Inversion Transforms of the Cube - IV

 

 image

(continued from here)

Inversion transforms of the cube may involve any number of dimensions ranging from zero to three. We have previously considered these changes in isolation. Here we intend to look briefly at how these various dimensional transforms are related one to another.

 

Recapitulating:

HEAVEN, the identity element trigram, produces no changes whatsoever when acting as operator upon any trigram as operand.

The daughter trigrams produce changes in a single dimension along the four edges of the cube that correspond to the dimension of the single yin line of the particular daughter trigram under consideration.(1)

The son trigrams, each having two yin lines, produce changes in two dimensions along two connected(intersecting) edges that correspond to the two dimensions of the two yin lines of the particular son trigram under consideration.(2)

EARTH, having three yin lines, produces changes in all three dimensions and therefore along three edges intersecting at two right angles not in the same plane.

Significantly, any of the three son trigrams produces changes in the two dimensions other than that involved in the changes produced by the antipodal (opposite/complementary) daughter trigram.(3)

 

Another perspective: There is another way to look at all of this.

 If the son trigrams are producing two related edge changes along two edges intersecting at right angles then, in effect, they are producing a single change along the diagonal connecting the non-intersecting ends of those edges. These diagonals are the twelve diagonals of the six faces of the cube.

Similarly, EARTH, in producing three related edge changes involving three edges intersecting at two right angles not in the same plane, in effect is producing a single change expressed along the diagonal which connects the non-intersecting ends of the first and third edges. These are the four diagonals of the cube, each of which passes through the center of the cube and connects one of the four pairs of antipodal trigrams.(4)

(to be continued)

 

(1) WIND, having its yin line in the x-dimension, acts solely along the corresponding horizontal edges of the cube. FIRE, having its yin line in the y-dimension, acts solely along the corresponding vertical edges of the cube. LAKE, having its yin line in the z-dimension, acts along the corresponding forward/backward edges of the cube and no others.

(2) THUNDER, having its yin lines in the y- and z-dimensions, acts along corresponding intersecting edges (vertical and forward/backward). WATER, having its yin lines in the x- and z-dimensions, acts along all corresponding intersecting edges (horizontal and forward/backward). MOUNTAIN, having its yin lines in the x- and y-dimensions, acts along any and all corresponding intersecting edges (horizontal and vertical).

(3) This seems an innocent enough observation, but I suspect it has highly significant repercussions in the quantum realm and in quantum logic. We likely will finding this relationship turning up not infrequently in future explorations. I believe (and this is highly speculative) that it has something to do with the structural logic that makes the existence of matter possible.

(4) In addition to the obvious connection these diagonal relationships have to the Pythagorean theorem there is the tantalizing possibility as well that they may be related in some obscure manner to Einstein’s special theory of relativity (or invariance, his preferred term for the theory.)

© 2014 Martin Hauser

Please note - This is a beta post meaning that the content and/or format may not yet be in finalized form. A reblog as a TEXT post now will contain this caveat to warn readers to refer to the current version which appears in the source blog. This note will not however appear in a LINK post which itself accomplishes the same. Thank you for not removing this statement and for your understanding. :)

Inversion Transforms of the Cube - III

 

image

(continued from here)

We are ready now to summarize the rules for trigram multiplication and the inversion transforms of the cube.(1) They are presented here as a kind of shorthand with a brief explanatory note attached elaborating the reason that these rules work as described. Failure to understand why they work as they do is a conflationary error. To avoid falling victim to such an error keep your eye on the bouncing balls, the yin lines that produce transformation.

HEAVEN as operator in the process of trigram multiplication always leaves any trigram operand unchanged due to its lack of any yin lines. Since trigram multiplication is commutative in the narrow mathematical sense, any trigram acting as operator upon HEAVEN as operand always yields itself.

EARTH as operator in the process of trigram multiplication always changes any trigram operand into the opposite (complementary) trigram. It does so by virtue of its three yin lines. Again due to the commutativity of trigram multiplication, any trigram acting as operator upon EARTH as operand will always yields its opposite (complementary) trigram.

Any trigram multiplied by itself gives HEAVEN as result. This arises from the fact that regardless of whether a line is yin or yang, negative or positive, the product is a yang or positive line.

Any trigram multiplied by its complementary, i.e., opposite, trigram gives EARTH as result. This arises from the fact that corresponding lines in all three dimensions are always opposite in sign and therefore give yin or negative lines as product.

