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pocossin

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Hexagrams may be interpreted as binary numbers, but is there an example at Clarity of binary used in divination?
 

pocossin

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I am now working on William Horden's I Ching Mathematics for the King Wen Version. In interpreting a hexagram as a binary number, there are four possibilities. Is the place of least value at the bottom or the top? Are 0 and 1 represented by a yin or yang line? Horden (hsin) has the place of least value at the top, and 0 is represented by a yang line and 1 by a yin. Hatcher also has the place of least value at the top, but 0 is represented by a yin line and 1 by a yang. For hexagram 48 (Yijing, Word by Word, p.341),

Code:
▄▄  ▄▄ 0
▄▄▄▄▄▄ 1 
▄▄  ▄▄ 0
▄▄▄▄▄▄ 1
▄▄▄▄▄▄ 1
▄▄  ▄▄ 0

Binary 011010 = decimal 26.

A yang line looks like a 1, and a yin line is hollow like a 0. This may be the reason for this popular assignment. William Horden would represent the hexagram as binary this way:

Code:
▄▄  ▄▄ 1
▄▄▄▄▄▄ 0 
▄▄  ▄▄ 1
▄▄▄▄▄▄ 0
▄▄▄▄▄▄ 0
▄▄  ▄▄ 1

Binary 100101 = decimal 37.

This assignment is probably symbolic. Yin implies substance and thus equals 1. Heaven is mostly empty space and thus equals 0. Further, this assignment accords with the practice of the taixuanjing of two thousand years ago. As for the location of the least significant digit, the taixuanjing has the least significant digit at the bottom, but Shao Yong's diagrams suggest that the least significant digit be at the top, and these diagrams grace the cover of Horden's book.
 
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bradford

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When writing 6-digit binary numbers you always write the big digit first, so the bottom line would be 32.
Also, if you take the shao yong arrangement and turn it clockwise 90 degrees, you get the numbers written in sequence the same way Chinese used to be written, top to bottom, right to left.
One, being odd, has to be Yang.
My Dimensions chapter uses the binary geometry throughout.
 

pocossin

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When writing 6-digit binary numbers you always write the big digit first, so the bottom line would be 32.

But if the Chinese wrote from top to bottom, then the top line would be written first.

One, being odd, has to be Yang.

However, Horden uses zero-based numbering, which puts qian in the first place. Cardinal 0 = ordinal 1. It's the same in the taixuanjing. A heaven line is taken as 0.
 

bradford

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But if the Chinese wrote from top to bottom, then the top line would be written first.

That hardly synchs up with the text calling the top line 'top' and the bottom "first" now does it?
Just turn the diagram 90 degrees and look at it. It might relate to how they learned not to build things from the roof down.
 

pocossin

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That hardly synchs up with the text calling the top line 'top' and the bottom "first" now does it?

I agree, but one of the motives for going to binary may be to escape the text. The linear figures of the taixuanjing are constructed from the top down, and that was two thousand years ago. My impression is that top down construction of linear figures became the norm for non-orthodox yijing divination, so I am not surprised that Shao Yong may have thought that way.
 

bradford

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If you have already made up your mind to go in this direction, then you will have to find a way to ignore Shao Yong's diagrams as well.
 

jzy369

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pocossin bradford:

May I ask what is the key benefits for assigning binary figures?

I have read a few academic papers on the "orders" of IC, statistics of IC text (classified into how may text are auspicious, how many are misfortune...). But so far I have not found them to be particularly useful, some paper even claimed the current orders to be "wrong" due to some symmetrical idiosynchrocies. I wonder if this assigning binary can benefit our IC understanding in some other way?
 

bradford

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jzy???

Iulian Shchutskii had an interesting take on looking backwards at the structures of the Yijing with
more modern understanding:
If we speak of beehives, then we must mention that their cells are in hexagonal
sections. This is a concept of geometry, but no one will suspect us of an inclination to
ascribe to the bees a knowledge of geometry. If the representatives of a culturally backward
people know how to construct huts out of logs, this still does not mean that they have a
knowledge of such engineering science as the statics of wooden structures or strength of
materials. However, if we were to study their technique, we could not avoid these scientific
terms to make the results of our studies understandable to contemporary readers. Finally,
how many millions of people have played various musical instruments without the slightest
knowledge of the laws of acoustics and the theory of music? ... . Thus, in studying the
techniques of thinking reflected in the Book of Changes, we must use our present day
techniques of thinking. p. 166


This was taken from my Volume 2, Chapter 1, Dimensions (free download at link below) which makes extensive use of the Xian Tian arrangement in looking at the more original dimensions of the Zhouyi and Wings, even though the Xian Tian came along more than 2000 years later. It's another window. I just never made the error of thinking that binary numbers (or even yin and yang) were part of the original Zhouyi thinking. You should also know that I am not taking this in the same direct as Pocossin.
 

pocossin

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If you have already made up your mind to go in this direction, then you will have to find a way to ignore Shao Yong's diagrams as well.

