Comparator of partitions
How alike are two ways of grouping the 64?
Choose two partitions of the 64 and compare how much they agree with the adjusted Rand index.
Original findingThe site splits the 64 hexagrams in several ways: Jing Fang's palaces, the basins of the nuclear map, the cosets of the subgroup of pure hexagrams, and the rows and columns of trigrams. How alike are any two of them? The adjusted Rand index (ARI) measures it: 1 if they are identical, 0 if they are independent, negative if they separate more than chance.
| A \ B | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 7 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
row = group of Jing Fang's palaces, column = group of Cosets of the pure hexagrams; cell = hexagrams in common. A clean diagonal would mean identical partitions.
The palaces and the cosets are both partitions into 8 groups of 8, but they are not the same: their ARI is −0.125 (they separate more than chance). The reason is algebraic: each palace is the pure hexagram plus a fixed set of 8 generating masks, {0, 2, 32, 48, 56, 58, 60, 62}, and that set is not closed under XOR. Not being a subgroup, the palaces cannot be cosets of anything. Two constructions that look like twins (8 × 8) turn out, when measured, to be almost opposite.
The adjusted Rand index measures the similarity between two partitions of the 64; palaces and cosets give ARI = -0.125, further apart than chance, because the 8 generating masks do not form a subgroup.
verificar_particionesSearch 2026-07-25: there is structural literature on Jing Fang's Eight Palaces (Mesker 2002; comparisons in Yijing Dao) but no quantitative comparison of partitions; the ARI = −0.125 between the palaces and the cosets of the subgroup of pure hexagrams does not appear published.
The types of claim and the full bibliography (APA) are in Foundations.