先天 · 後天 · the eight trigrams

The two heavens

The Earlier and Later bagua as permutations

Switch between Earlier and Later Heaven and watch each trigram travel to its new position.

The 8 trigrams have two classical circular arrangements. The Earlier Heaven (先天), attributed to Fu Xi, is the cosmological order: heaven above, earth below. The Later Heaven (後天), attributed to King Wen, is the seasonal order of the world in motion. Going from one to the other is a permutation of 8 elements: this whole laboratory in miniature. As in Chinese maps, south is up and east is on the left.

Each trigram faces its exact binary complement: the 4 axes add up to 7.

cycles (2 of 4)

fixed points

order (τⁿ = id)

14/28

inversions (50%)

The permutation τ, position by position

τ sends the trigram occupying a position in the Earlier Heaven to the one occupying that same position in the Later. It decomposes into 2 cycles of 4: no trigram stays put, and τ applied 4 times is the identity.

Kun→Kan→Dui→Xun→cycle of 4

Gen→Qian→Li→Zhen→cycle of 4

parity even · 14 of 28 inversions: here too, as in the King Wen of the 64, the disorder hovers around exactly half.

Where the symmetry breaks

In the Earlier Heaven, the 4 axes join binary complements: Kun (0) · Zhen (4), Gen (1) · Li (5), Kan (2) · Dui (6), Xun (3) · Qian (7). Each pair adds up to 7, which is the bitwise NOT over 3 lines. In the Later Heaven that property breaks on almost every axis: only Kun and Zhen (fire to the south, water to the north) remain facing each other as complements. Verified over the positions: 4 complementary axes in the Earlier one, exactly 1 in the Later.

τ	Kun	Gen	Kan	Xun	Zhen	Li	Dui	Qian
↓	Kan	Qian	Dui	Kun	Gen	Zhen	Xun	Li