The algebra of quaternions provides a descriptive geometry of elliptic space in which Clifford parallelism is made explicit.
The lines on 1 in elliptic space are described by versors with a fixed axis r:
For an arbitrary point u in elliptic space, two Clifford parallels to this line pass through u.
The right Clifford parallel is
and the left Clifford parallel is
Generalized Clifford parallelismEdit
Clifford's original definition was of curved parallel lines, but the concept generalizes to Clifford parallel objects of more than one dimension. In 4-dimensional Euclidean space Clifford parallel objects of 1, 2, 3 or 4 dimensions are related by isoclinic rotations. Clifford parallelism and isoclinic rotations are closely related aspects of the SO(4)symmetries which characterize the regular 4-polytopes.
Rotating a line about another, to which it is Clifford parallel, creates a Clifford surface.
The Clifford parallels through points on the surface all lie in the surface. A Clifford surface is thus a ruled surface since every point is on two lines, each contained in the surface.
Given two square roots of minus one in the quaternions, written r and s, the Clifford surface through them is given by