Octopartition pt. II

This uncommon term is used to describe the act of dividing a thing into eight sections, called octants; colloquially, "pieces of eight."  I Ching lends itself well to illustrating the notion since it can be octo-partitioned twice in succession as the following graphic depicts.

The largest cube represents I Ching as a holism.  The mid-sized cube represents Ba Gua (one of the Eight Diagrams), a three-line figure or trigram, two of which, when arranged in superposition, produce an hexagram.  The smallest cube, its six faces standing for the six lines, represents an individual hexagram out of the 64 that comprise the I Ching.

Taoist Cosmogony pt. II

To recap, the two First Powers interact to produce the Four Symbols.  The Four Symbols intermingle with the two First Powers again to produce the eight diagrams, or Ba Gua.

arbitrary Ba Gua

Observe that the Ba Gua effectively combine to produce an octo-partitioned space in three dimensions.  The binary metric space allows for two values per dimension.  One of the Ba Gua is depicted here, with each of its corner given as a triplet of binary digits.

Three binary axes combine to yield 2 x 2 x 2 = 8 partitions in the metric space we've defined.  Each of the eight partitions may then be visualized as a cube.  The eight partitions also can represent the Ba Gua as eight cubes.

Eight Ba Gua comprise an hypercube, 

These 8 cubes come together at a common vertex (ORIGIN) to form a composite cube of a higher order, or hypercube.    Each of the eight Ba Gua cubes contributes one distinct corner to the higher-order composite cube.  A vector (path) drawn from the ORIGIN to each of the 8 corners of the hypercube defines direction in space and the bounds of the space.  If all eight paths measure equally, a spherical space results.

Hexagram, Trigram, I Ching

Movement between corners indicates motion, or CHANGE.  Note that all observable phenomena are intrinsically based on motion.  Consider that we can't really observe something that does not change, and if we were to try, what would be the point?  Most interesting matters involve CHANGE, and understanding CHANGE is the whole point of I Ching.

The Canon of Change

King Wen sequence

 Our jaunt through the Taoist version of Creation had a goal: it teaches the basic structure of the Book of Change:
Sixty-four six-line figures, each composed of two three-line figures, positioned one atop the other.
The 8 three-line figures, Ba Gua, or trigrams, enumerate the eight partitions of space.  As they are responsible for creating and supporting space, they can be assumed capable of representing anything that space can contain, thus they are archetypal.  They combine in pairs to account for the time dimension as represented by Change.


Each line can 'hold' one of two possible values, yin or yang, '0' or '1', up or down.  The lines can be understood as representing the Three Operations that produced them: Earth, Mankind, and Heaven.  At top right are presented the 64 Changes in King Wen's ordering.  The Canon of Changes, as it is sometimes called, contains 6 x 64 = 384 lines in total, each bearing a decision on its significance in the context of the hexagram in which it appears and its position within that hexagram, and the trigrams of which hexagram is composed.  This framework provides for a flexible and fertile interpretative system.

Taoist Cosmogony pt III: Juxtaposition

The version of Chinese cosmogony we've been discussing comes from Ta Chuan, the Great Treatise; it is an appendix to the Chinese Book of Changes.


After the Four Symbols (xiang) produce the Eight Diagrams (Ba Gua), the Eight Diagrams interact to produce the Canon of 8 x 8 = 64 Changes (I Ching).  By crossing or combining each of the Eight Diagrams (Ba Gua) with itself and each of the others, all possible combinations of Change are represented.


Recall that Three Operations produce akasha, upon which all matter and phenomena depend.  Given akasha, a world and everything in it can be modeled, but we are left with just a static model -- a dead world.  To create a real world, we require Change.  To account for Change, we require a second model to represent the outcome of the Change.

In another discussion of Change, we suggested that vectors or paths between corners of the Ba Gua hypercube could be used to represent Change.  The option to juxtapose two individual Gua (trigrams: three-line figures) is simply an explicit presentation of the finished state, rather than using a vector to suggest the approached state.  Given the notation in the figure at top, we can represent movement from (1,1,1) --> (0,1,1) with an arrow pointing from the first corner to the second as just shown, or simply enumerate it as (111,011).  This interaction between two of the Eight Diagrams produces Change #9, Small Harvest.  The remaining sixty-three Changes may be likewise produced.