Each of the three daughter trigrams (WIND, FIRE, LAKE), because they all contain a single yin or negative line, acting as operator upon any trigram as operand always transforms or moves the trigram in the dimension that contains the yin line. The corresponding line in the operand trigram, if yin, becomes yang and if yang, becomes yin.

Each of the three son trigrams (THUNDER, WATER, MOUNTAIN), because they all contain two yin or negative lines, acting as operator upon any trigram as operand always transforms or moves the trigram in the two dimensions that contain the yin line. Each corresponding line in the operand trigram, if yin, becomes yang and if yang, becomes yin.

(to be continued)

 

(1) The rules described here apply to the 3-dimensional context of the trigrams. It should be understood though that analogous rules apply to any imaginable setting in any number of dimensions. This is the case because all transforms are based ultimately on the yin/yang complementary dichotomy that occurs at the 1-dimensional level with the first division of non-polar reality into interactive polar halves.

© 2014 Martin Hauser

Please note - This is a beta post meaning that the content and/or format may not yet be in finalized form. A reblog as a TEXT post now will contain this caveat to warn readers to refer to the current version which appears in the source blog. This note will not however appear in a LINK post which itself accomplishes the same. Thank you for not removing this statement and for your understanding. :)

Inversion Transforms of the Cube - II

 

image

(continued from here)

Referring back to the two simple rules of multiplication set forth in the preceding post we note that a transformative change occurs only when one or more yin(negative) lines are involved, whether as operator, operand or both. Therefore the best strategy for following the transformative changes is to keep an eye on the yin lines and let the yang lines fend for themselves, which they will.

The three daughter trigrams (WIND, FIRE, LAKE), all having a single yin(negative) line, are capable of producing change only in one dimension, the dimension in which the yin line appears in the particular trigram. WIND, with its negative line in the first (horizontal) dimension can produce change only in that dimension. Similarly, FIRE with its solitary yin line in the second (vertical) dimension can produce change only in the vertical dimension, and LAKE with its solitary yin line in the third (forward/backward) dimension, can produce change only in that dimension.

The three son trigrams (THUNDER, WATER, MOUNTAIN) all have two yin lines and can produce change in two dimensions. THUNDER, with its yin lines in the second and third dimensions, produces changes in both of those dimensions. WATER, with its yin lines in the first and third dimensions will produce changes in those dimensions. MOUNTAIN, with yin lines in the first and second dimensions will produce changes in those dimensions.

EARTH, the mother trigram, has yin lines in three dimensions and will produce inversions in all three. HEAVEN, the father trigram, contains no negative lines and therefore brings about no inversive changes whatsoever which is precisely what makes it the identity operator of multiplication.(1) 

(to be continued)

 

(1) It is well to keep in mind that what the transformative changes described involve is essentially the multiplication of unit vectors throughout the eight octants of Cartesian space.

© 2014 Martin Hauser

Please note - This is a beta post meaning that the content and/or format may not yet be in finalized form. A reblog as a TEXT post now will contain this caveat to warn readers to refer to the current version which appears in the source blog. This note will not however appear in a LINK post which itself accomplishes the same. Thank you for not removing this statement and for your understanding. :)

Inversion Transforms of the Cube - I

 

 image

(continued from here)

As we begin our exploration of the inversion transforms of the cube it is well to keep in mind that from the Taoist point of view this is the same as the 3-dimensional inversive transforms of the trigrams. (1) The broken(yin) line and solid(yang) line serve respectively as the inversion (2) and identity elements of multiplication or transformation throughout all dimensions. In practical terms this means that two simple rules suffice to characterize all transformations through any number of dimensions:

  1. The broken yin line produces inversion through a point of reference(3) when acting upon either a solid yang line or a second yin line.
  2. The solid yang line produces no inversion, leaving the operand unchanged regardless of its sign.

 

This can be summarized as

image

which are the same rules we all learned in grade school using minus(-) and plus(+) in place of yin and yang.

As noted previously, one of the great advantages of using the Taoist yin/yang notation over the Cartesian +/- notation is that manipulation of the trigrams is far easier than the analogous manipulation of the Cartesian ordered triads. And when we come to consideration of manipulations through 6 dimensions using the 64 hexagrams, which is the true meat of the matter, there is no real Cartesian analogue available to use.(4)

(continued here)

 

(1) Which is to say, the transforms of yin and yang through three dimensions. Taoist inversive transforms can encompass any number of dimensions. The hexagrams of the I Ching involve transformations which occur throughout six dimensions. At some future time we will consider those transformations. Here, however, we are restricting our focus to the three dimensions of the ordinary cube and of the Taoist trigrams.

