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Fermat cubic surface

In geometry, the Fermat cubic, named after Pierre de Fermat, is a surface defined by x3+y3+z3=1.{\displaystyle x^{3}+y^{3}+z^{3}=1.\ }Methods of algebraic geometry provide the following parameterization of Fermat's cubic: x(s,t)=3t−13(s2+st+t2)2t(s2+st+t2)−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)=3s+3t+13(s2+st+t2)2t(s2+st+t2)−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−(s2+st+t2)(s+t)t(s2+st+t2)−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 w3+x3+y3+z3=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.

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