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Structural Mathmatics of The I Ching

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I am not sure if this is an actual term, but I was wonding about the Structural Mathmatics of the I Ching.

Mainly I am wondering about how the 64 hexagrams cames about.
Is it because 1 is Heaven and 2 is Earth?
I feel like I have heard that before.


Here is what is in my head as far as how 64 hexagrams cames to be (Starting at the bottom and using 6 for the 6 lines of a hexagram).

32x2 = 64
16x2 = 32
8x2 = 16
4x2 = 8
2x2 = 4
1x2 = 2

So I guess I have 2 more questions, IF this is how I should be looking at how the 64 are formed...

1. Why 6?
2. Why multiply by 2?
 

bradford

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There is half of a structure but no mathematics to the King Wen arrangement.
There is a set of geometrical properties that is present throughout Yijing, and all of the relevant dimensions of the Yijing have a geometry.
The study of this is known as Xiangshu Jia, Image and Number School.
The Primal or Early Heaven arrangement is purely mathematical, and perfectly geometrical, but this did not come about until ten centuries ago..
 

russell

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Why multiply by 2? This is the easy one; each line has two possibilities (solid or broken), so the number of gua has to be two to the power of the number of lines.

Why 6? This may ultimately be arbitrary. Richard Rutt (something of a heretic to some) says there is evidence that the hexagrams may have originally been pentagrams. And there are a number of hexagrams in which one of the lines seems to have little to do with the other five. At some point, someone wanted more possibilities, and so added a sixth line, doubling the number of gua from 32 to 64. This could go on forever, but becomes impractical at some point; 64 (2^6) has stood the test of time.

The pentagram theory depends on a feature of the line statements of fifteen hexagrams (1, 3, 4, 5, 7, 13, 15, 18, 27, 31, 33, 36, 56, 58, and 59), where a key word or theme occurs in only five of the statements. In Hexagram 36 ‘Crying pheasant’, for instance, the word for ‘crying pheasant’ occurs in only five lines. The sixth line statement seems to be an intrusion and may be a later addition. When the intrusion is removed from each of the fifteen, the resulting pentagram drawings are all different from each other.
—Richard Rutt, Zhouyi (2007, p.96)
p.s. The pentagram idea is consistent with the idea that the complete gua are historically primary, and the trigrams are a later addition. I myself don’t pay as much attention to the trigrams as some people do.

—Russell
 

pocossin

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The pentagram idea is consistent with the idea that the complete gua are historically primary, and the trigrams are a later addition.
Couldn't the pentagram theory be chance occurrence rather than deliberate design? For example, the "key word or theme" may occur from zero to six times. Are there occasions where it occurs two or three times? To sort all this out is a bit of trouble, and one would have to look for repetition of Chinese characters.
 
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What I get:

Duality symbolized:
___
_ _

The interation of those 2 symbolized in all possibilities:

___
___

_ _
_ _

___
_ _

_ _
___

What I do not get:

Why didn't the structure then go from the interacting of those possibilities creating Quadgrams?


Thanks for everyone's input thus far.
 

bradford

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Couldn't the pentagram theory be chance occurrence rather than deliberate design? For example, the "key word or theme" may occur from zero to six times. Are there occasions where it occurs two or three times? To sort all this out is a bit of trouble, and one would have to look for repetition of Chinese characters.
It is in the use of Chinese characters, particularly reiteratives, that Rutt's pentagram theory is shown to be incorrect and the early existence of Trigrams, in pairs, is vindicated. This is not to say that Trigrams are primary, only that they were acknowledged within the Hexagrams when the text was written. It is also Not to say that most of the meanings for the Trigrams, as found in the Shuo Gua, were fully developed when the Zhouyi was written.
 

bradford

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Hi bradford
I typed Xiangshu Jia in google and am not getting anything good.
Let me know if you know of any specific books on the subject.
This is a huge field and it will be easy to get overwhelmed.
And to uncover big steaming piles of doodoo.
There is a good but very expensive book by Bent Nielsen, A Companion to Yi Jing Numerology. See intro at Google Books.
Also a doctoral thesis available at UMI through your library, by George Fendos Jr. “Fei Chih’s Place in the Development of I-Ching Studies.” Ph.D. dissertation in Chinese Studies: University of Wisconsin, Madison, WI: 1988.
Both are very technical.
But I would of course recommend starting with my own Dimensions chapter in Volume 2
(free download) at the link below.
 

sparhawk

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p.s. The pentagram idea is consistent with the idea that the complete gua are historically primary, and the trigrams are a later addition. I myself don’t pay as much attention to the trigrams as some people do.

