| Line | Journey | Notes | Line | Journey | Notes |
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This journey equals parity. | | |
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Reduces to static yang | |
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Reduces to moving yin | |
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Reduces to moving yang | | |
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Reduces to static yin | | |
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It is said, when the student is ready, the master appears.
But I say, when the student is ready to become a master, his journey appears before his feet.
-- Lao Tse Kaud
In the traditional way of reading a hexagram with moving lines, the figure is reduced to two basic hexagrams of basic lines only, a before and after figure. The moving lines are said to cause the situation to "move" from one reading to the other. With the usage of hyperemblematic lines in the hexagram, the entire figure undergoes a complicated journey that tells many more details of the entire story. The refinement to the oracle that this change produces is quite profound, since it points to many "moments of decision" along the way.
For the sake of concreteness, let us assume we are creating a hexagram with lines from HL(2). The yarrow stick technique is easily modified to produce such lines, although the technique would have to be totally redefined for orders of 3 or higher. Let us assume the following sequence of lines were thrown, in bottom to top order: (1001), (0110), (0000), (0110), (0101), (0110). We will number the lines of the hexagram in traditional notation, with numbers from 1 to 6. The first line is simply static yang, as is seen by refering to Table 3. Lines 2, 4 and 6 are all static yin lines -- no surprises here. Line 5 is a moving line with a journey of (0011), which starts out yang and turns to yin. In fact, this line is reducible to moving yang. Line 3, however, is a whole different kind of animal. The journey of (0000) is (0110): it switches from yin to yang and then back to yin, again. Each of these line journeys is 4 steps long, so the hexagram itself goes through a journey of 4 steps, as illustrated in Figure 2.
Figure 2. A Hexagram Journey
This journey of 4 hexagrams actually describes a very plausible scenario for the mid-level programmer at the Oracle Temple. He arrives to work on Monday morning to find some code he put into the system last week is crashing the system. Before diving into the problem reports and test data, he throws the yarrow sticks and receives the answer just discussed. He would get the following preliminary commentary from the oracle on his situation.
Initial Situation: Hx. 10, Bad Fix. This is stating the obvious, that error has been commited and the ministers are badgering him for a new fix to get the system working again.
Next Step: Hx. 9, Infinite Loop. In the haste to create a quick patch, further problems are encountered, possibly even hanging the system. Additionally, attempts to fix one problem seem to cause a cascade of other problems, blocking further progress. There is no chance for success for many days.
Next Step: Hx. 15, Burn Out. It's Friday and despite many hours of overtime all week, a solution seems to fade into the distance. Frustration and fatigue are mounting, clouding the mind and causing the creative juices to evaporate. The programmer's attention drifts off to some time away from work. A brief vacation can clear the mind and remove blockages.
Final Outcome: Hx. 4, IBT. It's late Friday evening. With no progress in sight, the programmer takes the only sensible path available and joins his friends at the beer hall. An evening of levity and lubrication, followed by a weekend of rest, should make for a better solution the following Monday. The Tao of Code is like that sometimes.
Notice how much more information is available than when you move directly from Bad Fix to IBT. Such superficial advice could easily encourage lazy working habits among the programmers, which would not be advantageous.
The question of how to incorporate the moving line texts is a bit tricky. Obviously, any line that is non-static at any time in the journey should be consulted when reading the initial hexagram. And while not all practitioners would agree, it seems prudent to me to include the moving line texts in any intermediate hexagrams, if that line changes state when moving from that hexagram to the next one. While these intermediate moving lines are of a secondary nature in the reading, they often supply an uncanny commentary to the overall themes. The usage of the moving line texts in the current example is left as an exercise for the reader.
(The increased complexity of a reading like this may have prompted the priests to raise their fees when doing divinations for the public. Normally, the yarrow stalks would be thrown for the price of a chicken. Shortly after the Trigram Project closed down, some of the higher masters started tacking on a "moving line surcharge" of a cup of rice per moving line in the hexagram. Perhaps this was to compensate for the extra work required to read a complex figure, or that the duration of consultations nearly doubled in length due to these lines.)
