# 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 hyperbolic quadric, *Q* known as the **Klein quadric**.

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 Λ^{2}*V* of *V*. The line coordinates obtained this way are known as Plücker coordinates.

These Plücker coordinates satisfy the quadratic relation

defining *Q*, where

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
and
. 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 *A*_{3} and *D*_{3}.

## References

- Albrecht Beutelspacher & Ute Rosenbaum (1998)
*Projective Geometry : from foundations to applications*, page 169, Cambridge University Press ISBN 978-0521482776 - 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 S_{5}, page 331, Ginn and Company. - Ward, Richard Samuel; Wells, Raymond O'Neil, Jr. (1991),
*Twistor Geometry and Field Theory*, Cambridge University Press, ISBN 978-0-521-42268-0.