I read the page and went through the "verify the cycles for yourself" sequence and I still have no earthly idea when defining the cycles, what is the rule that says "if you're currently on hexagram X, you can calculate the next hexagram Y by doing..."
Each hexagram has two positions: one in the binary natural order (0-63), and one in the King Wen sequence. The rule is: a hexagram
moves from its natural order position to its King Wen position. For example, Qian is at position 63 in the natural order and position
0 in the King Wen sequence, so σ(63)=0. Then look at position 0 (Kun), which is at position 1 in the King Wen sequence, so σ(0)=1.
Follow this chain until you return to the start. There is no formula — σ is defined by the mapping table between the two orderings.
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.
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.
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.
Well... you could describe it that way, if you wanted to.
The Wolfenstein 3D code implements a function from a coordinate pair (x₁, y₁) to a new coordinate pair (x₂, y₂) which has the property that, if you start with the pair (0, 1), repeatedly applying the function will take you through every coordinate pair that represents a valid pixel on screen and then return to (0, 1).
"Cycle" in the context of permutations still refers to the process of applying a function repeatedly and ultimately returning to the original value you started with, but there is no concept of "visiting every valid value in between".
(And this is not a necessary part of the phenomenon in Wolfenstein, either; it could, theoretically, have used a function that painted three different points red and never visited the rest of the screen. "Cycle" directly refers to the fact that repeated application of the function will eventually produce a result that has been seen before, which Wolfenstein uses as the condition to break out of a while loop. The fact that every pixel on screen has been visited at that point is a fact about the "cycle length".)
I wrote up something of a description of permutation cycles before looking up the Wolfenstein thing, so here it is:
---
Permutations are usually considered in terms of the "cycles" that make them up.
Intuitively, you can describe a permutation by explicitly listing the position to which it assigns everything. In this method,
[1, 4, 2, 5, 3]
is a permutation of 5 objects which places the first object first, the second object third, the third object fifth, the fourth object second, and the fifth object fourth.
This is cumbersome, and it obscures the internal structure of the permutation. It is more conventional to describe a permutation as a collection of cycles; our example permutation would be given as
(1)(2 4 5 3)
This tells us that the first element is in a 1-length cycle with itself, and the other four elements share a 4-length cycle. Specifically, after one application of the permutation, element 4 will conceptually metamorphose into element 5 (which follows "4" in the cycle), element 2 will become element 4, element 3 will become element 2, element 1 will stay right where it is, and so forth.
This representation, among other virtues, makes it pretty easy to compute the order of the elements after one, or more, applications:
After four applications, we've come back to the original order of the elements. This is because our permutation contains a 1-cycle and a 4-cycle, and the least common multiple of 1 and 4 (the cycle lengths) is 4 (the number of applications required to return to the original order). You can see the 1-cycle running down the column at position 1, and you can see the 4-cycle running down the columns at positions 2, 4, 5, and 3, which are the elements contained in that cycle.
Armed with this, we can go a little further: after 75 applications, 1 will advance through its cycle 75 times (remaining "1"), and each other position will advance through its cycle 75 times. Since that cycle is 4 elements long, this is the same as advancing 75 mod 4 (= 3) times, giving us
(75) [1, 3, 5, 2, 4]
It turns out that every permutation arranges elements into cycles like this. We could consider a permutation on two objects:
(1 2)
which swaps the objects. We can consider what is essentially the same permutation on 200 objects:
(1 2)
which swaps the first two objects while leaving the other 198 objects in place. (Formally, those objects are all in 1-cycles, and we just don't bother writing them all down.) With a slightly more complex permutation on 200 objects:
(3 70 54 159)
we will see 196 objects stay in place while the objects in positions 3, 70, 54, and 159 rotate through those four positions. If we considered a different permutation on 200 objects for which (3 70 54 159) was one of the cycles it contained, the behavior at those four positions would remain the same as in this example, but the other 196 objects would behave differently. Every four applications, the object which started at position 3 will return to position 3, but the overall permutation might contain cycles of other lengths, so the period of the overall permutation will probably not be four.
You can think of a permutation as a collection of wheels of different circumferences, with each application rotating every wheel through a constant-across-the-wheels arclength. We call those wheels 'cycles'.
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.
>> 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.