The only thing I have made up my mind to do is to try to understand William Horden's book well enough to write a book review as I promised. Apparently Shao Yong did not explain his square arrangement in writing. The hexagrams are systematically arranged. My guess is that Shao took Qian as the first item and Kun as the last, so in binary interpretation, is it 63 to 0 or 0 to 63? I prefer 0 to 63. I am influenced by similar number patterns in the taixuanjing.
 

pocossin

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I Ching Mathematics gives a handy way to make binary-decimal conversions. Using Horden's assignment of yang line = 0, yin line = 1, and the place of least significance at the top, the trigrams as binary numbers have the values 0☰, 1☱, 2☲, 3☳, 4☴, 5☵, 6☶, 7☷.

Decimal to binary: Divide by 8. The remainder is the upper trigram, and the quotient is the lower trigram.

What is the binary of decimal number 49?

49 ÷ 8 = 6 with a remainder of 1.

1 = ☱
6 = ☶

What is the binary of decimal number 30?

30 ÷ 8 = 3 with a remainder of 6.

6 = ☶
3 = ☳

Binary to decimal: Multiply the lower trigram by 8 and add the upper trigram.

What is the decimal value of hexagram 31?

☱ = 1
☶ = 6

6 X 8 = 48
48 + 1 = 49

What is the decimal value of hexagram 54?

☳ = 3
☱ = 1

1 X 8 = 8
8 + 3 = 11

This conversion procedure is also given at the bottom of this webpage.

http://www.russellcottrell.com/VirtualYarrowStalks/little-endian.htm
 

jzy369

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Music metaphor was a good one. Could you post the free download link again? It seems to disappear from my web page display...
 

myanon0001

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Apparently Shao Yong did not explain his square arrangement in writing. The hexagrams are systematically arranged. My guess is that Shao took Qian as the first item and Kun as the last, so in binary interpretation, is it 63 to 0 or 0 to 63? I prefer 0 to 63. I am influenced by similar number patterns in the taixuanjing.

Although Shao's arrangement can be obtained by treating the hexagrams effectively as binary (base-2) numerals and so placing them in numerical order, exactly the same arrangement is also naturally produced without imputing any such mathematical significance to the hexagrams (i.e., without interpreting the lines as digits whose products with powers of two are to be summed).

In particular, by simply regarding each hexagram as a string or "word" (read from bottom-to-top) from an alphabet whose only "letters" are ▄▄▄▄▄▄ and ▄▄ ▄▄, Shao's hexagram sequence is just the 64 possible words of length 6, placed in an ordinary alphabetical order[SUP]*[/SUP]. This ordering depends, obviously, upon which of the two "letters" is supposed to be first in the alphabet. If ▄▄▄▄▄▄ comes before ▄▄ ▄▄ in the alphabet, then the hexagrams in natural alphabetical order are as follows:
Code:
▄▄▄▄▄▄      ▄▄  ▄▄      ▄▄▄▄▄▄      ▄▄  ▄▄      ▄▄▄▄▄▄                   ▄▄▄▄▄▄      ▄▄  ▄▄
▄▄▄▄▄▄      ▄▄▄▄▄▄      ▄▄  ▄▄      ▄▄  ▄▄      ▄▄▄▄▄▄                   ▄▄  ▄▄      ▄▄  ▄▄
▄▄▄▄▄▄      ▄▄▄▄▄▄      ▄▄▄▄▄▄      ▄▄▄▄▄▄      ▄▄  ▄▄                   ▄▄  ▄▄      ▄▄  ▄▄
▄▄▄▄▄▄      ▄▄▄▄▄▄      ▄▄▄▄▄▄      ▄▄▄▄▄▄      ▄▄▄▄▄▄                   ▄▄  ▄▄      ▄▄  ▄▄
▄▄▄▄▄▄      ▄▄▄▄▄▄      ▄▄▄▄▄▄      ▄▄▄▄▄▄      ▄▄▄▄▄▄                   ▄▄  ▄▄      ▄▄  ▄▄
▄▄▄▄▄▄      ▄▄▄▄▄▄      ▄▄▄▄▄▄      ▄▄▄▄▄▄      ▄▄▄▄▄▄        ...        ▄▄  ▄▄      ▄▄  ▄▄

On the other hand, if ▄▄ ▄▄ comes before ▄▄▄▄▄▄ in the alphabet, then the corresponding ordering is as above except each ▄▄▄▄▄▄ is replaced by ▄▄ ▄▄, and vice versa.

