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... octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative.

Comes from a a quote by John Baez. Clearly, the sucessor to quaterions from the Cayley-Dickson process is a numerical beast, but has anybody found any real-world uses for them? For example, quaterions have a nice connection to computer graphics through the connection to SO(4), and that alone makes them worth studying. What can be done with a nonassociative algebra like the octonions?

Note: simply mentioning that they

have applications in fields such as string theory, special relativity, and quantum logic.

is not what I'm looking for (I can read wikipedia too). A specific example, especially one that is geared to someone who is not a mathematician by trade would be nice!

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One might want to note here that there exist nonassociative operations where hardly anyone seems to have any wonder about their use. Examples include subtraction, division, exponentiation, material implication, etc. Also, nonassociative operations, in some sense, happen a lot more often than associative operations. –  Doug Spoonwood Feb 14 '12 at 14:05
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The difference here is that multiplication is an operation that we'd expect to be associative, and all of a sudden it isn't. –  Joe Z. Feb 13 at 19:46
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4 Answers 4

up vote 7 down vote accepted

John Baez has a long online article about uses of the octonions, at least some of which is concerned with their relationship to physics. You might also want to read his papers with Huerta, Division Algebras and Supersymmetry I and Division Algebras and Supersymmetry II.

I don't think you'll be able to find an easy application to explain to a layman, since the octonions are naturally connected to geometry in higher dimensions than most people can be bothered to care about.

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A way of guaranteeing that real (so phase an integer multiple of $\pi$) fading radio signals from 8 transmit antennas will, crudely speaking, always interfere constructively, is based on octonions. See this article by Tarokh et al for more background. In their formula (5) you see the matrix representing multiplication by a generic octonion.

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