(2) We are speaking here literally of inversion of direction or sign and are concerned exclusively with the direction or sign aspect of vectors. Magnitude is of no concern within this focused context.

(3) For both Cartesian geometry and mandalic geometry this point of reference is either the origin point of the coordinate system or the origin point of a dimensional subdivision thereof (i.e., center of cube, center of cube face or center of cube edge for a 3-dimensional coordinate system.)

(4) One could be easily enough constructed but why go out of one’s way to make life difficult?

© 2014 Martin Hauser

Please note - This is a beta post meaning that the content and/or format may not yet be in finalized form. A reblog as a TEXT post now will contain this caveat to warn readers to refer to the current version which appears in the source blog. This note will not however appear in a LINK post which itself accomplishes the same. Thank you for not removing this statement and for your understanding. :)

A Teaser

 

image

 

A possible solution to the still-hanging-in-air question as to whether we uncovered a broken symmetry in an earlier post has occurred to me. Here is a provisional answer and a teaser of sorts:

  • A broken symmetry cannot exist where there was no symmetry to break in the first place.

We will elaborate on that idea after our look at the group of inverse transforms of the cube which begins in the post immediately following.

As we progress through the inverse transforms of the cube keep your eye on the bouncing ball and see if you can distinguish any symmetries from asymmetries. Mandalic geometry views the latter as being just as important as the former, if not more so, particularly where the asymmetries are related to one another in specific recurring manners.

Classic 4beats passing 2juggler 6balls side

(continued here)

Image: World record holding club passers Christoph and Manuel Mitasch passing 11 clubs in Linz, Austria. By Cmitasch (Own work) [CC-BY-SA-3.0 or GFDL], via Wikimedia Commons

Animation: Classic_4beats_passing_2juggler_6balls_side.gif By Own work [GFDL or CC-BY-SA-3.0-2.5-2.0-1.0], via Wikimedia Commons

© 2014 Martin Hauser

Interlude 5

 

image

 

In a sense the rationalists of the Enlightenment were like children playing with new toys not fully understood.(1) They were enchanted with the new geometric method of representing dimensions and positive and negative numbers offered by Cartesian geometry but they didn’t yet quite comprehend the intricacies involved or the awesome power unleashed by the methodology. Descartes himself likely did not.

Though Descartes’ method of geometry offered what seems from our modern perspective a ready-made access to dimensional numbers and to the number plane and number 3-space, the bait appears not to have been taken. Thought processes remained rooted for an inordinately long time in the well-worn tracks of linearity, and the number line, in spite of the obvious superiority of the more inclusive approach of higher dimensionality, reigned supreme in the minds of Enlightenment thinkers.(2)

We will soon see that the number line as we know it is just one variant in a spectrum of simultaneously existent species of number lines which all dovetail together to form a higher-dimensional geometry. Quite possibly spacetime itself is constructed so at Planck scale. And in some paradoxical sense, at that scale uniformity and non-uniformity of space are one and the same once one manages to get beyond the conventions of labeling and unitary perspective. Symmetry and asymmetry also combine at some level and relate to one another singularly in some higher dimensional sense. That takes us well beyond the usual perspective by which we view things. Possibly the most important rule to keep in mind throughout all of this is the one that states, “Nature is never simplistic but always, wherever and whenever it can, takes the most simple direct route available.”(3)

There are more loose ends still to be tied up and they will be eventually. But next we will tackle the large challenge involving the inversion transforms of the cube. With that accomplished we will be better equipped to evaluate the question left hanging in mid-air regarding whether or not we unearthed a broken symmetry during our recent explorations. 

Image: Tunnel vision, a focus too narrow. License: CC0 Public Domain/FAQ

 

(1) The same, of course, could be said today about both theoretical and experimental physicists and any number of other categories of contemporary thinkers. Likely it is always so in times of rapid change and innovation.

(2) Though perhaps the perspective expressed is anachronistic as rationalists of the 17th and 18th centuries were still largely uncomfortable with holistic syncretic approaches to investigation and understanding of natural processes. Their view was that comprehension of nature required breaking it into its component parts. Possessed of such an analytic view of things they must have felt more comfortable with lines and points than with cubes and squares. I see no convincing evidence that they ever conceived of the number line as existing in the larger contexts of 2- and 3-dimensional space. Someone more familiar with the history of mathematics than myself may view this matter differently. I am simply trying to understand from a cultural/psychological perspective how it came about that this opportunity was missed, at least until much later when rationalism no longer held sway.

