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Why is 8 so special?

First of all let me state that I am not a mathematician but I work in computer science and engineering.

I was reading about hypercomplex numbers, and in particular about quaternions and octonions.

My question is is there anything special with the 4 dimensions of the quaternions and with the 8 of the octonions? Why can't we also do hypercomplex numbers with any other arbitrary sum of imaginary parts? For example in 5, 6 or 3 dimensions?

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    See http://math.stackexchange.com/questions/39522/why-is-8-so-special and the links within.2012-04-25

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