Composing Rotations via Hamilton Turns

Sir William Rowan Hamilton, of quaternion fame, also proposed a geometric construction to compose three-dimensional rotations using, what he called, turns. A turn is a directed arc on a great circle of the unit sphere. Such turns can be added like vectors, by sliding them along their great circle to place them head-to-tail. A rotation operation corresponds to a turn that spans one-half of the rotation angle on the great circle perpendicular to the rotation axis. The turn corresponding to two successive rotations is the sum of the turns of the two rotations, in order of application. These operations are not commutative.