Octonionic Algebra¶
The four normed division algebras¶
There are exactly four number systems where multiplication preserves norms (\(|ab| = |a||b|\)) and every non-zero element has an inverse. Each is built by doubling the previous one (the Cayley-Dickson construction):
| Algebra | Dim | Lost property |
|---|---|---|
| Reals \(\mathbb{R}\) | 1 | -- |
| Complex \(\mathbb{C}\) | 2 | Ordering |
| Quaternions \(\mathbb{H}\) | 4 | Commutativity |
| Octonions \(\mathbb{O}\) | 8 | Associativity |
Hurwitz's theorem (1898) proves no further normed division algebra exists.
Non-associativity¶
The defining feature of octonions is that \((ab)c \neq a(bc)\) in general. The associator measures this:
Key properties:
- Totally antisymmetric: swapping any two arguments flips the sign
- Alternativity: \([a, a, b] = [a, b, b] = 0\) (vanishes when any two arguments are equal)
- Flexibility: \([a, b, a] = 0\)
- Norm bound: \(\|[a,b,c]\| \leq 2\|a\|\|b\|\|c\|\)
Despite non-associativity, octonions satisfy the Moufang identities, which constrain how parenthesization affects results.
The Fano plane¶
The multiplication table of the 7 imaginary octonion units is encoded by the Fano plane \(\mathrm{PG}(2,2)\) -- a finite projective plane with 7 points and 7 lines. Each line \((e_i, e_j, e_k)\) satisfies \(e_i e_j = e_k\) (with cyclic permutations positive and anti-cyclic negative).
The 7 lines define 7 quaternionic subalgebras \(\mathcal{S}_\ell = \mathrm{span}\{1, e_i, e_j, e_k\}\). Within each subalgebra, the algebra is associative.
See the API Reference for full documentation of Octonion, associator, and all operations.