—Russell
You mean primary as in one hexagram only, no zhi gua? Well, depends on how far you want to go back in history but zhi gua were already present in the Zuo Zhuan, hailing from the Warring States period and documenting the Spring and Autumn one, albeit those zhi gua were obtained in a different way from the present yarrow method. So, we can safely assume that the zhi gua have been around for at least 2500 years. Of course, zhi gua are a resulting observation of yarrow manipulation, thus a mathematical method, which was concurrent with the cracking of bones, as those records show. Those early records are unclear about how yarrow was used but, the possibility itself of the observation/invention of zhi gua that early cannot be fully denied. Lack of hard proof for their early existence only proves there's no hard proof...
 
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bradford

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Hi Luis-
He's talking about the hypothesis that the hexagrams (or pentagrams) were discovered first and the trigrams were developed conceptually only much later, as an afterthought. As in Steve Moore's Trigrams of Han.
 

sparhawk

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Thanks... ADHD flare up... :D
 

boyler

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Sorry for interrupting, but isn't this 1, 2, 3, 4, 6 thing already "explained" in 'Ten Helpers'?
 
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boyler,

You are not interrupting. I am trying to figure this out any way I can. Honestly I am not arriving at a conclusion that is suitable thus far. :mad: so anything will help.

Sorry for interrupting, but isn't this 1, 2, 3, 4, 6 thing already "explained" in 'Ten Helpers'?
You are talking about how some people have been saying that there were originally 5 lines and that the 5th line is something that sometimes does not fit in?? :confused:

What is Ten Helpers? (I am sure this is right infront of my face, but I cannot see beyond my nose right now)
Are you 1 of 10 of my helpers?:rofl:

Simply:
I am trying to figure out why there are 6 spots to each hexagram. I think I need math and philosophy to complete my understanding..

Please see my previous posts on what I DO understand.


Take Care,
Courtney
 
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What I get:

Duality symbolized:
___
_ _

The interation of those 2 symbolized in all possibilities:

___
___

_ _
_ _

___
_ _

_ _
___

What I do not get:

Why didn't the structure then go from the interacting of those possibilities creating Quadgrams?
Quoting myself
 

rodaki

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hi Courtney,

[What I do not get:

Why didn't the structure then go from the interacting of those possibilities creating Quadgrams?
I'm guessing it's because in six places all possible yin/yang combinations are covered? (yang/yang, yin/yang, yin/yin) I'm not an expert though . .
 

sparhawk

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What is Ten Helpers? (I am sure this is right infront of my face, but I cannot see beyond my nose right now)
Are you 1 of 10 of my helpers?:rofl:
Robert refers to the 十翼 shí yì, "Ten Wings"...

I am trying to figure out why there are 6 spots to each hexagram. I think I need math and philosophy to complete my understanding..
Do you have the Wilhelm translation? It is in the Ta Chuan, Part 1, chapter 2. An explanation of sorts is in there.
 
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Do you have the Wilhelm translation? It is in the Ta Chuan, Part 1, chapter 2. An explanation of sorts is in there.
Yes. I have that. Thanks. I am an idiot sometimes. Plus, I spend most of my time on this site at work without any references. I think I just want to chat about it. I will look those chapters over and discover the secrets to the universe! :mischief: Moo haha

Don't tell my boss about this post
 

russell

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Some “nice” things about 64: besides being a power of 2 (2^6), it is a square (8^2) and a cubic (4^3) number.

Chessboards have 64 squares (8^2). And, relevant or not, there are 64 DNA codons (4^3). It is a conveniently-sized chunk of information.

One wonders if the Tao Te Ching has 81 chapters because Lao Tzu was trying to “outdo” the I Ching, 9 squared vs. 8 squared.

—Russell
 
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Thanks for everyone who chimed in. I am starting to understand now based on the Wilhelm translation and the link to bradford's book.
 

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