What are some real-world uses of Octonions?

Solution 1:

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.

Solution 2:

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.

Solution 3:

The Freudenthal–Tits magic square

Solution 4:

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.