The King Wen Permutation

A mathematical observation on a 3,000-year-old Chinese text

52 cycle
|
10 cycle
|
2 cycle

The 64 hexagrams of the I Ching have two canonical orderings: the binary natural order (0–63) and the King Wen sequence (~1000 BCE). Treating the map between them as a permutation in S₆₄, its cycle decomposition yields [52, 10, 2] with zero fixed points (P≈0.37 for random permutations). 81% of hexagrams are in one orbit, meaning the reordering is highly coupled rather than a collection of small local swaps. We have not found this cycle type discussed in the literature we surveyed.

Verify It Yourself → Read the Paper →

Key Facts

Properties of the King Wen permutation σ ∈ S₆₄

3
cycles
random expectation: 4.7
0
fixed points
no hexagram stays in place
81.25%
in one cycle
52 of 64
260
order of σ
σ²⁶⁰ = identity
3.349
mean Hamming distance
random: 3.0
3.2:1
even:odd ratio
parity of cycle positions

Interactive Pages

22 explorations — everything runs client-side in your browser

The Discovery

Verify Permutation
Verify the [52, 10, 2] cycle decomposition yourself
Algorithm Animation
Watch the algorithm trace cycles step by step
Research Paper (English)
Full interactive paper with reproducible code
Research Paper (Chinese)
中文交互式论文

The Journey

Derivation Journey
13-stage interactive derivation: from Taiji to 384 line texts
Retrospective
Project retrospective: how we got here
Binary Line Texts
384 line texts in binary representation

Cross-Disciplinary

Genome Rearrangement
Genome rearrangement analysis (Hannenhalli-Pevzner framework)
Game Theory
Game theory × I Ching mapping
Four Minds
Leibniz, von Neumann, Shannon, Petoukhov: independent convergence on binary × hexagram structures
Coupled System
Individual vs. institutional paradox engine
Coupled Cycles
Coupled cycle dynamics (early version)
Shang Timeline
Shang Dynasty 600-year timeline correlation

Wave Analysis

Kondratieff Waves speculative
Kondratieff waves × I Ching phase mapping — exploratory pattern comparison, not a causal claim

Binary Explorations

Bit Arrangements
Hexagram bit arrangement patterns
Bit Forms
Visual forms of binary structures
Bit × Morse
Hexagram binary mapped to Morse code
Bit × Music
Hexagram binary mapped to musical notation
Bit × Voltage
Hexagram binary as voltage signal waveforms
Historical Note: The binary natural order was formalized by Shao Yong (1011–1077 CE), approximately 2,000 years after King Wen. We do not claim King Wen "rearranged" the binary order. Our analysis characterizes the permutation between two independently defined orderings.