Fano plane — octonion multiplication mnemonic
Test non-associativity:
Choose basis elements a, b, c:
Choose basis elements a, b, c:
Computation:
(ab)c = —
a(bc) = —
Associator [a,b,c] = —
|[a,b,c]| = —
Octonions O:
Normed division algebra over ℝ, dim=8.
By Hurwitz 1898: only ℝ,ℂ,ℍ,O.
Non-associative: eᵢ(eⱼeₖ) ≠ (eᵢeⱼ)eₖ
for most i,j,k.
Alternative law holds:
(aa)b = a(ab), (ab)b = a(bb)
Automorphism group: G₂
(exceptional Lie group, dim=14)
|ab| = |a||b| (normed!)
Connection to S⁷ via Hopf map S¹⁵→S⁸
Normed division algebra over ℝ, dim=8.
By Hurwitz 1898: only ℝ,ℂ,ℍ,O.
Non-associative: eᵢ(eⱼeₖ) ≠ (eᵢeⱼ)eₖ
for most i,j,k.
Alternative law holds:
(aa)b = a(ab), (ab)b = a(bb)
Automorphism group: G₂
(exceptional Lie group, dim=14)
|ab| = |a||b| (normed!)
Connection to S⁷ via Hopf map S¹⁵→S⁸