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 quadric. Thus, the space of lines is a 4-dimensional projective variety. Its subvarieties are line complexes. If the underlying vector space of S is the 4-dimensional vector space V, then T has as the underlying vector space the 6-dimensional exterior square Λ2V of V. The line coordinates obtained this way are known as Plücker coordinates. These Plücker coordinates satisfy the quadratic relation
p 12 p 34 + p 13 p 42 + p 14 p 23 = 0 {\displaystyle p_{12}p_{34}+p_{13}p_{42}+p_{14}p_{23}=0}
defining Q, where
p i j = u i v j − u j v i {\displaystyle p_{ij}=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 classes, 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 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 of the Dynkin diagrams A3 and D3.
References Albrecht Beutelspacher & Ute Rosenbaum (1998) Projective Geometry : from foundations to applications, page 169, Cambridge University Press ISBN 978-0-521-48277-6 Arthur Cayley (1873) "On the superlines of a quadric surface in five-dimensional space", Collected Mathematical Papers 9: 79–83. Felix Klein (1870) "Zur Theorie der Liniencomplexe des ersten und zweiten Grades", Mathematische Annalen 2: 198 Oswald Veblen & John Wesley Young (1910) Projective Geometry, volume 1, Interpretation of line coordinates as point coordinates in S5, page 331, Ginn and Company. Ward, Richard Samuel; Wells, Raymond O'Neil Jr. (1991), Twistor Geometry and Field Theory, Cambridge University Press, Bibcode:1991tgft.book.....W, ISBN 978-0-521-42268-0.
