Clifford parallel

Generalized Clifford parallelism

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.

Clifford surfaces

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

${\displaystyle \{e^{ar}{e}^{bs}:0\le a,b<\pi \}.}$

History

Clifford parallels were first described in 1873 by the English mathematician William Kingdon Clifford.

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

In 1931 Heinz Hopf used Clifford parallels to construct the Hopf map.

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