Tuesday, October 12, 2010

Quantifying Change: Number Magic

Quantifying Change: Number Magic
The Master said: "He who knows the method of change and transformation may be said to know what is done by that spiritual power."
This post specifically treats a section of Ta Chuan (the Great Treatise) chapters 52 through 58. This text may be found in James Legge's translation of I Ching, and in Stephen Karcher's (2000) translation of Ta Chuan, where the corresponding chapter is entitled "Number Magic and Consultation." The goal of this work is to solidify the means of quantifying and measuring change. Central to this approach is adopting a new look at the 64 familiar hexagram figures.

"Number Magic" begins by designating even numbers as YIN and odd numbers as YANG. To this point, the He Tu diagram depicts the counting numbers one through ten with light-colored (YANG) and dark-colored (YIN) dots.

The He Tu (diagram) is traditionally said to originate from the emergence of a dragon-headed horse with carp-like scales that emerged from the Yellow River. Its body was covered in strange markings which were noted by the sages of the time and studied in depth. Eventually, these markings were codified into a set of dots (see the diagram at right), called the He Tu, or River Map. Many observations about nature were derived from study of the He Tu, including the existence of the north-south axis of the Earth, and the idea that heat rises while cold descends. Gradually, associations with directional and Five Phase energies were incorporated into the He Tu.

NB: Note the enumeration of the four ritual numbers [6,7,8,9] on the periphery of the He Tu diagram.
(Excerpt attributed to http://www.maelearning.com/onlinecourses/store/details.asp?id=4, graphic courtesy of http://www.8whitestarfengshui.com/The_Luo_Shu_and_He_Tu.html; referenced June 2010)

After a description of the yarrow-stalk oracle, also called "the operation by threes and fives" we are given:
The numbers (required) for Ch'ien (or the undivided line) amount to 216; those for K'un (or the divided line), to 144. Together they are 360, corresponding to the days of the year.The number produced by the lines in the two parts (of the Yî) amount to 11,520, corresponding to the number of all things. Therefore by means of the four operations is the Yî completed.
Legge's commentary on this passage:
"The actual number of the undivided and divided lines in the hexagrams is the same [192 of each]. But the representative number of an undivided line is 9, and of a divided line 6. Now 9 x 4 (the number of the emblematic figures) x 6 (the lines of each hexagram) = 216; and 6 x 4 x 6 = 144. The sum of these products is 360, which was assumed, for the purpose of working the intercalation, as the standard length of the year. But this was derived from observation, and other considerations;--it did not come out of the Yî."


Ch'ien appears in a divination when one casts nines (greater YANG) in all six places
K'un appears in a divination, when one casts sixes (greater YIN) in all six places

We propose the following derivation for the numbers
216 = 9 (symbolic for greater YANG) * 4 (xiang) * 6 (positions)
144 = 6 (symbolic for greater YIN) * 4 (xiang) * 6 (positions)
216 + 144 = 360 (days in a sacred year)

The number produced by the lines in the two parts (of the Yî) amount to 11,520, corresponding to the number of all things.

In the quoted passage above, “The number of all things,” 11,520, approximates the "ten thousand things," a common reference in the Tao Te Ching. The ten thousand things indicate the products of the interaction of Heaven and Earth, namely, all beings and phenomena between Heaven and Earth. The author admits to some uncertainty on this point, however, because no known references include Heaven and Earth among the ten thousand things. The term applies only to those things that are produced by the interaction of Heaven and Earth. The number of all things, 11,520, is shown by Legge to comprise the yao-numbers of Heaven and Earth, including the ten thousand things, the latter symbolized by the other sixty-two hexagrams.

Legge's notes:

The number in paragraph 53 (11,520) arises thus: 192 (the number of each series of lines in the sixty-four hexagrams) x 36 (obtained as above) = 6912, and 192 x 24 = 4608, the sum of which = 11,520. This is said to be 'the number of all things,' the meaning of which I do not know. The 'four operations' are those described in paragraph 31.

I Ching is composed of 384 (6 x 64) lines, half of which are YANG, half are YIN.  This means that 11,520 is the sum of:
6912 = 192 (YANG lines) x 4 (xiang) x 9 (symbolic for greater YANG)
4608 = 192 (YIN lines) x 4 (xiang) x 6 (symbolic for greater YIN)

We should note, however, the subtle clue to a construct inside the Changes: the yao-numbers of hexagrams #1 and #2 are 216 and 144, and the sum of the yao-numbers of all the hexagrams is 11,520 (derived above).  

If Heaven and Earth have yao-numbers resulting from divination ("operation by threes and fives"), we may confidently infer that each of the hexagrams has its own yao-number.  Our goal then, is to find an heuristic to produce an hexagram to yao-number assignment in accordance with the clues given in the text.

We find from the following exercises below that not just any substitution will work, summing yao-numbers over the entirety of I Ching must equal 11,520, and hexagrams #1 and #2 must have yao-numbers 216 and 144 respectively.  We observe that the mean value of any given yao is 7.5 (the average of [6,7,8,9]).  Therefore the lower limit of distributional variance would result from assigning 7.5 to all lines.  Thus, the sum of YANG yao-numbers would equal the sum of YIN yao-numbers.  

The median case of distributional variance results from example A in the table below.  The upper limit of distributional variance of results from example B following.

A: Substituting 7s for YANG and 8s for YIN:
All lines in the CHANGES: 6 x 64 = 384
Lines allotted to YANG and YIN: 192 + 192 = 384
4 * 7 * 192 = 5376 (7 representing YANG lines)
4 * 8 * 192 = 6144 (8 representing YIN lines)
5376 + 6144 = 11520 (
BSubstituting 9s for YANG and 6s for YIN:
All lines in the CHANGES: 6 x 64 = 384
Lines allotted to YANG and YIN: 192 + 192 = 384
4 * 9 * 192 = 6912 (9 representing YANG lines)
4 * 6 * 192 = 4608 (6 representing YIN lines)
6912 + 4608 = 11520 (Legge's solution)

Note that the Changes corresponding to hexagrams #1 and #2 would not have the correct stick-numbers under scheme A, though the sum of the numbers yielded is 360.  These explorations demonstrate that the means of representing YIN and YANG over the sixty-four Changes has consequences for the balance of YIN and YANG.  

From these results we confirm that the heuristic we sought for generating a yao-sum from a Change is equivalent to multiplying by four the appropriate ritual number at each of the six positions.

Returning to our example of Change #28 which emerges from hexagram #55 and hexagram #56, we enumerate the ritual numbers of each position, to yield [987786].  Multiplying each of these six ritual values by four produces [45 * 4 =] 180, which is the yao-sum for Change #28 (which we derived from the Change operation).

Conclusion: While Legge's method is mathematically correct, it alters the distribution of Change by limiting its consideration of ritual numbers to greater YIN [6] and greater YANG [9], while ignoring lesser YIN [8] and lesser YANG [7].

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