thought maybe dating you would lead to some grand awakening, that I’d realise what I was missing and finally find people attractive like everyone else. But it didn’t. And I’m done trying to avoid that.
And you know what else I’ve realised? I don’t wanna fit in. All this romance, and kissing and other nonsense has just made everyone’s lives worse, and god knows things aren’t bad enough around here already! In case you’ve forgotten, Betty, Jason’s still dead, and whoever killed him is still out there. The last thing we need is some partyruining things even more.”
The prompt was about Jughead figuring out he’s aro/ace, so here’s a comics!verse mini fic of sorts about it. This is my first go at writing fic, so please excuse (or at least, constructively criticise) any glaring errors, lmao.
My favourite scenes fall under three categories: (1) Relatable Aro/Ace Stuff, (2) Wholesome Friendship Moments, and (3) Extremely Specific Callouts Of Myself (though i guess the ace stuff could go under that last one too, heh). Under a readmore since it’s a bit long - click through for Me Being Extra About Comics.
@jugheadweek
