Fermat cubic - Alchetron, The Free Social Encyclopedia

In geometry, the Fermat cubic, named after Pierre de Fermat, is a surface defined by

x 3 + y 3 + z 3 = 1.

Methods of algebraic geometry provide the following parameterization of Fermat's cubic:

x ( s , t ) = 3 t − 1 3 ( s 2 + s t + t 2 ) 2 t ( s 2 + s t + t 2 ) − 3 y ( s , t ) = 3 s + 3 t + 1 3 ( s 2 + s t + t 2 ) 2 t ( s 2 + s t + t 2 ) − 3 z ( s , t ) = − 3 − ( s 2 + s t + t 2 ) ( s + t ) t ( s 2 + s t + t 2 ) − 3 .

In projective space the Fermat cubic is given by

w 3 + x 3 + y 3 + z 3 = 0.

The 27 lines lying on the Fermat cubic are easy to describe explicitly: they are the 9 lines of the form (w : aw : y : by) where a and b are fixed numbers with cube −1, and their 18 conjugates under permutations of coordinates.

References

Fermat cubic Wikipedia

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