Show HN: The King Wen Permutation: [52, 10, 2]

▲ Show HN: The King Wen Permutation: [52, 10, 2](gzw1987-bit.github.io)

31 points by gezhengwen 5 hours ago | 16 comments

I analyzed two orderings of the 64 I Ching hexagrams and found the permutation cycle decomposition between them is [52, 10, 2] with zero fixed points. Nobody has done this kind of analysis before and this cycle type has not been reported in the literature. You can verify it yourself.

▲ kazishariar 19 minutes ago | parent | next [-]

How/Can you compare this to Magic Squares?

▲ gezhengwen 5 hours ago | parent | prev | next [-]

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?

▲

seanhunter 25 minutes ago | parent | next [-]

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.

▲

dmos62 5 hours ago | parent | prev [-]

Fascinating. I've barely any knowledge of I Ching. What motivated you to explore this and I Ching in general?

▲

gezhengwen 3 hours ago | parent [-]

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.

	▲	IAmBroom an hour ago \| parent [-]
		I love you. Marry me.

▲ busfahrer 2 hours ago | parent | prev | next [-]

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?

	▲	gezhengwen 2 hours ago \| parent [-]
		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.

▲ casey2 3 hours ago | parent | prev | next [-]

Random shuffles usually have a big loop

	▲	gezhengwen 2 hours ago \| parent [-]
		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.

▲ chordbug 4 hours ago | parent | prev [-]

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.

▲ Someone 4 hours ago | parent | next [-]

> and is the result even that surprising?

Not as far as I can tell from skimming https://en.wikipedia.org/wiki/Random_permutation_statistics.

▲ gezhengwen 2 hours ago | parent | prev | next [-]

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.

▲ thaumasiotes 4 hours ago | parent | prev [-]

> 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:

    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?'.

▲

gezhengwen 2 hours ago | parent [-]

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.

	▲	mcphage an hour ago \| parent [-]
		> 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?