A quaternion is a vector in with a noncommutative product (see [1] or Quaternion). Quaternions, also called hypercomplexnumbers, were invented by William Rowan Hamilton in 1843. A quaternion can be written or, more compactly, or , where the noncommuting unit quaternions obey the relations .

A quaternion can represent a rotation axis, as well as a rotation about that axis. Instead of turning an object through a series of successive rotations using rotation matrices, quaternions can directly rotate an object around an arbitrary axis (here ) and at any angle . This Demonstration uses the quaternion rotation formula with , a pure quaternion (with real part zero), , normalized axis , and for a unit quaternion, , where the quaternion conjugate for is .