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It can be generalized, but doing so requires some subtlety. The naive approach (the Cayley-Dickson construction) can be repeated ad infinitum, but it doesn't continue to yield useful results for representing geometric interactions like rotations in high dimensions. Thankfully, this is a solved problem. The correct generalized structure for doing geometry is called a Clifford algebra. For n-space and any nonnegative integers p,q satisfying p+q=n, there is a corresponding real Clifford algebra Cl(R,p,q). Cl(R,0,1) turns out to be isomorphic to C (the complex numbers), and Cl(R,0,2) is a four-dimensional algebra that turns out to be isomorphic to Q (the quaternions). This is actually not that surprising, because the signature (p,q) more or less means the algebra is built by adjoining p generators that square to +1 and q generators that square to -1 in the base field. This is formalized by taking a quotient of the tensor algebra of the field. You might wonder though why we have (p,q) = (0,2) for the quaternions. That's because if the two generators that square to -1 are i and j, then we can build the third as k = ij, so we get it for free. A real Clifford algebra is known as a geometric algebra, and these give rise to objects called rotors. Rotations in an arbitrarily high-dimensional space can then be written as conjugation by a rotor. |