Yarrow oracle depicted graphically by the 8x8 grid

[saharan]

The graphic depicted here shows the so-called FuXi arrangement overlaid on a tableau of concentric squares, each discretely colored. This presentation is remarkable in that it explicitly encodes the yarrow oracle probabilities; i.e. each ring comprises a specific proportion of the hexagrams as shown in the table above.

⚏ dynamic yin, represented by '6', probability = 0.0625 ( 4 in 64)

⚌ dynamic yang, represented by '9', probability = 0.1875 (12 in 64)

⚎ static yang, represented by '7', probability = 0.3125 (20 in 64)

⚍ static yin, represented by '8', probability = 0.4375 (28 in 64)


​

It does appear that the oracle and its probabilities are derived from this 8x8 grid, and that the ritual as is commonly understood may have been a way to produce the probabilities from the grid. Any proper quadrant of this 8x8 square likewise encodes the same probabilities at one-quarter scale: 1/16, 3/16, 5/16, 7/16.

 

[bradford]

Interesting find, delightful in fact, but the conclusion doesn't follow.
The Xian Tian is pure binary geometry so any binary expression or algorithm is going to exhibit some sort of pattern on the grid. This merely shows that the Yarrow process that Zhuxi left us with eight centuries ago was built on a binary operation.
For other examples, see figures 1-34 in my Volume Two.

 

[saharan]

Thank you, Bradford. To be sure, the speculation is limited in scope to the oracular function (divination ritual) particularly as it acts as index to the content.

If I am following your critique correctly, the verity of the speculation is independent of any particular hexagram overlay; correctness depends more so on the geometry of any bounding field and its divisions. Admittedly, I erred by including the kua in the diagram (now corrected).

Given the Platonic seniority of the 8x8 geometrical form to mathematical formulae/equations, it seems more reasonable that the yarrow divination ritual (oracular function) was an attempt to mathematically recreate the geometry of the four ring-shaped regions than to infer the converse, or to infer that no causative relationship exists between the 8x8 form and the yarrow ritual.

 

[bradford]

I'm not sure they were that sophisticated. Even Shao Yong, who came up with the Xian Tian arrangement, might have arrived at the sequence using little more than a doodle.
But the question I've been asking myself is whether these Yarrow probabilities date any further back than the Song Dynasty, and I haven't seen any evidence that they do. In fact, I'm in the process of going another 1400-1800 years further back and collecting all of the pre-Qin divination records I can find (Guoyu, Zuozhuan, etc.). So far the lines I have found spread as follows:
6 - 51
7 - 134
8 - 153
9 - 44
This looks to me like it's converging on the 1/8, 3/8. 3/8, 1/8 that we see with today's coin method. But I've never really been happy with the Yarrow probabilities over the coins.

 

[saharan]

Most of the modern Yijing editions seem to pose the yarrow as the more authentic (and senior?) oracular ritual than the coin method, which makes sense since it is preserved in the text of Ta Chuan. I would be interested to learn more of the historical context of mathematical divinatory methods.

 

[bradford]

Some yarrow method originated alongside the Zhouyi, and one is at least mentioned in the Da Zhuan, if not detailed. I only doubt that the one we have now is that method, or that this method is any older than Zhuxi and eight centuries. If my suspicions are correct I am afraid that the original method is long lost, at least barring some new archaeological discovery.

 

[Sparhawk]

Most of the modern Yijing editions seem to pose the yarrow as the more authentic (and senior?) oracular ritual than the coin method, which makes sense since it is preserved in the text of Ta Chuan. I would be interested to learn more of the historical context of mathematical divinatory methods.

The bias is true but incorrect and misguided by poor scholarship. Unfortunately, all those modern and popular editions have been using the Wilhelm/Baynes translation as the standard, the edition to go to for historical information and opinions.

Indeed, the use of yarrow in divination is even older than the Zhouyi proper. Its use appears in oracle bone inscriptions from the Shang Dynasty, for example. What's not certain though is how they were used. As it is, coins use for divination go back to the Han Dynasty, a good thousand years before the Song Dynasty, when the current, received method, was "reconstructed" by Zhuxi. There's even evidence that Zhuxi manipulated the text of the Da Zhuan to make it fit his reconstruction and/or in trying to understand what it meant. Alas, Zhuxi's method makes no sense when matched with some of the earliest divination records from the Zhou Dynasty, where yarrow was used, those in the Zuo Zhuan and Guo Yu.

There's an interesting article that talks about this and tries to reconstruct, once again, a method that would match the results of those earliest records:

How to Form a Hexagram and Consult the I Ching
by Shih-chuan Chen
Journal of the American Oriental Society, Vol. 92, No. 2. (Apr. - Jun., 1972), pp. 237-249.

​

I have also wrote a simplified version of that method: "A proposed alternative milfoil divination method."

 

[Sparhawk]

Crossed posts with Brad...

 

[saharan]

I am pleased to be in the company of erudition and to be able learn from you; thank you both, Brad and Luis.

The utter simplicity of the concentric square rings on the 8x8 and the coincidence of their producing the same probabilities are the purported yarrow oracle ... it speaks to me. Truth often is betrayed that kind of straightforward simplicity -- at least in the stories.

Still, the historical record does not support the hypothesis, bummer. Truth is ever-elusive when treating with antiquity. I'm reminded of Tennyson's words "[Death] puts our lives so far apart we cannot hear each other speak."

