elliptic geometry Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines a ...

, two lines are Clifford parallel or paratactic lines if the perpendicular distance between them is constant from point to point. The concept was first studied by

William Kingdon Clifford William Kingdon Clifford (4 May 18453 March 1879) was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his ...

elliptic space Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines a ...

and appears only in spaces of at least three dimensions. Since

parallel lines In geometry, parallel lines are coplanar straight lines that do not intersect at any point. Parallel planes are planes in the same three-dimensional space that never meet. ''Parallel curves'' are curves that do not touch each other or inters ...

have the property of equidistance, the term "parallel" was appropriated from

Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...

, although the "lines" of elliptic geometry are

geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...

curves and, unlike the lines of

, are of finite length. The algebra of

quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...

s provides a descriptive geometry of elliptic space in which Clifford parallelism is made explicit.

Introduction

The lines on 1 in elliptic space are described by

versor In mathematics, a versor is a quaternion of norm one (a ''unit quaternion''). The word is derived from Latin ''versare'' = "to turn" with the suffix ''-or'' forming a noun from the verb (i.e. ''versor'' = "the turner"). It was introduced by Willi ...

s with a fixed axis ''r'':

Georges Lemaître Georges Henri Joseph Édouard Lemaître ( ; ; 17 July 1894 – 20 June 1966) was a Belgian Catholic priest, theoretical physicist, mathematician, astronomer, and professor of physics at the Catholic University of Louvain. He was the first to th ...

(1948) "Quaternions et espace elliptique", ''Acta''

Pontifical Academy of Sciences The Pontifical Academy of Sciences ( it, Pontificia accademia delle scienze, la, Pontificia Academia Scientiarum) is a scientific academy of the Vatican City, established in 1936 by Pope Pius XI. Its aim is to promote the progress of the math ...

12:57–78 :

\lbrace e^ :\ 0 \le 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 :

\lbrace u e^:\ 0 \le a < \pi \rbrace,

and the left Clifford parallel is :

\lbrace e^u:\ 0 \le a < \pi \rbrace.

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) In mathematics, the group of rotations about a fixed point in four-dimensional Euclidean space is denoted SO(4). The name comes from the fact that it is the special orthogonal group of order 4. In this article ''rotation'' means ''rotational dis ...

symmetries Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...

which characterize the

regular 4-polytopes In mathematics, a regular 4-polytope is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions. There are six convex and ten star re ...

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 In geometry, a surface is ruled (also called a scroll) if through every point of there is a straight line that lies on . Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, the ...

since every point is on two lines, each contained in the surface. Given two square roots of minus one in the

s, written ''r'' and ''s'', the Clifford surface through them is given by :

\lbrace e^e^ :\ 0 \le a,b < \pi \rbrace.

History

Clifford parallels were first described in 1873 by the English mathematician

. In 1900

Guido Fubini Guido Fubini (19 January 1879 – 6 June 1943) was an Italian mathematician, known for Fubini's theorem and the Fubini–Study metric. Life Born in Venice, he was steered towards mathematics at an early age by his teachers and his father, who w ...

wrote his doctoral thesis on ''Clifford's parallelism in elliptic spaces''. In 1931

Heinz Hopf Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician who worked on the fields of topology and geometry. Early life and education Hopf was born in Gräbschen, Germany (now , part of Wrocław, Poland), the son of Elizabeth ( ...

used Clifford parallels to construct the

Hopf map In the mathematical field of differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere. Discovered by Heinz Ho ...

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

Klein quadric In mathematics, the lines of a 3-dimensional projective space, ''S'', can be viewed as points of a 5-dimensional projective space, ''T''. In that 5-space, the points that represent each line in ''S'' lie on a quadric, ''Q'' known as the Klein q ...

.Hans Havlicek (2016) "Clifford parallelisms and planes external to the Klein quadric", ''Journal of Geometry'' 107(2): 287 to 303

Citations

References

* * Laptev, B.L. & B.A. Rozenfel'd (1996) ''Mathematics of the 19th Century: Geometry'', page 74,

Birkhäuser Verlag Birkhäuser was a Swiss publisher founded in 1879 by Emil Birkhäuser. It was acquired by Springer Science+Business Media in 1985. Today it is an imprint used by two companies in unrelated fields: * Springer continues to publish science (particu ...

. *

Duncan Sommerville Duncan MacLaren Young Sommerville (1879–1934) was a Scottish mathematician and astronomer. He compiled a bibliography on non-Euclidean geometry and also wrote a leading textbook in that field. He also wrote ''Introduction to the Geometry of N ...

(1914) ''The Elements of Non-Euclidean Geometry'', page 108 Paratactic lines,

George Bell & Sons George Bell & Sons was a book publishing house located in London, United Kingdom, from 1839 to 1986. History George Bell & Sons was founded by George Bell as an educational bookseller, with the intention of selling the output of London univ ...

* Frederick S. Woods (1917
Higher Geometry
"Clifford parallels", page 255, via

Internet Archive The Internet Archive is an American digital library with the stated mission of "universal access to all knowledge". It provides free public access to collections of digitized materials, including websites, software applications/games, music, ...