I found this by accident while analyzing the I Ching with code. 81% of hexagrams are locked in one chain, none stays in its original
position. You can verify it yourself in the browser. Has anyone seen this before?
I doubt they already had the King Wen order in the Shang dynasty. Manuscripts dated to as late as the Han dynasty have a totally different hexagram order. In any case traditionally the divination book for Shang is considered to be the Guicang, not the I Ching (=Zhouyi = Changes of Zhou), which according to tradition put kun before qian.
The Shang dynasty people knew the pairing structure of hexagrams (inverted/complementary pairs), but cycle decomposition is a modern
group theory tool that did not exist until the 19th century. These are two different levels of analysis.
They knew about the cycle. That's why it's called the King Wen sequence right? Not sure what part of this you think people didn't know about so we may be talking at cross purposes.
The I Ching has influenced China for over 3000 years. I believe there must be a reason for that. In China, the I Ching is often
treated as mysticism. But I believe in science. The end of mysticism must still be science. So I did a lot of research and found a
unique pattern inside. I searched all the literature and found nothing about it. So I shared it here.
You are right, the expected largest cycle of a random permutation is around 40. 52 is larger but not extreme. I did not claim this
result is statistically significant.
Yes, it is essentially the same mathematical concept — both are cycle decompositions of permutations. Carmack used a permutation to
ensure every pixel is visited exactly once.
The I Ching has a historical connection to magic squares — the Lo Shu is a 3x3 magic square traditionally linked to the I Ching. But
cycle decomposition analyzes the permutation between two orderings, which is a different mathematical structure from the
row/column/diagonal sums of magic squares. That said, it is an interesting direction worth exploring.
We truly live in an age where facts that are worth "maybe one sentence of space on Wikipedia" can be expanded into full-blown AI-coded interactive websites. I'm not sure how to feel about this. I think in this case it ascribes an inappropriate sense of grandeur: making a mathematical curiosity (and is the result even that surprising?) seem like some deep truth has been unveiled, or we finally found God's Number.
You are right, the presentation may be overdone. The result itself is a small mathematical fact. I made the interactive page so people
can verify it themselves, not to make it look grand. Thank you for the criticism, I will adjust.
>> Zero fixed points — not a single hexagram occupies the same position in both orderings. The structural difference is total.
As a mathematical matter, the expected number of fixed points for any permutation is 1. Some have more. For some to have more, others must have less, and all of those will have 0.
But as a logical matter, "the structural difference is total" is pure gibberish. Consider these two permutations on 5 elements:
1. [2, 3, 4, 5, 1]
2. [5, 1, 2, 3, 4]
"Not a single element occupies the same position in both orderings."
But of course these two permutations have a nearly identical structure (they are rotations in opposite directions, and are each other's inverses); they are far more closely related to each other than either is to
3. [4, 3, 2, 1, 5]
even though permutation 3 shares the assigned position of "3" with permutation 1, and the assigned position of "2" with permutation 2.
Then:
>> We reframe the question:
>> Transform the question "what is the structural distance between two orderings"
>> into the mathematical problem "what is the cycle structure of a specific permutation in S₆₄?"
This is nonsense. The 'question' cannot be transformed into the 'problem', because they are completely unrelated ideas. It's like transforming the question 'what is the Levenshtein distance between two strings?' into the problem 'if a specific string were in alphabetical order, how would it be pronounced?'.
You are right, zero fixed points does not mean total structural difference. Your counterexample is good. My wording was wrong, I will
fix it. What interests me is not the statistical rarity, but that 81% of elements are in one orbit — this means the reordering is
highly coupled, not a bunch of small local swaps.
> What interests me is not the statistical rarity, but that 81% of elements are in one orbit — this means the reordering is highly coupled, not a bunch of small local swaps.
But what is the significance of the reordering being highly coupled?
The observation itself is the value — it tells you the King Wen sequence is not a bunch of small local adjustments, but a holistic
rearrangement. But it cannot tell you why King Wen arranged it this way.
http://www.levity.com/eschaton/waveexplain.html
McKenna got deep into this...
https://www.fractal-timewave.com/articles/math_twz_10.htm
I found this by accident while analyzing the I Ching with code. 81% of hexagrams are locked in one chain, none stays in its original position. You can verify it yourself in the browser. Has anyone seen this before?
People have known about this since the Shang dynasty so yes it has been noticed before.