 

[bradford]

Still, you did find another cool pattern.

 

[Sparhawk]

The utter simplicity of the concentric square rings on the 8x8 and the coincidence of their producing the same probabilities are the purported yarrow oracle ... it speaks to me. Truth often is betrayed that kind of straightforward simplicity -- at least in the stories.

Still, the historical record does not support the hypothesis, bummer. Truth is ever-elusive when treating with antiquity. I'm reminded of Tennyson's words "[Death] puts our lives so far apart we cannot hear each other speak."

Hey, no reason to despair, just as Brad pointed out, you did find a cool pattern that, as far as I know or recall, wasn't observed before (and boy, after three millennia of exegesis, that's hard as hell). Which makes me wonder more about those Song scholars, not only Zhuxi but also Shao Yong, from whom, it appears, the binary arrangement of the hexagrams comes from. OTOH, I'm not as disciplined or skeptical as Brad when it comes to attributing the Chinese more mathematical knowledge than he does. I am an optimist of the kind of Richard S. Cook, who wrote a phone-book sized monograph, attributing ancient Chinese a lot more mathematical sophistication than anyone I know. Mind you, nobody I know with scholarly credentials, including Ed Hacker, whom is in his 80's and spent at least half of his life musing about the King Wen Sequence (KWS), appears to have taken Cook seriously. Who knows, perhaps Cook suffers from the same kind of dismissal Chris Lofting did: "theoretical suicide by obfuscation"... Or perhaps, as it appears to be the ongoing attitude towards it, the man just created a humongous scaffolding wrapped around the KWS like a glove, as in "observe a pattern and apply an until then inexistent meaning to it".

My point is, I've no strong reason to dismiss the idea that those ancient Chinese sages did indeed observe what you have but also, perhaps, arrange things around, such as divination methods, to fit those patterns.

 

[bradford]

Hi Saharan-
Got to foolin' around with this as fools are wont to do.
Here are three more Xian Tian expressions of your 1357 (or 4, 12, 20, 28) ratio.
http://www.hermetica.info/1357.jpg
Structurally you may find #35 to have the most interesting Gua involvement (assuming you are using the Xian Tian gua)

 

[peterg]

yarrow method

Given the parameters the originator was working under , its begins to become understandable why he choose the current yarrow method.

All 18 operations had to be the same , for practical purposes.

He got the idea of adding +1 to the remainders , by looking at the pattern that emerges when you don't add anything.

He choose division by 4 by trial and error. What else could he have chosen ?
Nothing else computes or produces wildly asymmetrical figures.

He needed 46 - 50 sticks to get bundles of 6789 at the end.

He deliberately choose asymmetry because he wanted to give more weight to moving yang ( the option of more weight to yin was available : use 47 instead of 49 )

He choose 1357 / 16 because thats all that was on the table.
He might have chosen symmetry of 2266 / 16 = 1133 / 8 , which was available ( use 48 instead of 49 sticks ) ( any multiple of 4 sticks produces odds of 1/ 2 )

Once a method existed it would naturally attract a rationale of ideas around it , not necessarily invalid ideas.

Voila.

 

[saharan]

On account of our common interest in hermetics, please peruse new post "Yao Numbers and the Tree of Life" on the Divination forum and render your thoughts.

 

[superman]

The graphic depicted here shows the so-called FuXi arrangement overlaid on a tableau of concentric squares, each discretely colored. This presentation is remarkable in that it explicitly encodes the yarrow oracle probabilities; i.e. each ring comprises a specific proportion of the hexagrams as shown in the table above.

⚏ dynamic yin, represented by '6', probability = 0.0625 ( 4 in 64)

⚌ dynamic yang, represented by '9', probability = 0.1875 (12 in 64)

⚎ static yang, represented by '7', probability = 0.3125 (20 in 64)

⚍ static yin, represented by '8', probability = 0.4375 (28 in 64)


​

It does appear that the oracle and its probabilities are derived from this 8x8 grid, and that the ritual as is commonly understood may have been a way to produce the probabilities from the grid. Any proper quadrant of this 8x8 square likewise encodes the same probabilities at one-quarter scale: 1/16, 3/16, 5/16, 7/16.

Eh, what do the colors mean?

 

[peterg]

yarrow

Analysing yarrow is easier than it might seem at first glance.

Since the divisor is 4 you need look at any 4 consecutive splits to see the pattern.

So 24+25 , 23+26 , 22+27, 21+28 , give remainders 5555 , which has no practical value.

And with the +1 (the ' one above ', 49+1, is essentially a reminder ? ) this becomes :

24+24+1 , 23+25+1 , 22+26+1 , 21+27+1 , giving remainders 9555 with odds 1/4 and 3/4

Since you want 6789 at the end , only 47, 48, 49, or 50 sticks applies , if the divisor is 4


Pascals triangle reminds me of yarrow , with its 'one above'. The lines of PT, (yin + yang )^ 0-6 ), develop in a similar way to yarrow. If you read the lines as digits of a number ( 1, 5, 10, 10, 5, 1 becomes 161051), rather than as coefficients , then each line is a power of 11 , thats 10 times the previous line , plus one time. PT might have helped inspire yarrow since it was around in 12c.AD China. Perhaps the I Ching inspired PT.