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| {{for|the Klein '''quartic'''|Klein quartic}}
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| 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'''.
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| 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<sup>2</sup>''V'' of ''V''. The [[line coordinates]] obtained this way are known as [[Plücker coordinates]].
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| These Plücker coordinates satisfy the quadratic relation
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| : <math>p_{12} p_{34}+p_{13}p_{42}+p_{14} p_{23} = 0 </math>
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| defining ''Q'', where
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| : <math>p_{ij} = u_i v_j - u_j v_i</math> | |
| are the coordinates of the line [[Linear span|spanned]] by the two vectors ''u'' and ''v''.
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| 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 <math>C</math> and <math>C'</math>. The [[geometry]] of ''S'' is retrieved as follows:
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| # The points of ''S'' are the planes in ''C''.
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| # The lines of ''S'' are the points of ''Q''.
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| # The planes of ''S'' are the planes in ''C''’.
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| The fact that the geometries of ''S'' and ''Q'' are isomorphic can be explained by the [[isomorphism]] of the [[Dynkin diagram]]s ''A''<sub>3</sub> and ''D''<sub>3</sub>.
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| ==References==
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| * {{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|isbn=978-0-521-42268-0}}.
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| {{geometry-stub}}
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| [[Category:Projective geometry]]
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| [[Category:Quadrics]]
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