Clifford parallel

 

The lines on 1 in elliptic space are described by versors with a fixed axis r:^[1]

${\displaystyle \lbrace e^{ar}:\ 0\leq a<\pi \rbrace }$ 

For an arbitrary point u in elliptic space, two Clifford parallels to this line pass through u. The right Clifford parallel is

${\displaystyle \lbrace ue^{ar}:\ 0\leq a<\pi \rbrace ,}$ 

and the left Clifford parallel is

${\displaystyle \lbrace e^{ar}u:\ 0\leq a<\pi \rbrace .}$ 

Clifford's original definition was of curved parallel lines, but the concept generalizes to Clifford parallel objects of more than one dimension.^[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.

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^[1][3]

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

Clifford parallels were first described in 1873 by the English mathematician William Kingdon Clifford.^[4]

In 1900 Guido Fubini wrote his doctoral thesis on Clifford's parallelism in elliptic spaces.^[5]

In 1931 Heinz Hopf used Clifford parallels to construct the Hopf map.^[6]

In 2016 Hans Havlicek showed that there is a one-to-one correspondence between Clifford parallelisms and planes external to the Klein quadric.^[7]

• Clifford torus
• 24-cell § Clifford parallel polytopes
• Regular 4-polytopes