(3) I should probably add parenthetically that a simple direct route is not always among the options available.

© 2014 Martin Hauser

Please note - This is a beta post meaning that the content and/or format may not yet be in finalized form. A reblog as a TEXT post now will contain this caveat to warn readers to refer to the current version which appears in the source blog. This note will not however appear in a LINK post which itself accomplishes the same. Thank you for not removing this statement and for your understanding. :)

Loose ends: The Components of Multiplication

 

image

(continued from here)

Mathematics has cooked up a complex conflationary brew in its operation of multiplication. This operation is nowhere near as simple or uniform as mathematics makes it out to be. Multiplication has at least ten distinct components. Let’s look at this curious conglomeration a little more closely. This brew is just begging for a little subversion.

Mathematics portrays and would have us understand multiplication in the following fashions

image

Such a presentation makes multiple conflationary errors. Positive numbers are given some sort of precedence over negative numbers such that it is considered unnecessary to explicitly show their sign. Although intended only as a form of shorthand this leads to significant confusions. Operators are treated as simple numbers in a manner similar to the way that operands are viewed. Sign(direction) and magnitude are treated in combination as though only one entity belonging to a single species of mathematical construct. The signs of operator and operand are viewed as being alike in all ways. All of this is dangerously misleading.(1)

The actual components of multiplication can be summarized as

  1. Scalar magnitude of the operand(multiplicand)
  2. Scalar magnitude of the operator(multiplier)
  3. Scalar magnitude of results (identical for both intermediate and final)
  4. Direction of the operand(multiplicand) expressed by its sign
  5. Instruction of the operand contained in its sign regarding the direction of the intermediate result expressed by the sign of that result
  6. Instruction of the operator contained in its sign regarding the direction of the final result expressed by the sign of that result 
  7. Direction of the transitional(intermediate) result expressed by its sign
  8. Direction of the final result expressed by its sign
  9. A scaling operation in which magnitude of operator acts upon magnitude of operand.(2)
  10. A sign upon sign operation performed by sign of operator upon sign of operand.(3)

The scalar magnitudes of operator, operand and result are pure numbers without direction or dimension in and of themselves. Rectilinear direction is specified by their sign components. Dimensions can readily be indicated by composites of multiplications if necessary or desired.

The sixth through tenth components are perhaps the most challenging to explicate. Mathematics treats the operator in multiplication as a vector (magnitude + direction) as it does the operand. In actuality it is an entity that performs two separate and distinct operations upon the operand. This entity is a complex operator which produces in turn a magnitudinal and a directional influence upon the operand. The first operation produces scaling of the operand; the second determines whether or not the sign of both the operand and the intermediate result of multiplication, which are one and the same, is inverted or reflected through the origin.

The scaling operation is simply a variant of the operation of addition. Six times four, for example, is actually the magnitude four scaled up six times (4+4+4+4+4+4). The plus signs here denote the operation of addition, not the sign of any number(s). The direction or sign of the intermediate result of multiplication is identical to the sign of the operand. The sign portion of the operator then determines whether inversion is to be performed at this stage. If the sign of the operator is positive(+) no inversion occurs, leaving the sign of the final result the same as that of the intermediate result. If, however, the sign of the operator is negative(-), inversion does occur, producing a final result with sign opposite to that of intermediate result and operand.(4)

The fact that positive times negative and negative times positive both yield a negative result is a matter of perspective and definition if the surface appearance is scratched. The equal balance of the ancients is redefined to become the polar co-valences(5) of the Enlightenment. The rationalists of the 18th century quickly progressed within a generation or two from denying the very existence of negative numbers to becoming intoxicated with their power. In course of these philosophical and mathematical developments a veil of confusion arose between symmetry and asymmetry which persists to this day.(6) The backstory to all of this involves both the determinants of conflation and convention as well as the very different routes taken by Western and Eastern philosophies and the repercussions of these differing worldviews on the way mathematics evolved.(7)

(continued here)

Image: "Can you do addition?" the White Queen asked. "What’s one and one and one and one and one and one and one and one and one and one?" "I don’t know," said Alice. "I lost count." 

- Lewis Carroll [Through the Looking Glass]; illustration by John Tenniel

 

(1) It may be that all these conflationary errors arose historically in a misguided attempt to create a practical shorthand. If so the original intents have long since been forgotten (at worst) or overlooked (at best).

