Quaternions

The quaternions are a four-dimensional division algebra with three anticommuting square roots of minus one, discovered by Hamilton in 1843 while trying to multiply triples. They compose rotations of three-dimensional space without the degeneracies of Euler angles, which is why graphics and robotics keep them. Writing here derives them as the even subalgebra of the geometric algebra of space, where the three imaginaries are bivectors and the half-angle is a record that a rotation is two reflections.