One of the curious phenomena that are produced by higher order hexagrams is the existence of "cycles" where the same hexagram appears more than once in the journey. (Cycles can never happen in the traditional paradigm.) It's as though the querent is asked to return to an initial situation at some point in order to make a different set of decisions and create a different outcome. These cycles very rarely imply an infinite loop that can never be escaped, emphasizing the role of intelligent choice to move beyond limiting circumstances. Further, after order 6 or more, it's theoretically possible (though vanishingly unlikely) to cast a figure that journeys through the entire set of 64 hexagrams in the Q-Ching! The strange tangled interpretive loops of such a figure would likely baffle even the Wisest One.
In order to quickly formalize the results of this section, we will introduce a formal notation for higher order hexagrams. Such a figure is composed of a large number of basic lines, organized in a two dimensional grid. One coordinate says which line (numbered 0 to 5) we are looking at, where each line is an n-tree or higher order line; the second describes which basic line of the n-tree we're refering to. The first coordinate thus varies over the domain 6 = { 0, 1, 2, 3, 4, 5 }, while the second has a domain of 2n, as defined in a previous section. When a function of two variables is being discussed, where the separate domains of the variables are over sets A and B, the combined domain (both dimensions simultaneously) is usually written A x B (called a "cross product"); the set theoretic details of this operation are superfluous for our current purposes.
This long, extended discussion is a complete solution to the interpretation problem.Def: Given a fixed order n, a hexagram of order n is a function h : 6 x 2n -> BL. The value h(m, x) says whether the x-th basic line in the m-th n-tree or hyperemblematic line of the hexagram is either yin or yang. The phrase "hexagram of order n" is often abbreviated "n-hx". The set of all hexagrams of order n is denoted Hx(n).
Def: Given an n-hx h, the journey corresponding to h, written journey(h), is a function j : 6 x 2n -> BL (another n-hx), such that for all m and x, j(m, x) = h(m, x) + parity(x).
Def: Given an n-hx h with journey j = journey(h), "step t of the journey" (where t is in 2n ), written step(h, t), is a basic hexagram function f : 6 -> BL such that for all m, f(m) = step(h, t)(m) = j(m, top(n) - t). Intuitively, each basic hexagram that is a step in the journey is read out of the journey of h by counting down each n-tree in j.
Def: Given an n-hx h, the m-th line (where m goes from 0 to 5), written line(h, m), is an n-tree f such that for all x, f(x) = h(m, x).
Def: An n-tree f with journey j = journey(f) is globally moving if and only if j is not a constant function, i.e., j is neither zero or one. Globally moving lines are all consulted in the moving line texts of the initial hexagram, step(h, 0).
Def: An n-tree f with journey j = journey(f) is locally moving at step t if and only if j(top - t) is not equal to j(top-t-1). A line in a hexagram h is locally moving at step t if the line changes state when going from step(h, t) to step(h, t+1). It is advised to read the moving line texts in hexagram step(h, t) for all locally moving lines.
While this model of HEL solves many problems in the original theory, problems nobody ancient or modern has handled before, it also suggests many new lines of inquiry for the future. The theory can be extended or generalized in quite a number of ways, many of which have not been explored by researchers in the field. Further, it suggests some new interpretation techniques, even within the traditional theory that is confined to HL(0) and HL(1). Let me briefly sketch out some of these extensions.
Every hexagram can be analyzed as a series of 4 trigrams. The outer trigrams consist of lines 1-2-3 and 4-5-6. The two inner trigrams are lines 2-3-4 and 3-4-5. By analogy, one can look at the "inner quads" of a hexagram, namely lines 1-2-3-4, 2-3-4-5 and 3-4-5-6. Each of these quads is equivalent to a a second order line from HL(2) and these 3 lines can be conveniently arranged as a second order trigram. This trigram will most likely have moving lines, so the entire trigram goes through a journey of 4 steps.
