# Why is 8 so special?

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. May 16 '11 at 23:33
@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 0:06
@Chaz: I'm on the fence, and will thus let other people vote on it. – J. M. May 17 '11 at 0:11
@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 0:37
@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. May 17 '11 at 0:42

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 $2^{n-2} = n^2$.