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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!

Pete L. Clark
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Hooked
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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 '14 at 19:46

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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.

Qiaochu Yuan
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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.

Jyrki Lahtonen
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There are lots of applications, and one of them is Glimpses of the Octonions and Quaternions History and Todays Applications in Quantum Physics.

More recently, some researchers have been motivated to formulate portions of the Standard Model in terms of octonions. A more extreme--but intriguing--view is that octonions are fundamental from which all "lower" number systems follow.

Joel DeWitt
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The Freudenthal–Tits magic square

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First lines of a poster on the october qunatum physics conference: Projective and projection geometry for a new kind of unification

Gudrun Kalmbach H E

For a unification of the four basic interactions I use an octonian vector space with suitable projections, add Moebius transformations to the $SU(1) \times SU(2) \times SU(3)$ symmetry and seven $3$-dimensional Fano measures of Gleason which include Euclidean space coordinates.

Everyone who believes today that the world of physics is not a projection like used for TV, having originaly $4$ dimensions, is kept ignorant like in the middle ages. Double up the $4$ coordinates to octonian vectors by including an input vector $e_0$, an output vector $e_7$, two energy coordinates: $e_5$ for mass and a Higgs field, $e_6$ for frequencies. $e_1,e_2,e_3$ and $e_4$ are for space and time coordinates.

Later
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gudrun kalmbach h.e.
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To my knowledge, octonions are big in machine/deep learning and computer vision. Here's an example, and another.

I'd also like to mention that they extend the quaternions the same way quaternions extend the complex numbers. Quaternions let you use {1,j} or {1,k} as a complex base instead of just {1,i}, so you can do complex math freely in any basis, but you should be able to move the whole basis in/out of the rotation plane, etc. The same goes for octonions: edges can still serve as complex bases, but the attached faces are valid quaternion bases too. If I understood octonions better myself, maybe I could be of more help.

There are also the integer/integral octonions to consider, there was something about them forming the E8 lattice, but it all went over my head.

John P
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