Friday, October 1, 2010

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:

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
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.

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. 

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.

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.

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?

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