Fermat cubic explained

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

Real points of Fermat cubic surface.

References

• • Web site: Complete cubic parameterization of the Fermat cubic surface. Noam. Elkies.