It would seem to be a matter of dispute whether Shao essentially invented a binary (base-2) numeration system six centuries before Leibnitz, for it seems at least as likely that he merely applied a binary alphabetical ordering.


[SUP]*[/SUP] Mathematically, the two approaches are perfectly equivalent as ordinal systems, as long as the strings/numerals involved are all of equal length (here 6). The numerals derive their order from a standard ordering given to the individual digits, and the strings derive theirs from a standard ordering given to the individual letters. This is clear by noting that the same ordering algorithm works, mutatis mutandis ...

Alphabetical ordering of equal-length strings: "To determine which of two strings comes first in alphabetical order, their first l
etters are compared. If they differ, then the string whose first letter comes earlier in the alphabet is the one which comes first in alphabetical order. If the first letters are the same, then the second letters are compared, and so on."

Numerical ordering of equal-length numerals: "To determine which of two numerals comes first in numerical order, their first digits are compared. If they differ, then the numeral whose first digit comes earlier in the digit-ordering is the one which comes first in numerical order. If the first digits are the same, then the second digits are compared, and so on."

 
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pocossin

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Shao's hexagram sequence is just the 64 possible words of length 6, placed in an ordinary alphabetical order.

I agree that the alphabetical order is the same as the numerical order when all the "words" have the same length. However, so far as I know, alphabetical order did not exist in ancient China. The square arrangement could have been produced by trigrams, using the Earlier Heaven number sequence. Upper trigrams in each row follow that sequence as do lower trigrams in each column. The square diagram may be described as a table coordinated horizontally and vertically by the Earlier Heaven number sequence.
 
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myanon0001

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However, so far as I know, alphabetical order did not exist in ancient China. The square arrangement could have been produced by trigrams, using the Later Heaven number sequence.

Although the ancient Chinese may have had no alphabetical order, it seems likely that scholars of the time would have been been familiar with the concept, e.g. via exposure to other languages (in which it was being used at least as early as the first millennium BCE).

I think it's the Earlier Heaven sequence that shows up in the trigrams in Shao's square; but in any case, deriving the hexagram ordering from that of the trigrams pushes back the question to how the trigram ordering was obtained. (The Earlier Heaven ordering also is alphabetical.)
 

pocossin

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Isn't Shao's circular arrangement of hexagrams evidence that he was thinking in terms of the Earlier Heaven arrangement of trigrams?
 

myanon0001

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Isn't Shao's circular arrangement of hexagrams evidence that he was thinking in terms of the Earlier Heaven arrangement of trigrams?

I was just reading the following article:

Leibniz' Binary System and Shao Yong's "Yijing"
James A. Ryan
Philosophy East and West
Vol. 46, No. 1 (Jan., 1996), pp. 59-90

In discussing Shao Yong's "proto-science", it reproduces some of the diagrams he used, the main one being called "the Diagram", which shows the circular binary arrangement of the hexagrams and also the Earlier Heaven arrangement of trigrams. Shao is described as a "latter-day exponent" of this Diagram, which it says preceded him by centuries, possibly going as far back as the 1st century BCE. The concluding paragraph asserts that "Shao's system amounts only to a proto-science in which the Diagram is not a number system."


 

bradford

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Shao is described as a "latter-day exponent" of this Diagram, which it says preceded him by centuries, possibly going as far back as the 1st century BCE.

This claim is not substantiated at all in Ryan's referenced work by Schuyler Cammann, "Some Early Chinese Symbols of Duality," so the assertion remains pure conjecture, both the centuries earlier part and the 1st Century BCE part. There remains no evidence that the arrangement, or its understanding, preceded Shao Yong.
On a side note, the possibility should be explored that the Early Heaven arrangement arose from a simple doodle, with no real mathematical concept at all.
xiantianhex.jpg
 

myanon0001

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bradford -- On pp. 66-67, Ryan also discusses a diagram that's effectively your "doodle" (except inverted), and asserts that the progression it illustrates is in fact how the sequence was derived.
 

pocossin

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the Diagram is not a number system

Of course. The ancient Chinese never added, subtracted, multiplied, or divided by using the hexagrams as binary numbers, but that isn't to say that the hexagrams cannot be advantageously used as binary numbers. William Horden's idea (as I understand it) is that by using the hexagrams as binary numbers patterns are revealed that disclose the mystical depths of the Yi.
 

myanon0001

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Of course. The ancient Chinese never added, subtracted, multiplied, or divided by using the hexagrams as binary numbers, but that isn't to say that the hexagrams cannot be advantageously used as binary numbers. William Horden's idea (as I understand it) is that by using the hexagrams as binary numbers patterns are revealed that disclose the mystical depths of the Yi.

I'm sure that's true. One can take the point of view that these are ways of extending the original numerology implicit in the oracle, enriching the language through which the spirit of Yi is enabled to speak.
 

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