By opposing an inner or initial situation with a second final or outer situation, a Change is produced.  We have now accounted for the three dimensions of space and the fourth temporal dimension (as represented by Change).

Xiantian: Commutativity, Complementarity, -(Certainty)

The Book of Changes comprises 64 lineal figures, where each of the figures may be represented alternatively as: 

two (2) superposed trigram figures, of which eight (8) exist, known as ba gua.

three (3) superposed di-gram figures of which four (4) exist, called xiang.

six (6) individual yao (lines), of which two (2) exist, open and closed or yin and yang.

While the ba gua, or eight trigrams have various presentations, a useful presentation is one that interprets them as complementary pairs as when represented in binary notation as given to us by Fu Xi in the so-called familial arrangement:

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The image below presents the traditional Fu Xi ordering, wherein the 64 figures are arranged on a 8x8 matrix, beginning with upper-left (Earth), permuting in an orderly fashion across the rows and through the series, finishing with lower-right (Heaven).  The numbers in the table cells are indices to the hexagram figures.  The numbers in black are binary values of the corresponding hexagram; the white numbers correspond to the King Wen sequence of hexagrams

Below, we present the FuXi in a mild variation of its traditional form.  We have rotated the picture of the sequence by 45 degrees anti-clockwise so that EARTH is at the top, and HEAVEN is its antipodal pair at the bottom. This positioning reveals a vertical axis passing through this pair of figures that is comprised of the "doubled" hexagrams.  Such hexagrams are named for the two identical trigrams  comprising the hexagram figure; these same trigrams are in the table above.  We term this the South-North axis.

The eight doubled hexagrams along this axis in order from top to bottom are:

2) EARTH, 52) MOUNTAIN, 29) WATER, 57) WIND, 51) THUNDER, 30) FIRE, 58) LAKE, 1) HEAVEN.

The horizontal axis oriented at 90 degrees to the South-North axis we will term the West-East axis.  Unlike the other axis, it is composed of hexagrams that are composed of two anti-symmetric trigrams.  In order from left to right these are:

11) PEACE, 18) DISRUPTION, 63) AFTER COMPLETION, 42) INCREASE, 32) PERSEVERANCE, 64) BEFORE COMPLETION, 31) SENSITIVITY, 12) OBSTRUCTION

superposed trigrams a before-after depiction, with CHANGE occurring between


superposed trigrams an external context, and an internal text, the former being a projection of the latter?]

 If a pair of trigrams depicts a situation both before and after a Change, we can characterize any single Change as a system composed of operator(s) and operand(s).  One such characterization posits that the operator is the arrangement, in superposition or juxtaposition, of the component trigrams so that the bottom, middle, and top lines of the two trigrams correspond. The operands are the trigrams themselves.

It should be noted that the primary axis comprises a set of Changes such that for each Change, the anterior or original condition is identical to the posterior or resultant condition.  In other words, the eight hexagrams along the primary axis are all internally commutative.  That is, if we were to reverse the order of the operands, the result is the same hexagram as if we had not.

Heisenberg showed that the commutation relation implies an uncertainty, or in Bohr's language a complementarity. Any two variables that do not commute cannot be measured simultaneously—the more precisely one is known, the less precisely the other can be known. (from http://en.wikipedia.org/wiki/Uncertainty_principle)

These complementary operands do commute; in the language of quantum mechanics this implies that the uncertainty condition described above does not apply to the hexagrams on the primary axis.  One might infer from this feature that the eight hexagrams on the primary axis somehow stand apart, are isolated, or insulated from Change.  They present a ground against which the others form figures.


[speculative] 
This analogy hints at the nature of a subspace, upon which stand the Ba Gua hexagrams as skeletal supports of 3-D space, around which akasha is wrapped.  The resulting figure is analogous to a cube, hence the use of the "hypercube," as a model for Being.

Alternatively, it suggests that the Ba Gua hexagrams, insofar as they serve as the "skeleton" or framework for akasha, are in a permanent state of superposition.  It follows from the latter that the Ba Gua are eternally non-manifest and unconditioned, since any observation or measurement of them would tend to collapse the superposition, thus, collapsing the 3-D space engendered. 

