⊑ · partial order

The Boolean lattice B6

The partial order of turning lines on

Choose a hexagram and look at its cone: how many stand above it and how many below.

There is a natural order among hexagrams, and it is not "has more yang lines" (that is only the rank). It is line-by-line dominance: x lies below y if every yang line of x is also yang in y. You climb by switching on one line at a time, from Kun (nothing on) to Qian (everything on). The resulting diagram uses exactly the same 192 edges of the hypercube, now oriented upward: same graph, another reading.

levels (rank 0 to 6)

192

coverings = edges of Q6

720

chains from Kun to Qian (6!)

1·6·15·20·15·6·1

sizes C(6,k), as in the spectrum

Hexagram↑ 16 above (2^4) · ↓ 4 below (2^2) · both cones include the hexagram itself

the edges point upward (switching on a line) · with a hexagram chosen: violet = its upper cone, green = its lower cone

The level sizes, 1, 6, 15, 20, 15, 6, 1, are the C(6,k): the same multiplicities as the spectrum of the hypercube that appears in the experiment on the symmetries. And from Kun to Qian there are exactly 6! = 720 maximal chains: one for each possible order of switching on the six lines.