Inelliptic geometry, two lines areClifford parallel orparatactic lines if theperpendicular distance between them is constant from point to point. The concept was first studied byWilliam Kingdon Clifford inelliptic space and appears only in spaces of at leastthree dimensions. Sinceparallel lines have the property of equidistance, the term "parallel" was appropriated fromEuclidean geometry, although the "lines" of elliptic geometry aregeodesiccurves and, unlike the lines ofEuclidean geometry, are of finite length.
The algebra ofquaternions provides a descriptive geometry of elliptic space in which Clifford parallelism is made explicit.Clifford bundle is a topological construction based onClifford parallel pointed out byHeinz Hopf (1931)[1]
The lines on 1 in elliptic space are described byversors with a fixed axisr:[2]
For an arbitrary pointu in elliptic space, two Clifford parallels to this line pass throughu.The right Clifford parallel is
and the left Clifford parallel is
Clifford's original definition was of curved parallel lines, but the concept generalizes to Clifford parallel objects of more than one dimension.[3] In 4-dimensional Euclidean space Clifford parallel objects of 1, 2, 3 or 4 dimensions are related byisoclinic rotations. Clifford parallelism and isoclinic rotations are closely related aspects of theSO(4)symmetries which characterize theregular 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 aruled surface since every point is on two lines, each contained in the surface.
Given two square roots of minus one in thequaternions, writtenr ands, the Clifford surface through them is given by[2][4]
Clifford parallels were first described in 1873 by the English mathematicianWilliam Kingdon Clifford.[5]
In 1900Guido Fubini wrote his doctoral thesis onClifford's parallelism in elliptic spaces.[6]
In 1931Heinz Hopf used Clifford parallels to construct theHopf map.[1]
In 2016 Hans Havlicek showed that there is a one-to-one correspondence between Clifford parallelisms and planes external to theKlein quadric.[7]