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 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"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 formula_3 and formula_4. The geometry of "S" is retrieved as follows: The fact that the geometries of "S" and "Q" are isomorphic can be explained by the isomorphism of the Dynkin diagrams "A" and "D". Klein quadric In