3 ms·

I'm not a mathematician, but I think it's about extending the idea of a scalar and a single rotation (Complex numbers) into a scalar + 3 rotations (Quaternions)

I'm not a mathematician, but I think it's about extending the idea of a scalar and a single rotation (Complex numbers) into a scalar + 3 rotations (Quaternions). The idea can be extended further to a scalar with 7 rotations - https://en.wikipedia.org/wiki/Octonion https://en.wikipedia.org/wiki/Octonion, but no further, for reasons I don't understand.

paulddraper 7y ago
You can actually go as far as you want to with the Cayley–Dickson construction [1] of algebras. 1. Complex numbers have associativity and communitivity of multiplication. (That is, (ab)c=a(bc) and ab=ba). 2. Quaternions have associativity but not communitivity. 3. Octonions have neither. 4. Sedenions [2], trigintaduonions, and not associative, commutative, nor even alternative [3]. (Alternative is associative specifically when the middle value is equal to one of the other's; i.e. a(ab)=(aa)b.) [1] https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_constru... [2] https://en.wikipedia.org/wiki/Sedenion https://en.wikipedia.org/wiki/Sedenion [3] https://en.wikipedia.org/wiki/Alternative_algebra https://en.wikipedia.org/wiki/Alternative_algebra
- pflats 7y ago
  Thanks for writing this! I referenced it in a reply upthread for why I like the equation. https://news.ycombinator.com/item?id=22204995 https://news.ycombinator.com/item?id=22204995
QuesnayJr 7y ago
You lose the ability to divide after the octonions.