Quaternions and Octonions

Abstract

We have all seen how useful complex numbers are for describing isometries of the plane. We will now look at the quaternionic numbers and how useful they are for describing rotations of ℝ³. Indeed computer-game programmers have taken to using quaternions as a quick way to turn the picture round when heroes need to defend themselves.

I pulled out on the spot a pocket book, which still exists, and made an entry there and then. Nor could I resist the impulse — unphilosophical as it may have been- to cut with a knife on a stone of Brougham Bridge, as we passed it, the fundamental formula with the symbols i, j, k: ${i^2} = {j^2} = {k^2} = ijk = - 1,$ which contains the solution of the Problem, but of course, as an inscription has long since mouldered away. W.R. Hamilton