Why is 8 so special?

Question

I have been reading about multi-dimensional numbers, and found out that it's been proven that the Octonions are the composition algebra of the largest dimension. I was wondering why, despite having infinitely many different dimensions of numbers, the only composition algebras are of 1, 2, 4, and 8 dimensions. What's so special about 8?

@J. M.: you are the link-master! Also, in light of "those three questions", would this be a duplicate topic? — The Chaz 2.0, May 17 '11 at 00:06
@Chaz: I'm on the fence, and will thus let other people vote on it. — J. M. ain't a mathematician, May 17 '11 at 00:11
@J.M., I have to agree with The Chaz. I usually have a hard time searching for question here and I end up using Google with site:math.stackexchange.com. Any tips on searching? The default OR in searches here is not convenient and I don't think it searches comments. — lhf, May 17 '11 at 00:37
@lhf: I did use Google (the built-in search is remarkably unhelpful); here the magic search-words are "quaternion" and "Frobenius". (Yes, Google can parse comments. Whodathunkit, eh?) Also, I happen to remember those three well... — J. M. ain't a mathematician, May 17 '11 at 00:42
http://www.youtube.com/watch?v=Tw8w4YPp4zM kinda both entertaining and informative — user2485710, May 14 '14 at 14:51

score 12 · Accepted Answer · answered May 17 '11 at 01:06

12

You might be interested in Hurwitz's proof of his theorem (which is not as strong as Wikipedia's statement). Here is the original German and an English translation. The maximal $n$ turns out to be the solution of $2^{n-2} = n^2$.

answered May 17 '11 at 01:06

Yuval Filmus

57,157

score -1 · Answer 2 · answered Mar 12 '21 at 13:06

-1

Your information is wrong: tessarines of any $2^n$ dimension (for instance, 16-dimensional tessarines) is a commutative, associative, composition algebra.

answered Mar 12 '21 at 13:06

Anixx

9,119

Why is 8 so special?

2 Answers2

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