The group (Z/2)⁶ and the Sierpinski
XOR, eight cosets and a fractal that comes out of Pascal
Tap a hexagram to see its coset (upper XOR lower) and the seven that keep it company.
Mutating lines is XOR, and with it the 64 hexagrams form the group (Z/2)⁶: Kun (all yin) is the identity and every hexagram is its own inverse (mutating it twice by the same thing brings it back). Inside live the 8 pure hexagrams (a doubled trigram), which form a subgroup; its 8 cosets split the 64 into a new partition.
The coset of a hexagram is all those that share the same upper ⊕ lower (the binary difference between its two trigrams). Coset 0 is the subgroup itself: the 8 pure hexagrams.
The matrix S[i][j] = 1 when j is a submask of i (all its yang lines are also in i) is, by Lucas's theorem, exactly C(i,j) mod 2: Pascal modulo 2. Drawn, it is the Sierpinski triangle, self-similar by blocks (S = [[S, 0], [S, S]]). And it is the same dominance relation of the Boolean lattice B6: the fractal is the shadow of the partial order.
64 × 64 · row i, column j · cell on when j ⊆ i · 729 ones
With XOR the 64 form the group (Z/2)^6; the 8 pure hexagrams are a subgroup whose 8 cosets partition the set, and the dominance matrix is Pascal mod 2 (Lucas), the Sierpinski triangle.
verificar_grupo_sierpinski · source: Lucas, 1878The types of claim and the full bibliography (APA) are in Foundations.