King Wen as a permutation
How much the book disorders the geometry
Tap a cycle to see where it travels, and compare the four historical orderings.
Original findingIf the Fu Xi ordering is the identity (position k holds value k), any historical sequence of the 64 hexagrams is a permutation σ of the 64 numbers. Every permutation decomposes into cycles: groups that rotate among themselves. Their structure is the exact measure of the disorder. Four of them run here: King Wen, the Mawangdui silk manuscript, Jing Fang's Eight Palaces and Fu Xi itself as the baseline.
The dotted line is Fu Xi (the identity). The scatter around it is the disorder of the chosen ordering; the points on the line are the fixed ones.
Each row is a cycle (King Wen numbers of the hexagrams); the chosen cycle is highlighted in the matrix.
| Ordering | Cycles | Longest | Fixed | Order σⁿ | Parity | Inversions | Cost in lines |
|---|---|---|---|---|---|---|---|
| King Wen the received book | 3 | 52 | 0 | 260 | odd | 1013 (50%) | 211 (3.35×) |
| Mawangdui c. 168 BCE | 4 | 50 | 0 | 600 | even | 1008 (50%) | 141 (2.24×) |
| Jing Fang first century BCE | 2 | 57 | 0 | 399 | even | 1008 (50%) | 93 (1.48×) |
| Fu Xi 11th century (Shao Yong) | 64 | 1 | 64 | 1 | even | 0 (0%) | 120 (1.90×) |
| Gray smallest possible | a walk of one line per step | – | 63 (1.00×) | ||||
Mawangdui groups into 8 blocks by upper trigram; Jing Fang walks its Eight Palaces (the same partition as the palaces experiment). And something notable happens: with the authentic historical ordering, the two tie at exactly 1008 inversions (50%), the same disorder that King Wen (1013) shows against Fu Xi. Three very different architectures that, measured against binary counting, end up at almost the same distance.
And that 1008 is no coincidence: it is exactly the expected number of inversions between two random orderings, n(n−1)/4, with standard deviation 86.3. The three historical orderings fall right there, so with respect to the binary ordering they sit at the distance of chance: uncorrelated with it. The family tree of the orderings then measures the other question, how much they resemble each other: and there King Wen and Mawangdui turn out to be siblings (759 inversions), with Jing Fang the solitary one.
But the cost in lines (the total Hamming distance, how many lines change when moving from one hexagram to the next) separates them again: although Jing Fang and Mawangdui tie in inversions, Jing Fang jumps much less (93 lines against 141), because each palace is almost a walk of one line per step. King Wen is the one that jumps the most (211) and the smallest possible is 63, that of a Gray walk. Two metrics, two different portraits of the same ordering.
Note: the palace order used is Jing Fang's traditional bagong (乾震坎艮坤巽離兌: father, three sons by age, mother, three daughters by age).
King Wen has no fixed point at all: at no position does it agree with binary counting.
The Mawangdui ordering arranges the 64 by upper trigram with a fixed rule for the lower trigrams; it is a scholarly reconstruction of the 1973 manuscript, not a direct datum.
As a permutation of Fu Xi, King Wen has 1013 inversions (order 260), Mawangdui 1008 (order 600) and Jing Fang 1008; the Hamming cost in lines is 211, 141 and 93.
verificar_ordenesSearch 2026-07-25: the published comparisons between the historical orderings are qualitative or visual (Yijing Dao; Cottrell; Cook 2006 as an attempt to derive King Wen); the inversion metric against binary counting, the 1013/1008/1008 tie and the table of line costs do not appear published.
The types of claim and the full bibliography (APA) are in Foundations.