If you find this interesting, I suggest you study group theory - this seems pretty much a direct consequence of the group structure.
I doubt they already had the King Wen order in the Shang dynasty. Manuscripts dated to as late as the Han dynasty have a totally different hexagram order. In any case traditionally the divination book for Shang is considered to be the Guicang, not the I Ching (=Zhouyi = Changes of Zhou), which according to tradition put kun before qian.
The Shang dynasty people knew the pairing structure of hexagrams (inverted/complementary pairs), but cycle decomposition is a modern group theory tool that did not exist until the 19th century. These are two different levels of analysis.
They knew about the cycle. That's why it's called the King Wen sequence right? Not sure what part of this you think people didn't know about so we may be talking at cross purposes.
Fascinating. I've barely any knowledge of I Ching. What motivated you to explore this and I Ching in general?
The I Ching has influenced China for over 3000 years. I believe there must be a reason for that. In China, the I Ching is often treated as mysticism. But I believe in science. The end of mysticism must still be science. So I did a lot of research and found a unique pattern inside. I searched all the literature and found nothing about it. So I shared it here.
The lines of a hexagram map to the edges of a tetrahedron.
I love you. Marry me.
Random shuffles usually have a big loop
You are right, the expected largest cycle of a random permutation is around 40. 52 is larger but not extreme. I did not claim this result is statistically significant.
Does cycle here mean the same thing as what Carmack used in Wolf3D to randomly fill the screen with red pixels without any of them repeating?
Yes, it is essentially the same mathematical concept — both are cycle decompositions of permutations. Carmack used a permutation to ensure every pixel is visited exactly once.
How/Can you compare this to Magic Squares?
The I Ching has a historical connection to magic squares — the Lo Shu is a 3x3 magic square traditionally linked to the I Ching. But cycle decomposition analyzes the permutation between two orderings, which is a different mathematical structure from the row/column/diagonal sums of magic squares. That said, it is an interesting direction worth exploring.
We truly live in an age where facts that are worth "maybe one sentence of space on Wikipedia" can be expanded into full-blown AI-coded interactive websites. I'm not sure how to feel about this. I think in this case it ascribes an inappropriate sense of grandeur: making a mathematical curiosity (and is the result even that surprising?) seem like some deep truth has been unveiled, or we finally found God's Number.
> and is the result even that surprising?
Not as far as I can tell from skimming https://en.wikipedia.org/wiki/Random_permutation_statistics.
You are right, the presentation may be overdone. The result itself is a small mathematical fact. I made the interactive page so people can verify it themselves, not to make it look grand. Thank you for the criticism, I will adjust.
> and is the result even that surprising?
No.
The exposition has its problems too. Consider:
>> Zero fixed points — not a single hexagram occupies the same position in both orderings. The structural difference is total.
As a mathematical matter, the expected number of fixed points for any permutation is 1. Some have more. For some to have more, others must have less, and all of those will have 0.
But as a logical matter, "the structural difference is total" is pure gibberish. Consider these two permutations on 5 elements:
"Not a single element occupies the same position in both orderings."But of course these two permutations have a nearly identical structure (they are rotations in opposite directions, and are each other's inverses); they are far more closely related to each other than either is to
even though permutation 3 shares the assigned position of "3" with permutation 1, and the assigned position of "2" with permutation 2.Then:
>> We reframe the question:
>> Transform the question "what is the structural distance between two orderings"
>> into the mathematical problem "what is the cycle structure of a specific permutation in S₆₄?"
This is nonsense. The 'question' cannot be transformed into the 'problem', because they are completely unrelated ideas. It's like transforming the question 'what is the Levenshtein distance between two strings?' into the problem 'if a specific string were in alphabetical order, how would it be pronounced?'.
You are right, zero fixed points does not mean total structural difference. Your counterexample is good. My wording was wrong, I will fix it. What interests me is not the statistical rarity, but that 81% of elements are in one orbit — this means the reordering is highly coupled, not a bunch of small local swaps.
> What interests me is not the statistical rarity, but that 81% of elements are in one orbit — this means the reordering is highly coupled, not a bunch of small local swaps.
But what is the significance of the reordering being highly coupled?
The observation itself is the value — it tells you the King Wen sequence is not a bunch of small local adjustments, but a holistic rearrangement. But it cannot tell you why King Wen arranged it this way.