mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the lines of a 3-dimensional

projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...

, ''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 In mathematics, a quadric or quadric surface is a generalization of conic sections (ellipses, parabolas, and hyperbolas). In three-dimensional space, quadrics include ellipsoids, paraboloids, and hyperboloids. More generally, a quadric hype ...

, ''Q'' known as the Klein quadric. If the underlying

vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...

of ''S'' is the 4-dimensional vector space ''V'', then ''T'' has as the underlying vector space the 6-dimensional exterior square Λ²''V'' of ''V''. The

line coordinates In geometry, line coordinates are used to specify the position of a line just as point coordinates (or simply coordinates) are used to specify the position of a point. Lines in the plane There are several possible ways to specify the position of ...

obtained this way are known as Plücker coordinates. These Plücker coordinates satisfy the quadratic relation :

p_ p_+p_p_+p_ p_ = 0

defining ''Q'', where :

p_ = u_i v_j - u_j v_i

are the coordinates of the line spanned by the two vectors ''u'' and ''v''. The 3-space, ''S'', can be reconstructed again from the quadric, ''Q'': the planes contained in ''Q'' fall into two

equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...

es, where planes in the same class meet in a point, and planes in different classes meet in a line or in the empty set. Let these classes be ''C'' and ''C''′. The

geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

of ''S'' is retrieved as follows: # The points of ''S'' are the planes in ''C''. # The lines of ''S'' are the points of ''Q''. # The planes of ''S'' are the planes in ''C''′. The fact that the geometries of ''S'' and ''Q'' are isomorphic can be explained by the

isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

of the

Dynkin diagram In the Mathematics, mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of Graph (discrete mathematics), graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the ...

s ''A''₃ and ''D''₃.

References

* Albrecht Beutelspacher & Ute Rosenbaum (1998) ''Projective Geometry : from foundations to applications'', page 169,

Cambridge University Press Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...

Arthur Cayley Arthur Cayley (; 16 August 1821 – 26 January 1895) was a British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics, and was a professor at Trinity College, Cambridge for 35 years. He ...

(1873) "On the superlines of a quadric surface in five-dimensional space", ''Collected Mathematical Papers'' 9: 79–83. *

Felix Klein Felix Christian Klein (; ; 25 April 1849 – 22 June 1925) was a German mathematician and Mathematics education, mathematics educator, known for his work in group theory, complex analysis, non-Euclidean geometry, and the associations betwe ...

(1870) "Zur Theorie der Liniencomplexe des ersten und zweiten Grades",

Mathematische Annalen ''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, ...

2: 198 *

Oswald Veblen Oswald Veblen (June 24, 1880 – August 10, 1960) was an American mathematician, geometer and topologist, whose work found application in atomic physics and the theory of relativity. He proved the Jordan curve theorem in 1905; while this was lo ...

& John Wesley Young (1910) ''Projective Geometry'', volume 1, Interpretation of line coordinates as point coordinates in S₅, page 331, Ginn and Company. * {{citation, title=Twistor Geometry and Field Theory, first1=Richard Samuel, last1= Ward, first2= Raymond O'Neil Jr., last2= Wells, author2-link=Raymond O. Wells, Jr., publisher=Cambridge University Press, year= 1991, bibcode=1991tgft.book.....W, isbn=978-0-521-42268-0. Projective geometry Quadrics