Curiously, the basic trigrams in this journey exactly equal the outer trigrams and the negations of the inner trigrams in order! (The inner trigrams are negated because of parity considerations.) Evidentally, there is a very deep relationship between trigrams and quads in a hexagram.
Despite the complexities introduced by the journey concept, it must be admitted that journeys are still completely deterministic. Even though each step of the hexagram's journey represents a moment of choice that can change the situation, there is no overt mechanism for the oracle to represent alternative outcomes. One possible extension of this theory would be to introduce a new kind of "line" that allows user input at each step and changes state according to its own internal "program". It's not too hard to program a line to have behavior similar to the hyperemblematics we have just examined, but with the ability to adjust this behavior based on yin-yang inputs. Essentially, each hexagram in the journey is a new situation, allowing a yin or yang style response from the querent. By choosing yin or yang at each step, or by possibly abstaining from any choice, many alternative journeys can be examined. (It's also possible to program the line to ignore user input at times, when his choices make no difference to the outcome.) For centuries, the oracle has always been a one way communication, with the oracle telling the user what the situation is. With programmable lines, the oracle at last becomes interactive.
This approach, modelled after a famous interpretation of quantum mechanics, has become known as a "many worlds oracle". While nobody has examined this approach in the literature to date, the actual details shouldn't be that difficult to work out.
One of the fallouts of the Trigram Project was a vision of a fractal hexagram, where each line is a trigram, each line of the trigrams are trigrams, and so on recursively. Much effort was spent trying to find a satisfying correspondence between lines and trigrams, similar to the tried and true mapping between emblematic lines and basic digrams. Obviously, there are some "day one" problems here that need to be worked out, probably requiring a redefinition of emblematic lines to mirror the trigrams more closely.
Assuming these hurdles can be overcome, a theory of lines based on recursion by creating trigrams instead of digrams would follow naturally. In large measure, such a theory would mirror the approach in this paper, although certain details would, of course, be different. It's my personal opinion that the "interpretation problem" for this theory would lead to some bold and innovative ways of reading the oracle that would make traditional approaches pale in comparison.
One of the practical problems with using hyperemblematic lines is their sheer number. I hinted earlier at how fast the size of the HL(n) sets increase, but consider the following. HL(7) exceeds the number of picoseconds since the Big Bang by many orders of magnitude, while HL(9) dwarfs most estimates of the total number of subatomic particles in the entire universe. Obviously, there's little use in practical, everyday terms to even consider more than second or third order lines. However, mathematics today is rarely about what is practical. Researchers prefer to let their theories go where they may, even if they go straight towards castles in the conceptual sky. There's a "dirty secret" about all the lines we've discussed in this paper: despite the astounding growth of the HL(n), they are still all finite. Even if you lump all the HL(n) together, the total collection is countably infinite in size, but that's the smallest infinity that mathematicians work with. There's an extension that's a no brainer staring us in the face: transfinite hyperemblematic lines. More precisely, if N = { 0, 1, 2, ...} (the set of all natural numbers), a transfinite line is "simply" a function f: N -> { 0, 1 }. Intuitively, we can talk about lines that go through an infinite number of changes.
As a theoretical research program, transfinite lines are a quick jump over a cliff. This set of lines is by definition isomorphic with the real number line, a mathematical object that in the last century has driven set theorists to the breaking points of their discipline. While I haven't worked with this theory much to date, it's evident an entirely different approach from the finite theory is required. I suspect the discipline will resemble topology more than logic, though the outlines of the theory aren't even dimly visible yet.
As I said, a theory of transfinite lines has absolutely no utility in the "real world" of consulting the oracle. But these theories are profoundly useful if they reveal to us deeper qualities of the Tao than finite musings can show us. Physicists pursue their TOE ("theory of everything"). We can certainly pursue our own "Theory of All Objects", the Tao.