(2) The nature of the result of this operation varies depending upon whether operator and operand are respectively greater than one(1), equal to one(1), greater than zero(0) but less than one(1), or zero(0).

(3) This operation involves a directive of the operator which by means of its sign determines whether or not inversion of the sign of the operand occurs and defines the sign(direction) of the final result of the multiplication.

(4) As an example, take -6 times +4. First, the scalar portion(6) of the operator(-6) scales the magnitude(4) of the operand(+4) up to magnitude 24. The directional/sign portion(+) of the operand initially imparts to this magnitude 24 a positive value (+24). Next the directional/sign portion of the operator(-) changes the initial transitory positive result (+24) to a negative one (-24) by means of inversion or reflection through a point (here the zero point of the number line.) In the case where the sign of the operator is negative and the sign of the operand also negative, say -6 times -4, the scalar portion(6) of the operator(-6) again scales the magnitude(4) of the operand(-4), up to magnitude 24. The sign portion of the operand(-) initially gives this result a negative sign (-24). Finally, the negative sign portion(-) of the operator(-6) changes this intermediate result to a positive one (+24) by means of inversion or reflection through the zero point of the number line.

(5) For more on poles and polarity, valence, and bonding see (1,2,3,4,5,6,7).

(6) This is a topic for future discussion.

(7) The complementary nature of yin and yang in Chinese philosophy stands in stark contrast to the traditional role of negative and positive in Western thought.

The yin and yang symbol in actuality has very little to do with Western dualism; instead it represents the philosophy of balance, where two opposites co-exist in harmony and are able to transmute into each other. In the yin-yang symbol there is a dot of yin in yang and a dot of yang in yin. This symbolizes the inter-connectedness of the opposite forces as different aspects of Tao, the First Principle. [Continue»]

© 2014 Martin Hauser

Please note - This is a beta post meaning that the content and/or format may not yet be in finalized form. A reblog as a TEXT post now will contain this caveat to warn readers to refer to the current version which appears in the source blog. This note will not however appear in a LINK post which itself accomplishes the same. Thank you for not removing this statement and for your understanding. :)

Loose ends: Routes, results and commutation

 

image

(continued from here)

As an example of a context in which commutation fails consider the transforms of cube inversions. To clarify, we have already demonstrated that commutation in the narrowly focused mathematical sense does hold true in this context. But here we are considering the larger more inclusive case of objective reality that physics and the sciences in general are concerned with.

Referring to the diagram above and focusing here on the forward facing plane we can see that WIND times MOUNTAIN equals FIRE and also that MOUNTAIN time WIND equals FIRE as well. So the purely mathematical definition of commutativity is satisfied. But hold on a moment. Consider how we got from here to there and whether that makes a difference in the real world, the world we actually live in.

When WIND as operator acts upon MOUNTAIN as operand in the operation of dimensional multiplication it causes MOUNTAIN to move horizontally one step to the right to FIRE. When MOUNTAIN as operator acts upon WIND as operand, however, it causes it either to move first horizontally one step to the right to HEAVEN then down one step to FIRE or down first one step to MOUNTAIN then horizontally one step to FIRE. Another alternative, given sufficient force, would be to cause WIND to move along the diagonal down and to the right simultaneously to FIRE.

Pure mathematics would have us believe that none of this makes a difference. In other words, all these real world differences make no difference. Really? How is that? Solely because pure mathematics has decreed it so in its rule book which it is ever so careful to maintain as internally self-consistent.

Mandalic geometry, however, is concerned with that real world that pure mathematics would have us ignore because it is a hybrid discipline the allegiance of which is to the way things actually work rather than to the preconceived notions of a self-consistent book of tautological rules. For that reason it insists that the route taken in arriving at a destination can be as important or more so than the destination itself. Whether or not this is true in a particular case can only be determined by experiment and experience. There is no rule book that will reveal real world truth for every occurrence or all parameters of exploration.

Science often marvels at how mathematics so successfully applies to the real world. It would do well to take note of the many situations where this platitudinal truism falls far short of the truth that science seeks.

(continued here)

 © 2014 Martin Hauser

Please note - This is a beta post meaning that the content and/or format may not yet be in finalized form. A reblog as a TEXT post now will contain this caveat to warn readers to refer to the current version which appears in the source blog. This note will not however appear in a LINK post which itself accomplishes the same. Thank you for not removing this statement and for your understanding. :)