Fermat cubic (original) (raw)

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Geometrical surface

3D model of Fermat cubic (real points)

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

x 3 + y 3 + z 3 = 1. {\displaystyle x^{3}+y^{3}+z^{3}=1.\ } {\displaystyle 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 {\displaystyle x(s,t)={3t-{1 \over 3}(s^{2}+st+t^{2})^{2} \over t(s^{2}+st+t^{2})-3}} {\displaystyle x(s,t)={3t-{1 \over 3}(s^{2}+st+t^{2})^{2} \over t(s^{2}+st+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 {\displaystyle y(s,t)={3s+3t+{1 \over 3}(s^{2}+st+t^{2})^{2} \over t(s^{2}+st+t^{2})-3}} {\displaystyle y(s,t)={3s+3t+{1 \over 3}(s^{2}+st+t^{2})^{2} \over t(s^{2}+st+t^{2})-3}}

z ( s , t ) = − 3 − ( s 2 + s t + t 2 ) ( s + t ) t ( s 2 + s t + t 2 ) − 3 . {\displaystyle z(s,t)={-3-(s^{2}+st+t^{2})(s+t) \over t(s^{2}+st+t^{2})-3}.} {\displaystyle z(s,t)={-3-(s^{2}+st+t^{2})(s+t) \over t(s^{2}+st+t^{2})-3}.}

In projective space the Fermat cubic is given by

w 3 + x 3 + y 3 + z 3 = 0. {\displaystyle w^{3}+x^{3}+y^{3}+z^{3}=0.} {\displaystyle 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.

Real points of Fermat cubic surface.