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.
I read the page and went through the "verify the cycles for yourself" sequence and I still have no earthly idea when defining the cycles, what is the rule that says "if you're currently on hexagram X, you can calculate the next hexagram Y by doing..."
Each hexagram has two positions: one in the binary natural order (0-63), and one in the King Wen sequence. The rule is: a hexagram moves from its natural order position to its King Wen position. For example, Qian is at position 63 in the natural order and position 0 in the King Wen sequence, so σ(63)=0. Then look at position 0 (Kun), which is at position 1 in the King Wen sequence, so σ(0)=1. Follow this chain until you return to the start. There is no formula — σ is defined by the mapping table between the two orderings.
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.
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.
http://www.levity.com/eschaton/waveexplain.html
McKenna got deep into this...
https://www.fractal-timewave.com/articles/math_twz_10.htm
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.
Well... you could describe it that way, if you wanted to.
The Wolfenstein 3D code implements a function from a coordinate pair (x₁, y₁) to a new coordinate pair (x₂, y₂) which has the property that, if you start with the pair (0, 1), repeatedly applying the function will take you through every coordinate pair that represents a valid pixel on screen and then return to (0, 1).
"Cycle" in the context of permutations still refers to the process of applying a function repeatedly and ultimately returning to the original value you started with, but there is no concept of "visiting every valid value in between".
(And this is not a necessary part of the phenomenon in Wolfenstein, either; it could, theoretically, have used a function that painted three different points red and never visited the rest of the screen. "Cycle" directly refers to the fact that repeated application of the function will eventually produce a result that has been seen before, which Wolfenstein uses as the condition to break out of a while loop. The fact that every pixel on screen has been visited at that point is a fact about the "cycle length".)
I wrote up something of a description of permutation cycles before looking up the Wolfenstein thing, so here it is:
---
Permutations are usually considered in terms of the "cycles" that make them up.
Intuitively, you can describe a permutation by explicitly listing the position to which it assigns everything. In this method,
is a permutation of 5 objects which places the first object first, the second object third, the third object fifth, the fourth object second, and the fifth object fourth.This is cumbersome, and it obscures the internal structure of the permutation. It is more conventional to describe a permutation as a collection of cycles; our example permutation would be given as
This tells us that the first element is in a 1-length cycle with itself, and the other four elements share a 4-length cycle. Specifically, after one application of the permutation, element 4 will conceptually metamorphose into element 5 (which follows "4" in the cycle), element 2 will become element 4, element 3 will become element 2, element 1 will stay right where it is, and so forth.This representation, among other virtues, makes it pretty easy to compute the order of the elements after one, or more, applications:
After four applications, we've come back to the original order of the elements. This is because our permutation contains a 1-cycle and a 4-cycle, and the least common multiple of 1 and 4 (the cycle lengths) is 4 (the number of applications required to return to the original order). You can see the 1-cycle running down the column at position 1, and you can see the 4-cycle running down the columns at positions 2, 4, 5, and 3, which are the elements contained in that cycle.Armed with this, we can go a little further: after 75 applications, 1 will advance through its cycle 75 times (remaining "1"), and each other position will advance through its cycle 75 times. Since that cycle is 4 elements long, this is the same as advancing 75 mod 4 (= 3) times, giving us
It turns out that every permutation arranges elements into cycles like this. We could consider a permutation on two objects: which swaps the objects. We can consider what is essentially the same permutation on 200 objects: which swaps the first two objects while leaving the other 198 objects in place. (Formally, those objects are all in 1-cycles, and we just don't bother writing them all down.) With a slightly more complex permutation on 200 objects: we will see 196 objects stay in place while the objects in positions 3, 70, 54, and 159 rotate through those four positions. If we considered a different permutation on 200 objects for which (3 70 54 159) was one of the cycles it contained, the behavior at those four positions would remain the same as in this example, but the other 196 objects would behave differently. Every four applications, the object which started at position 3 will return to position 3, but the overall permutation might contain cycles of other lengths, so the period of the overall permutation will probably not be four.You can think of a permutation as a collection of wheels of different circumferences, with each application rotating every wheel through a constant-across-the-wheels arclength. We call those wheels 'cycles'.
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.
> 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.
You didn't explain or prove there is a reason for it.
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.