Clifford parallel

The algebra of quaternions provides a descriptive geometry of elliptic space in which Clifford parallelism is made explicit.

[2] 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.

[4] In 1900 Guido Fubini wrote his doctoral thesis on Clifford's parallelism in elliptic spaces.

[6] In 2016 Hans Havlicek showed that there is a one-to-one correspondence between Clifford parallelisms and planes external to the Klein quadric.

Two Clifford parallel great circles on the 3-sphere spanned by a twisted annulus . They have a common center point in 4-dimensional Euclidean space , and could lie in completely orthogonal rotation planes.