[close]

When we inspect the secondary axis, however, we observe a different situation.  In fact, none of the remaining fifty-six hexagrams commute internally; reversing the positions of the trigrams for these hexagrams produces a different hexagram in each case.  This non-commutativity relation tells us that the internal condition of these hexagrams is not complementary. In the language of quantum mechanics, the constituent trigrams may be said to represent incompatible observables, meaning that the two trigram components of the Change under investigation cannot be simultaneously evaluated with arbitrary precision.

[Incompatible observables are conveniently interpreted as wholly disparate "views" of the same underlying reality, as in the case of waveform--corpuscular manifestations of electrons]  

These fifty-six Changes may also be understood as forming twenty-eight pairs, each half being the complement of its mate.  Being complementary, each of these twenty-eight pairs of Changes enjoy a commutative relation.  Such a relation, as we have suggested, does not invoke the quantum mechanical uncertainty condition. Fourteen of these pairs are assumed to represent conditions in the Celestial realm, with the Mundane realm accounting for the remaining twenty-eight Changes.

[speculative]
In the case of an entangled pair of subatomic objects, state-change of one entails instantaneously equal and opposite state-change in its twin, regardless of their respective locations. This condition implies a dimension of identity shared across the entangled pair.   The quantum states of the entangled objects are said to "instantaneously commute"  or "teleport" across the pair.  Through the [suspected] underlying identity, no signal is passed "between" the pair, thus the luminal barrier is not violated.

In the language of quantum mechanics, this complementarity across pairs suggests that one may know with precision the states of either member of the pair, though one may not know the internal quantum states of either member.  This implies that uncertainty may not apply to the entangled pair--this is the same situation we suggested might obtain from the Ba Gua hexagrams along the primary axis.

[close]

Remaining to be determined:

Given that the function of the oracle is to quantize or measure the uncertainty surrounding the diviner's inquiry: 

• What does uncertainty mean within the context of divination or Change?  
• What does the notion of "incompatible observables" mean in the context of Change?

Octopartition pt. I

The figure on the right is a representation of the Mayan One God Hunab-Ku.  According to Dr. C.J. Calleman, author of Purposeful Universe, other names given to this supreme entity of the Maya include Eight-Partition Place, the One Giver of Movement and Measure, and the One Giver of Energy and Boundary.  This notion of eight partitions or octopartition would present itself again.  It should be mentioned that Hunab-Ku is not regarded as an individual being, but as all BEING, the fundament of all existence.

Similarity between this hand-fashioned wireframe model at left and the 2x2x2 binary hypercube suggests itself.  Likewise, there is similarity between the process of fashioning the wireframe model and Taoist cosmogony:

• Measuring, closing a strand of wire (differentiation, singularity)
• Twisting (Great Axis) a loop yields two loops (Two First Powers)
• Two loops intermingle to produce quadrants (Four Symbols)
• Twisting a orthogonal third axis results in octopartition (Eight Diagrams or Ba Gua)

Binary hypercube

Three loops, three axes, three operations are needed to create space from a singularity.  It should be mentioned that the Taoist conception of nature is represented by Man, positioned or crucified "at the crossing" of Heaven and Earth.  These three are explicitly termed the Three Operations in The Great Treatise; we now have some idea of the reason for this terminology.

Octopartitioning is arguably special because it encompasses all spatial directions -- up, down, and all the cardinal points.  If one considers the Origin (the point at the center of the cube, within is also represented.

from "Purposeful Universe"

Astrophysicists have reportedly determined that the Milky Way galaxy is similarly octopartitioned by the orbit of the Earth as it travels about the galactic center.  




Thus, a galaxy, our Earth, or any physical object may be visualized or modeled by means of the octopartition.  Indeed, navigation makes extensive use of this feature.

Hand-wrought copper wire

In theory, the number of partitions could have continued to double indefinitely; this is one reason why they do not: with only Three Operations on a simple closed loop, we have the basis of matter and phenomena to be allotted discrete locations in space and enter Being from non-existence.  No additional dimensions are necessary to support the emergence of Humanity with its power of Consciousness coupled with its ability to undergo Experience.

Thus, an octopartition represents akasha, or space, but what about time?  Time implies Change, else it is pointless to measure time.  
In order to represent the fourth dimension, we must account for Change, but how?  
The answer: repetition.  That is, to arrange two similar objects to represent the first object both before and after a change as depicted in the graphic at right.